Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask

By Sean Carroll | July 7, 2008 1:07 pm

Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.

So — what do you want to know? What are the really interesting questions about QM that we should be talking about?

One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.

Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?

(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)

CATEGORIZED UNDER: Science
  • Matt

    Ooh, I’m looking forward to this. As a layman, one thing I’d love to see covered is the measurement problem. It seems like a lot of scientists that know better can fall into lazy language that makes it sound to the layman like consciousness itself changes reality. But some lucid, repeatable-by-someone-like-me explanation of why that’s definitely not the case would be very well appreciated.

  • Matt

    Also, some discussion of the different “interpretations” of QM would be great. I’ve never gotten my head around the Copenhagen interpretation.

  • Tim Tesar

    (if this is not a question about QM, then please ignore.)

    So-called “wave-particle duality” is sometimes described as a paradox. Is it?

  • Tim Tesar

    What about decoherence? How many physicists today believe it is correct? Does it solve the so-called “measurement problem”? Can we finally be sure that Schroedinger’s Cat is either alive (yay!) or dead (Oh, dear!), and not in a state of superposition?

  • Janus

    I’m looking forward to this, Sean, but honestly what I want to know is if there is a book about quantum mechanics written for laymen that actually explains the theory in detail _without_ using analogies that oversimplify and _without_ constantly resorting to poetic language? I wouldn’t mind having to think and work a bit to understand what’s being said, I wouldn’t mind having to read a few equations, I wouldn’t even mind having to learn some math I don’t know. I’d just like to read a book from an author that really tries to make me understand QM, as opposed to giving us a rough idea of what it might be like if I understood it. I want The Blind Watchmaker of quantum mechanics. :)

  • Kurt

    I would like to hear more about what a Hilbert space is vs vector space?
    Also, maybe you can start with the double slit experiment and what it means for a photon to interfere with itself leading to path integral representation of QM.
    either way can’t wait!

  • http://www.discovery-news.com LoCut

    I’m a complete layperson, but I’ve always been intrigued by the inability to directly see quantum states (maybe what Matt means by ‘the measurement problem’?). Why is it that they can’t be directly measured? Pretend I’m the grandmother. Without the physics degree.

  • Christopher M

    Here are a few ideas. I’m a somewhat educated layperson, so probably the right target audience for Bloggingheads.

    1. Do any of the proposed solutions to the measurement problem (or “interpretations” of QM more generally) have consequences that would allow us to distinguish them empirically, or get evidence one way or another?

    2. What is the math behind quantum mechanics like? I’m not asking for a math class, of course — maybe just an impressionistic sense of the major pieces & how they fit together. The heart of Newtonian mechanics seems to be calculus and differential equations — are those equally central to QM or is there something more? Is group theory involved? (The point isn’t to understand the math — it’s that knowing what mathematical tools a theory involves can help give a sense of what the theory itself is like.)

    3. Is it accurate or inaccurate to say that the “quantum” in QM means that the world is digital rather than analog?

    4. Quantum computing. Revolutionary or overhyped, and why?

    And finally, an anti-request. There are so many good discussions out there of the basics on the uncertainty principle, wave-particle duality, and the two-slit experiment that you won’t be doing much of a service if you spend a lot of time on yet another layman’s version of those things. Explain them of course, but don’t get too bogged down in the details!

  • Elliot

    I’d like a cogent discussion of Aspect’s experiment and it’s relationship to Bell’s inequality. Did it firmly prove Bell’s inequality?

    Bell’s inequality itself is a pretty intereting topic.

    e.

  • jeffw

    I also like some discussion of Bell and very recent experiments – Leggett, Zeilinger, realism vs locality, etc.

  • Joe Renes

    Perhaps you can work in the “church of the larger Hilbert space” into the discussion about the wavefunction of the universe.

    For those outside quantum information theory, the church of the larger Hilbert space is a joking reference to the fact that any quantum states which are ‘mixed’ in the sense of being a probabilistic combination of two or more ‘pure’ states (pure states being those which live in the Hilbert space) can always be thought of as part of an even larger system which is pure. So one speaks of “purifying” a mixed state, or, in cases of extreme need, going to the church of the larger Hilbert space. So it’s relevant to the discussion of the wavefunction of the universe. sorta.

  • Phil

    When one talks about a quantum of stuff, you are talking about the smallest possible piece of something. Quantum Mechanics basically states that there IS a smallest piece of energy. Quantum Mechanics also makes the assertion that one can say that there is a smallest distance, measure of time, velocity, force, etc. When one is talking about the smallest pieces of space does it make sense to think about these piece of space as having a shape?

  • Sam Taylor

    From a layman, perhaps you could discuss the practical applications of QM. For example, I conceptually understand why GPS clocks have to be adjusted for Special and General Relativity. Yet, my understanding of the only detectable effect of QM in the “macro” world occurs with a precision device like a comb 1/10th as thick as a human hair at a temperature of about 78 above absolute 0.

    I would enjoy being enlightened.

    Thanks,

    Sam

  • andy.s

    Question 1: Does “measurement” of a particle mean the same thing as “the world line of some particle interacts irreversibly with the world line of another particle”?

    Question 2: Can any other type of physics be formulated with this weird-ass probabilities-in-Hilbert-space formalism? Could you, for example, rework the Kepler problem with the position and momentum observables as the eigenvalues of some Hamiltonian? Would there be any advantage to it?

    Question 3: In a delayed choice two-slit experiment, a particle knows when it’s emitted whether its path is going to be through one of two slits or a superposition of both paths based on how it’s going to be measured, even if the measurement happens 50,000,000 years later. HOW THE HELL … ahem. Excuse me. How does it GOD DAMN KNOW HOW … ahem excuse me again.

    If that damn thing can somehow look 50,000,000 years into the future and see the laboratory it’s going to wind up in and see the scientist with his finger on a button and it makes its decision on how to propagate based on that, then the universe is rigidly deterministic to an extent that makes me want to just go and slit my wrists.

    Like Clive Bruckman says in the X-Files: “How can I see the future if it doesn’t already exist?”

  • John Merryman

    Is it reasonable to say; Wave=analog, particle=digital?

    Is a photon, as a quanta of light, analogous to a drop of water, in that its consistency of size is a function of factors operating on it, much like the surface tension of water vs. gravity makes drips of water similarly sized. Or is it a fundamental…I wouldn’t say structure, but possibly internal constraint?

  • John Merryman

    Quantum Mechanics also makes the assertion that one can say that there is a smallest distance, measure of time, velocity, force, etc. When one is talking about the smallest pieces of space does it make sense to think about these piece of space as having a shape?

    To further Phil’s question, is it that these past these smallest units of measurement, is it simply too blurred to distinguish one unit from the next, or are there clear units which cannot be further subdivided?

  • Moshe

    I have to protest (again…) that I don’t see the linearity of the Schrodinger equation (i.e the fact that you can add wavefunctions) as a distinguishing property of QM. In both classical and quantum mechanics the observable quantities (say, expectation value of operators) obey non-linear equations, and there is something describing the “state” of the system (wavefunction, or phase space distribution function), which is interpreted as probability and obeys a linear equation (Liouville or Schrodinger equation).

    It so happens that in classical mechanics you typically deal with the former set of objects, and in QM (but not QFT…) you usually deals with the latter. That is a matter of pedagogy, nothing else.

  • anonymous

    1) What does it really mean that there is a measurement problem? (Looking for good speculation that has some depth to it, not textbook recitation…)

    2) Where does ‘empirical data’ stop and ‘speculation’ begin when it comes to the problem of state selection?

    3) What’s up with the quantum Zeno effect? (Please relate answer to answers to one or both of above questions.)

    4) Can the equations of QM be modified to accomodate a)probabilities that change with time, and/or b) bi-directional flows of time with non-equivalent influences?

    5) Why default to equivalent starting probabilities when using QM to make predictions? Are there instances when previous data allows for fore-knowledge of a bias?

    6) What’s the point of quantum field theory? Why/How was it developed? What does it allow you to do, exactly? Why has it not been reconciled with gravity? What are its limitations – ie, it’s good for predicting behavior of a large set of data, but not an individual observation, etc.?

    This is fun! I’ll be thinking of some more questions…

  • UWbio

    I’m a biochemist so my understanding of quantum mechanics is limited so hopefully this doesn’t stray too far from what you were hoping to see:

    1. What are virtual photons? Why can virtual photons cause a force of attraction between + and – charges but regular photons cannot do the same?

    2. If we have, say, 8 qubits of entangled electrons trapped in holes on a chip of some sort and we also have a way of measuring the array of quibits then how is any information transmitted and stored? For instance, if the first and third electrons were excited to be spin-up but all the rest were untouched and spin-down then what would the superposition that resulted be and how would you use that to calculate something like the factorization of the number 15? Oh so confusing!

    3. Half silvered mirrors are interesting. If a single wavelength of light is emitted and half of the photons will pass through the silvered mirror and continue on until they reach another silvered mirror, in which half again will pass through, then a detector on the other side will read 1/4 the number of photons as were presumably emitted. Now if a full reflective mirror is used to reflect the photons that were reflected off the first half slivered mirror and then reflected again to hit the second half silvered mirror such that the path length is equal to 1/2n times the wavelength of the light (where n is odd) then all of a sudden the detector will read 0 because the photons have canceled each other out. Does the size of n matter? If n were absolutely huge then the time it takes for the photon to get to the detector along the direct path is much much shorter than the time it takes the photon to get around the more circuitous path to detector, but yet they are still able to cancel each other out. (I don’t even know if this is true the way I have written it, so if anyone knows what I am talking about please correct my question to make more sense than I assume it does in this form). Therefore the photon’s path is determined at the time it is emitted. Could this be used for faster than light communication? For instance, if an emitter on a stationary satellite near earth passed light through two silvered mirrors as explained above and there were two mirrors on a satellite millions of miles away that could interrupt the signal by moving one of the mirrors into place then suddenly there would be a signal way back at the earth satellite instantly…again, this is assuming I know at all what I am talking about.

  • http://tyrannogenius.blogspot.com Neil B.

    What I find very odd in QM is that we can make a reliable specific superposition, like for a photon, but no one is supposed to be able to find out the details later. IOW, we can make a photon equivalent to elliptical polarization, given the wave function: A |R> + Be^(i theta) |R> and e.g. 0.6 for A and 0.8 for B. The phase then provides an angle, not just an ellipse shape, so we can be sure a filter tuned to that wave would let the photon through (as it would also have ideal 100% transmission for the equivalent classical polarized light beam.) If I am a confidante of the photon’s creator, I can know just how to orient the right filter (say, combo of QWP and LPF) to get all “hits” etc. But if I don’t already know, I can’t find out for sure: all I can do is try a filter and orientation and I might get transmission or not. Either way, the photon is “ruined” by either being absorbed or changed into the new filter’s base. (Projection postulate? I wonder why that doesn’t have its own Wikipedia article.) All I really know is, that photon couldn’t have been the orthogonal to the filter base if it went through. But if the trait is “real” (unlike the literal contradiction in Fourier analysis of exact momentum and exact position), why can’t I find out? (I know, doing so might lead to weird effects in entangled states, like FTL communication, but suppose it didn’t?)

    This seems silly, like a kid saying “If you don’t know, I’m not going to tell you!” In some other comments around, I explained how we might circumvent that restriction by using the accumulation of angular momentum: Keep reflecting a photon around with mirrors, sending it through the same half-wave plate over and over (re-flipped by a second HWP if needed.) The HWP reverses the rotational sense (spin) but maintains the specific proportions of A and B (you may be surprised, but it does – known fact, and to be consistent with the affect on the classical wave.) If we did it enough, all those transits would build up detectable angular momentum in the HWP. It would be along a range, not an either/or because the result needs to be consistent with sending many many “separate” photons through (indistinguishability.) IOW, if the photon came out of a linear polarizer, the many transits wouldn’t build up net spin since the average effect is no rotation. Maybe it wouldn’t work, but it’s worth mulling over. It seems to resemble “weak measurements” as propounded by Yakir Aharonov.

  • JCF

    What are the implications of QM on the arrow of time? When reality splits under Everett’s Many Worlds interpretation into a “parallel universe” does this constitute a third direction of time and, subsequently with another split, a fourth, and then a fifth … (ad infinitum)?

  • http://tyrannogenius.blogspot.com Neil B.

    Correction, I mean the HWP swaps the values of A and B, so the proportion is in a sense “preserved” but saying it that way is confusing. Hence 0.6 for A and 0.8 for B turns into 0.8 for A and 0.6 for B, etc., which is then restored by a second transit through a HWP (and so on, hence the intuitively expected but legally challenging result …)

  • Kevin

    For Matt and LoCut, asking about the measurement problem, I think you should look up and read about the Stern-Gerlach experiment. I tend to use this experiment whenever a non-physicist friend/family member asks me about QM, simply because I think its an experiment that is fairly transparent and can be well understood from a good explanation about the actual setup and results, without needing to analogize about quantum mice traveling through a cheese field; it also provides an excellent experimental demonstration of many important and fundamental principles of QM.

    1) quantization of particle properties (angular momentum in this case.)
    2) How, after measurement, we don’t know all the information about the initial state (only magnitudes of constants, no phases), and
    3) Multiple Stern-Gerlach apparati in a row demonstrate measurement changing a state. There’s also a lot of information here about how a state can be a combination of two states when described in a particular way (basis), but be conveniently described as a pure state in another basis.

  • lee

    Collapse of the Wave Function. This is related to the measurement problem (in fact it is fundamental to it!). As a physicist, I am mostly comfortable with QM, but I can’t shake the overwhelming feeling of guilt that I get whenever I refer to wavefunction collapse! It is the only example I can think of where a fundamental physical process is described by a non-unitary operation.

    By the way – you are clearly going to have to expand this episode to a full semester…

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Yeah, there’s clearly no way we’re getting to all of this in an hour. If we’re lucky, we might be able to agree on a definition of “quantum mechanics.”

    But it’s good stuff, and we’ll try. Certainly we’ll talk a lot about the measurement problem and related issues. Bell’s inequality, maybe — that’s another thing for which I don’t yet have a good 30-second explanation.

    Moshe, I’m not sure I get the heart of your objection. I wouldn’t think that we should draw an analogy between wave functions in QM and classical distribution functions; wave functions are analogous to points in phase space, and density matrices are analogous to distribution functions. But I suspect this isn’t the place to resolve that.

    Janus, David himself has written a book which might be just what you are looking for. A couple of equations at the level of algebra, but most of the focus is on concepts, and it’s unfailingly precise.

  • Tom

    Love physics… but not very knowledabe other than at the popular science publication level.
    Not sure if this is directly related to QM, but I have always been perplexed by “virtual” particles.
    What problem are they created to address? Are they “real”, or just a mathematical contruct to help in understanding and analzing? Do virtual particles and the forces they represent have any analogy to the wave/particle duality?

  • Chris W.

    Speaking of questions and answers, Dennis Overbye of the New York Times is responding to questions by email throughout this week (7/7 – 7/11). One already answered is relevant to the topic of this post:

    Q. The ancient Greeks used these terms to distinguish ordered being, on the one hand, from being without order, on the other. My question: In your judgment, is our universe a chaos or a cosmos? — Greg Oakes

    See Overbye’s thoughtful response.

  • TimG

    Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?

    Here’s my best effort to explain this to lay person, off the top of my head.

    In classical mechanics, you can’t have a superposition of two different states. You can either say “The particle is over here”, or “the particle is over there”, but not “the particle is simultaneously somewhat over here and somewhat over there.”

    However, with (classical) waves (such as sound or light), you [i]can[/i] have a mix of two different states. You can play two different notes on the piano at the same time, or mix two different colors of light. If I ask “What frequency is this sound (or light)?” the answer might very well be “It’s a little bit this, and a little bit that.”

    In quantum mechanics, particles also have wave-like behavior, so it’s possible to get a mix of two states. Just like the mix of two notes (two frequency states) can’t be said to have a single definite frequency, in QM you can have a mix of multiple position states without a single definite position. The particle is somewhat in one place and somewhat in another.

    The real weirdness is that when you observe the state (measuring the particle to have some particular position) you change the state so that the particle is [i]definitely[/i] in that position. (At least, that’s the Copenhagen view on what you’re doing.) So you can’t [i]first[/i] measure the particle to be in one position and then immediately afterwards measure it to be in some far away position.

    How well does that explanation hold up in terms of readability/accuracy?

  • Josh

    I’m a non-laymen, but I still would like to throw my two cents into the hat and suggest two things that I’ve never seen covered in a “popular” exposition of QM that I think should be.

    1) Since you mentioned the wave function of the universe, how about the fact that the measurement problem is no problem from that point of view? In other words, the wave function of the universe never collapses. (Right? And if I’m wrong, then what’s doing the measuring that causes the collapse?) I feel like a lot of philosophers would have avoided misusing quantum mechanics if they understood this aspect, which I think is not too hard to convey.

    2) Quantum cryptography. Not *too* difficult to explain (although maybe too difficult for a 1-hour show), but also avoids all the hype and questions about the engineering reality of quantum computation. Quantum crypto is real, and has been done at surprisingly high speeds over (to me, at least) surprisingly long distances. Maybe this fact alone should be mentioned during a discussion of quantum computing.

    That’s it. Best of luck to you. I’m looking forward to it.

  • TimG

    What is the math behind quantum mechanics like?

    Perhaps most important is linear algebra — the math of vector spaces and linear operators — which (for those who don’t know what they are) can be thought of as generalizations of the vectors and matrices from high school algebra. In particular, quantum mechanical states are vectors in a particular kind of vector space called a Hilbert space, and the “observables” (things you can measure like “where the particles are”) are represented by operators acting on these states.

    Differential equations are also very important. Much of non-relativistic quantum mechanics boils down to solving a particular differential equation — Schrodinger’s equation.

  • http://rantingnerd.blogspot.com/ JD

    I second the motion for Collapse of the Wave Function. I’m a former physicist (ABD), and I never got a handle on all the various (weird!) consequences of wave function collapse.

    As near as I can tell, most of the weirdness of QM is in wave function collapse, not wave function evolution. That’s where all the observer effects (and the Deepak Chopra-esque misappropriations of QM) show up….

    Thanks!

  • Jason Dick

    TimG,

    I think you forgot to mention that operators in quantum mechanics are matrices.

    JD,

    If you’re interested in reading up on it, I’d suggest taking a look at quantum decoherence, as well as some of the derivative concepts like einselection. The basic idea is that wave function collapse is just an observer effect due to how quantum mechanical systems interact with one another. Specifically, it comes about because observers like ourselves are also described by quantum mechanics, and our interactions with our surroundings prevent us from observing any other states our overall wavefunction might be in (e.g. if we observe ourselves being in the state that observed outcome A, we can’t experience the state that observed outcome B: that state has decohered from us).

  • http://www.loblog.de Lore

    In Roger Penrose’ “The Road to Reality” is a nice chapter about the measurement problem. He gives about six different possible “solutions”. His own pov is, that a new theory incooperating strings, quantum fields, supersymmetry etc. will give an answer to the measurement problem.

    You shouldn’t forget, that mathematical formalism called QM ist about 70 years old now. It’s already a classic theory, which is very constrained (non relativistic, flat spacetime, not a multiparticle theory and it has the measurement problem…).

  • John Merryman

    From my attempts to understand the topic, it seems particles and waves of light are treated as two descriptions of the same state, but are they? Energy and matter are different but interchangeable states of energy, but while matter is gravitationally collapsing, energy, as radiation, is expanding. Since the ignition of matter is where matter turns to energy, where is it that energy turns to matter? Is it when radiation as a wave becomes a photon? Say the process of measuring this wave amounts to grounding it to the measuring device. So the energy wave collapses to this point of contact, like a miniature lightning bolt.

    That way the wave collapses as a function of contact and measurement is only a potential aspect of that, while there is no universal collapse, as matter is also igniting and expanding back out as waves of radiation.

  • John R Ramsden

    Janus (#5) if you’re comfortable with linear algebra, I reckon the best book on QM is “Principles of Quantum Mechanics” by Ramamurti Shankar (2nd ed, 1994).

    Basic linear algebra is a doddle, once you get the hang of it, and there are plenty of books. But like almost everything in maths and science these days, even the simplest topics can be developed and elaborated to incredible depths. So I’d be careful not to get bogged down and discouraged by some huge algebra tome beyond your needs.

    As one answer to Sean’s question, I’d like it confirmed (if true!) and emphasised that “observation” need not imply a sentient observer, as the word is conventionally understood. So a rock on Pluto could just as well be said to “observe” a photon collapsing onto it. This would eliminate a lot of mystical mumbo jumbo about the special role of sentient observers, for example bringing about their own existence by influencing past events.

  • Anne

    For a nice layperson’s description of QM, I like Feynman’s book “QED”: no formulas, no semimystical mumbo-jumbo, frank acknowledgement of the puzzling aspects.

    My questions are maybe a little more advanced:

    * Why should observables be operators on Hilbert space, and why should they both yield an eigenvalue and leave the system in an eigenstate?

    * Is there a tractable example of decoherence? That is, some system where you can work out the behaviour explicitly on paper, but in which there’s a tunable “peeking” parameter which you can turn from small (observable superposition effects) to large (classical behaviour)? For example a double-slit experiment with an electron where you have a charged test mass near one of the slits – close and you can “see” which slit the electron went through, so you get no discernible interference, far and you can’t “see” but you get superposition effects…

  • Aaron

    I wouldn’t think that we should draw an analogy between wave functions in QM and classical distribution functions; wave functions are analogous to points in phase space, and density matrices are analogous to distribution functions. But I suspect this isn’t the place to resolve that.

    I suspect it’s not either. It’s very possible to think of density matrices as fundamental, and wave functions as a very useful special case. Consider
    (a) density matrices already have the global phase symmetry “modded out”, and (b) the decomposition of a density matrix into pure states is not unique.

    It’s already a classic theory, which is very constrained (non relativistic, flat spacetime, not a multiparticle theory and it has the measurement problem…).

    It’s Galilean relativistic, just not Einsteinian relativistic. There’s nothing better for understanding just how much the phase is mathematical bookkeeping than realizing that not only is the global phase irrelevant, but that changes in phase between one component of the wave function and another are observer dependent. In one sense this is obvious — different observers will naturally use different basis vectors. For spin and rotations these are connected by representations of the rotation group. For Galilean boosts, these are connected with representations of the translation(*) group, e^(ipx/hbar}. But I find this to be somewhat surprising.

    It’s also a fine multiparticle theory. The distinction is that it’s a fixed particle-number and not a variable particle-number theory

    (*) Ok, technically, not translation, but translation + boosts, the translation bit is near trivial, and the boost bit looks like translation, since it’s just addition of vectors.

  • Debbie

    Can I put in a request for at least a paragraph on how the Schroedinger equation was first formed? Or the other two variations. Whenever I took quantum courses the equation was just presented without any commentary on how he came up with the equation. I usually find that understanding the derivation of an equation is more useful in helping people understand what an equation means. In some ways it seems the equation existed before scientists had an interpretation of what the wave function means and that just struck me as wacky.

  • cecil kirksey

    Sean:
    I would suggest that you and David discuss in detail the two-slit experiment. This is not a passe topic. It involves all the essential characteristics of QM. Topics to be covered:
    1. What do we really mean when say that we detect the entity in the two-slit experiment be it an electron, photon or other?
    2. Is the entity destroyed in this detection process?
    3. Do the apertures “detect” the entity? If not why not?
    4. Do the apertures have an effect on the entities? If yes what effect. If not why not?
    5. What constitutes a “real” entity?
    6. Why cannot the “wave function” be considered real?
    7. If the “wave function” is real then doesn’t this resolve all of the issues?

  • Michael Bacon

    There is an interesting discussion of the “wave function” and other matters by Feynman from a conference on the role of gravity and the need for its quantization, held at the University of North Carolina in Chapel Hill in 1957:

    http://arxiv.org/ftp/arxiv/papers/0804/0804.3348.pdf

  • JimV

    Like #35, I am frustrated by people who use QM to support dualism by claiming that minds can alter physical reality by providing a conscious observer to create QM effects such as the results of the double-slit experiment. I believe, like #35, that a mechanical observer would get the same results, but of course it is difficult to prove this without a conscious observer being involved at some point.

    The closest I can come to a counter-argument is to conceive of a adouble-slit experiment in which automated mechanical systems measure and record both which slit photons go through and the resulting interfrence or non-interference pattern. Then before a human reviews the latter results, the former results are physically wiped from computer memory without review.

    Anyway, I would appreciate the thoughts of someone who is smarter and more more knowledgable on this – seconding #35’s request.

  • http://www.soulphysics.org Bryan

    To what extent do quantum phenomena need explanations beyond existing quantum theory?

    I mean this question even apart from the measurement problem. Suppose (like Short et al) you can describe non-local correlation using a formalism that involves classical binary inputs and outputs, plus some randomness. Or suppose you can do the same thing using game theory.

    Are such accounts “explanations” that teach us something new about the quantum world? Or are they just examples of clever formalism?

  • Reginald Selkirk

    What is the quantum equation for consciousness? I frequently see physicists (e.g. Andre Linde) talk about how a conscious observer is needed for “observation’ of an event/particle, so I figure this must be well-characterized physical quantity.

  • http://tyrannogenius.blogspot.com Neil B.

    I have a beef against “decoherence” as a supposed solution for semi-solving (even at best) the problem of the collapse of the wave function. Maybe or not I truly appreciate the concept, but in any case: regardless of whether the wave superpositions are coherent or incoherent, waves would just stay waves unless there was some additional influence or principle forcing the sudden localizations that we call “collapses.” There is nothing intrinsic to the math of waves that enables or even describes “collapses” (not to be confused with the coming and going of e.g. thick spots due to shuffling of frequencies and the resultant Fourier composition of peaks, etc. – that’s still just a pure wave effect per se.) The collapse thing has to be “put in by hand.” You can believe in multiple worlds (hypocritical for anyone otherwise professing positivism and “falsifiability, BTW), but even then collapse events are gratuitous: we are just making multiple examples of all possible ways the collapses could occur. But by contrast, a bunch of true waves would just stay waves and not collapse at all, whether in some possible way or all possible ways!

    Maybe the question I asked in #20 is pertinent since half-wave plates interact with transiting photons in a peculiar way. Each transit must, on average, impart some spin (conservation of angular momentum) since the transit of many photons adds to measurable net spin. Yet oddly, transit does not “collapse” a photon, the photon’s circular base state composition is switched. Hence it stays a coherent superposition. You’d need to re/read that post to appreciate the full implications. In any case, this whole thing is perhaps the most challenging intellectual and metaphysical issue connected to “the universe” that the human mind has to ponder.

  • John Merryman

    Neil,

    In any case, this whole thing is perhaps the most challenging intellectual and metaphysical issue connected to “the universe” that the human mind has to ponder.

    I dunno. Every summit climbed only seems to leave us looking at a larger one in the distance.

  • James G

    What are the main “everyday” applications of Quamtum Mechanics? ie How useful is QM outside of academia?

    I can think of the lasers used in optical storage, bar-code readers and medicine, but are there any other everyday applications? (AFAIK the transitors used in current computers can be explained by classical physics)

  • gingerly

    The thing that gets me is the “sums of history” idea. It sounds incredibly cool and every time I read about it, I think, “That can’t really be what they mean!”

    What does it mean?

  • TimG

    Anne writes:

    Why should observables be operators on Hilbert space, and why should they both yield an eigenvalue and leave the system in an eigenstate?

    I think there’s probably a lot of different levels on which this question can be answered. Here’s my attempt at a basic answer:

    Why a Hilbert space? A Hilbert space is a particular kind of vector space, with some additional structure (particularly and inner product). A vector space is a set of elements (the vectors) with some operations defined on those elements. Specifically, you can add two vectors together, or you can multiply them by scalars (that is, numbers). The addition and multiplication also have to satisfy certain axioms.

    In quantum mechanics, you can have a superposition of two states. That is, if you have a state A and another state B, then it is also possible to have a state that’s, say, 30% A and 70% B.
    state = sqrt(0.3)*A + sqrt(0.7)*B
    So clearly our states can be multiplied by numbers and added together. This leads us to vector spaces and, more specifically, Hilbert spaces (once we define an inner product and check that all the appropriate axioms hold.)

    Why are observables operators? I guess the simplest answer is that the act of observing changes the system being observed, so the observables can’t merely be numbers, they have to operate on the state. Studying the properties off these observations one sees that they satisfy the necessary axioms to be linear operators, and, in fact, self-adjoint operators.

    Why eigenvalues and eigenvectors (a.k.a. eigenstates)? Well, if you have a state with a 30% chance of X = 1 and a 70% chance of X = 2, the expectation value of X is 0.3*1 + 0.7*2 = 1.7 You’re multiplying the different components of the state by the value of the observable in that component, which is exactly what would happen if they’re eigenvectors of the X operator with the eigenvalues 1 and 2.

    Why does it leave the system in an eigenstate? That’s an empirical observation: if you measure a particular observable, the state is changed so that further measurements yield the same value for the observable. Why that happens is a deep question, but we can see that it does happen and the math has to reflect that.

  • John Wendt

    What would chemistry be like if there were four dimensions (with very little curvature, like the three we know so well)? Would there be four 2p orbitals? Could we have quadruple bonds? Everyone of whom I ask this question starts out “Well, I”m not a quantum chemist…”, and I haven’t met any actual Q-chemists yet…

  • bcamarda

    These are all great questions. I’d be interested in your reaction to the new work by Zeilinger et al to test realism (following in the footsteps of Bell’s and Aspect’s work). From: http://www.seedmagazine.com/news/2008/06/the_reality_tests_1.php

  • http://www.geocities.com/aletawcox/ Sam Cox

    Andy S said…

    “Question 3: In a delayed choice two-slit experiment, a particle knows when it’s emitted whether its path is going to be through one of two slits or a superposition of both paths based on how it’s going to be measured, even if the measurement happens 50,000,000 years later. HOW THE HELL … ahem. Excuse me. How does it GOD DAMN KNOW HOW … ahem excuse me again.

    If that damn thing can somehow look 50,000,000 years into the future and see the laboratory it’s going to wind up in and see the scientist with his finger on a button and it makes its decision on how to propagate based on that, then the universe is rigidly deterministic to an extent that makes me want to just go and slit my wrists.”

    …all of which ties in with Seans brief comments on the wave function of the universe!

    This is going to be a very interesting discussion. I get the impression we are already in a postion to draw some (at least) preliminary conclusions!

    QM, like SR and GR, is awesome!

    Sean has plenty of material for the upcoming discussion!

  • Mike M

    What are the main “everyday” applications of Quamtum Mechanics? ie How useful is QM outside of academia?

    How about giant magnetoresistance, without which no hard-disk iPod would be complete? Although arguably many bits of modern technology rely on quantum mechanics, GMR is a nice one to use as an exemplar, because it is a blatantly quantum-mechanical phenomenon that went astoundingly rapidly from discovery (1988) to universal industrial adoption and Nobel prize (2007)

    It is also a good one to wave under the noses of those idiot politicians who think all money should go on “applicable” science rather than “useless” blue skies science (MRI is another good one there, co-invented in my department on the basis of an entirely pointless experiment, and now worth a billion or so a year).

  • andy.s

    What are the main “everyday” applications of Quantum Mechanics? ie How useful is QM outside of academia?

    Well, if the movie “What the Bleep…” is any indication, you can use it to sink free throws and to make yourself a great photographer or something.

    wow! more than 50 comments! Usually the science stuff get 10 or 15, tops. Good going, Sean.

  • http://guidetoreality.blogspot.com Steve Esser

    Re: the wave function of the universe. Wouldn’t this idea only make sense from the perspective of an observer standing outside the universe (which doesn’t make sense)?
    Thanks.

  • andy.s

    Re: Debbie #38

    Can I put in a request for at least a paragraph on how the Schroedinger equation was first formed? Or the other two variations. Whenever I took quantum courses the equation was just presented without any commentary on how he came up with the equation.

    That’s always bugged me, too. To date, I’ve got 4 Intro level Quantum texts in my library. Not one of them derives the equation.

    The only one who makes a stab at it, is –you guessed it– Feynman in one of his Lectures on physics volumes.

    It’s something along these lines: (note – there’s no preview, so my Latex’d equations might not be legible).

    (also note: I’m only a B.S., so my equations might also be gibberish).

    $latex |Psi(t^prime)rangle = U(t, t^prime) |Psi(t)rangle $
    meaning the wave function at a future t’ is some operator times the wave function at t – the operator depends on t and t’.

    Obviously,
    $latex U(t, t) = I $

    So in the infinitesimal region around t, U is probably something like,
    $latex U(t, t+delta t) = I + delta t Omega + (delta t)^2 (whatever) $

    That makes
    $latex |Psi(t+delta t)rangle = (I + delta t Omega + (delta t)^2(whatever) ) |Psi(t)rangle $

    Getting the derivative gives you:
    $latex |dot{Psi}rangle = Omega |Psirangle $

    Big whoop right? We had some operator U that we didn’t know anything about, now we have some operator $latex Omega$ that we don’t know anything about. But hit it on the left with $latex langle Psi |$ and you get:

    $latex langle Psi |dot{Psi}rangle = langle Psi |Omega |Psirangle $

    Adding the complex conjugate,

    $latex langle Psi |dot{Psi}rangle + langle dot{Psi} |Psirangle = frac{partial}{partial t}langle Psi |Psirangle = 0 = langle Psi |Omega +Omega^dagger|Psirangle $

    Which implies that $latex Omega^dagger = -Omega$.
    Cool. So $latex Omega$ is anti-hermitian.

    But we like Hermitian operators, so let’s multiply both sides by i. This gets us

    $latex i|dot{Psi}rangle = i Omega |Psirangle $

    That makes the operator on the right side Hermitian. That’s where the “i” in the Schroedinger equation comes from.

    Next is the units. $latex |Psirangle $ always has some weird units like length^-1/2, and naturally $latex |dot{Psi}rangle$ would be length^-1/2 s^-1. That means that $latex Omega$ will have units of s^-1 or frequency.
    Lucky we called it $latex Omega$ then.

    So $latex iOmega$ is Hermitian and has units of frequency, so it must be some kind of frequency observable. Well, the energy levels of atoms are always associated with absorption and emission frequencies so we can get the operator to have energy units by multiplying both sides by $latex hbar$ to get:

    $latex ihbar|dot{Psi}rangle = i hbarOmega |Psirangle $

    Which is just the Schroedinger equation, with $latex ihbarOmega$ identified with the Hamiltonian.

  • James G

    How about giant magnetoresistance, without which no hard-disk iPod would be complete? Although arguably many bits of modern technology rely on quantum mechanics, GMR is a nice one to use as an exemplar, because it is a blatantly quantum-mechanical phenomenon that went astoundingly rapidly from discovery (1988) to universal industrial adoption and Nobel prize (2007)

    That’s a great example, thanks! I really wasn’t aware of that, I assumed the modern computer was a product of classical physics (except for the laser in the optical drive), amazing that something as crucial as the hard drive depends on this “mysterious” science!

    Basically, everytime you access a database, anywhere in the world, the results returned require Quantum Mechanics to be correct.

    http://www.research.ibm.com/research/gmr.html

  • viggen

    Just glancing through some of the comments here, Sean, I would recommend that you take at least a moment during BloggingHeads to mention that the Periodic Table of the Elements is QM at its finest. Everybody seems so interested in the off-the-deep-end questions that nobody has looked closer to home–how do we know the shape of water or CO2, or one of a hundred thousand important biomolecules? Basic QM and the Periodic table gave us all the necessary tools. I say this because the Periodic Table is something that nearly every Layman will have passing familiarity with and because it’s one of the greatest, most impactful, most tangible victories of quantum theory. Moreover, while I know it’s not as sexy as Quantum information theory and entanglement, nor as hardcore as QED or QCD or the Standard Model, it is accessible.

  • Neal J. King

    andy.s,

    What you describe is interesting as far as it goes, but there are some problems in taking it as an explanation of the Schrödinger equation:

    – Schrödinger developed his theory in the context of solutions to partial differential equations, not state vectors in Hilbert space. What you are describing seems to be closer to Dirac’s presentation of QM.

    – The logical predecessor to the Schrödinger equation was of course Heisenberg’s theory. My one-minute summary of that invention was that Heisenberg was thinking about focusing on the Fourier components of the dynamical variables, and got the idea of limiting the frequencies in the spectrum of components to the allowed frequencies in the energy spectrum. This is described in Max Born’s book, The Dynamical Theory of Crystal Lattices, based on lectures given very shortly after Heisenberg’s invention. (The first part of the book is on lattices, the second is on Heisenberg’s brand-new QM.)

  • andy.s

    So why doesn’t any textbook ever derive it?

    I’ve got several introductory texts and they all talk around it in different ways. It gets kind of annoying after a while.

  • Doug

    @49 This one is actually not that hard! You’d have no chemistry at all, because the coulomb solutions don’t have stable orbits. I believe without a stable classical orbit the quantum business still couldn’t conspire to give you any chemistry.

    I’m also not a quantum chemist, but I’m not sure you really need one here.

  • Matt

    Following up on Doug’s answer (60), I think I also remember from partial diff-e, that any linear extra dimensions making the total number of linear dimensions an even number (4, in the case of the original question) results in every event echoing into infinity. Or something like that. I couldn’t even recognize a differential equation to save my life any more.

  • Patrick Dennis

    I’m guessing that if someone has a fairly clear-cut idea of what Hilbert Space is they are not in your target audience. I like #39 Cecil’s idea of using the two-slit experiment as a point of departure (a la Feynman). The explication thereof would give you the opptunity to explore any number of streams to any desired depth. I’d also second the suggestions those who have asked for some discussion of practical applications.

  • weichi

    andy s.,

    Are you equally bothered that textbooks don’t derive Maxwell’s equations? Or F=ma?

  • Paul Murray

    Can someone please explain bra-ket notation to me? The explanation on wikipedia just goews around in circles. I get as far as “Every key has a dual bra”, and follow the link on dual, and it’s off into chaosland.

    How do you get from bra-ket to actual numbers? I see lost of psis and thetas, but at what point do you plug actual numbers into these things and get some sort of result?

    Those spherical harmonics that are the orbitals of the hydrogen atom – does that apply to other atoms? What about molecules? I mean … as I understand it, the state of an electron is basically a complex number field which extends through all space, except that it’s very nearly zero almost everywhere except where the “location” of the electron is. If you have two electrons, then you multiply their wave functions together and integrate it, and the square modulus of the result gives you the probablility that they will interact. If they interact (exchange a virtual photon), the the momentum transfer means that they move apart, and that’s what electrical repulsion is. Or something. I’m still not sure how this interacts with time – the probability that they interact has got to be a “chance per second”, and all this stuff has got to be symmetrical WRT relativistic boosts.

    Now, was that kind of right? Or is the field some sort of thing where at each “point” in space there is actually a matrix of complex numbers?

    If an electron hits an antielectron and is anihalated, does the wave equation field thing dissapear everywhere simultaneously? Or does the dissapearance of it sort of propagate outward at the speed of light?

    Anyway.

    How did they figure out that Buckminsterfullerene was going to be yellow? I mean – what numbers to you plug in, and where do you plug them? In order to come up with that result, it couldn’t all have been algebra – they’d have to have dome some adding and multiplying to come up with the absorption spectrum. Sorry to carp on about it, but from the wikipedia page on bra-keyt notation, I can’t see how any of this stuff gets from math to reality.

    Why does a water molecule have that mickey-mouse shape? How do you get from “The hydrogens have one electron, the oxygen has 8″ to that particular shape?

    That’ll do.

  • http://tyrannogenius.blogspot.com Neil B.

    48. TimG:

    Why does [measurement] leave the system in an eigenstate? That’s an empirical observation: if you measure a particular observable, the state is changed so that further measurements yield the same value for the observable. Why that happens is a deep question, but we can see that it does happen and the math has to reflect that.

    Yes, we see it does happen, but that is all from examples of traditional “one shot” measurements where a single interaction results in either an eigenstate left over, or destruction of the particle (e.g., photon either passing or being absorbed by a polarizing filter. But that is not what happens when a photon passes through a half-wave plate! (Or, similar birefringent element.) Like I said, the photon gets its RH and LH bases swapped, but not collapsed. Of course, one pass through a HWP doesn’t measure anything anyway. Yet many passes should build up detectable angular momentum along a range, not just binary results, as I explained above. I can’t say for sure and must be humble about something that pushes the envelope, just asking for it to be considered.

  • andy.s

    weichi

    My copy of Reitz and Milford on E&M has a quite extensive discussion on Maxwell’s equations and I think most students would be uncomfortable with a text that stated them right off the bat with no discussion of their roots in Coulomb’s law, Ampere’s law, etc.

    As for F=ma it is, considerably more intuitive than $latex -frac{hbar^2}{2m}nabla^2Psi + VPsi = EPsi$.

    Intuition is a fairly useful faculty. It certainly leads me to suspect that you don’t have a lot of friends.

  • Christopher M

    I think this is probably off-topic, because I guess it probably has more to do with relativistic theories than with QM per se. But I have yet to read a really good explanation of how time dimension(s) and spatial dimensions are different. The whole “time as a fourth dimension” thing has made intuitive sense to me since I was a little kid. I can picture the whole of spacetime as the 4-D analogue of a 3-D block with world-lines running through it, a block that one can slice on different angles into different relativistic viewpoints on the world. But I’ve never quite understood, okay, time is a dimension like the left-right axis, but then why does it seem so different?

    One explanation I could make sense of would just say, well it’s only different in that the universe had this very low-entropy state at one end (the beginning) of its time dimension, so we have the “arrow of time” and thus the notion of experience, the “course” of our lives, etc. But I’ve also read in several places that, no, it’s not that easy, temporal and spatial dimensions really are just different.

    I don’t expect solid answers; I imagine this might be an area where different theories (general relativity, string theory, QM?) have different things to say (whether or not their answers are mutually contradictory). But is there a good discussion of “time as a dimension, but not a spatial dimension” anywhere?

  • Christopher M

    It certainly leads me to suspect that you don’t have a lot of friends.

    Sometimes I wish that I could enjoy the whole game of internet put-downs and flame-throwing, because it seems like something that would be a lot of fun if you were the kind of person who could enjoy it. Other times, I’m glad that it just seems ridiculous.

  • Jordan

    Talk about the classical analog of the Heisenberg Uncertainty Principle in classical waves- it makes sense when you’re thinking of water and not electrons, and the HUP is one of the things that makes QM so mysterious to the layman.

  • chemicalscum

    On the wave function of the universe I love Gell-Mann’s comment to Jim Hartle about the Hartle-Hawking wavefunction. “Hey Jim, if you know the wavefunction of the universe how come you’re not rich ?” or words to that effect.

    I watched Jim Hartle’s talk at Oxford for the Everett 50 years of MW meeting on the web. It was very interesting, I can’t remember the URL of the back of my head but its well worth a view together with the accompanying slides.

    Anyway as a chemist my interests in QM are more immediate and practical. Yesterday I compiled GAMESS (General Atomic and Molecular Electronic Structure System) on my system. So I guess I should shut up and calculate.

  • http://www.yaxom.com/gjb/blog/ Greg Black

    As a non-physicist, I’d add a vote for David Albert’s book as a nice introduction to QM.

  • weichi

    andy s,

    I apologize if I struck a nerve – it wasn’t my intention.

    Are you really looking for a derivation – i.e., logical deduction from some “more fundamental” foundation? I don’t think you are going to find it – Schrod eqn is just the god-given way that the world works (*). Sure, you can offer plausibility arguments like in your post 55 (and I agree they are helpful!) but that’s not what I think of as a “derivation”.

    In particular, I don’t think it’s correct at all to think of Maxwell’s equations as derived from Coulomb & Ampere laws; in fact I prefer to think of Coulomb’s law as a consequence of Maxwell eqns (in the electrostatic case). The roots of maxwell in coulomb and ampere are historical, not logical.

    Anyway, some places to look:

    * have you looked at Shankar’s path integrals chapter? 8.6 in particular shows how path integral approach is equivalent to schrod eqn, and maybe you’ll find path integral a more intuitive starting point?

    * Landau & Lifschitz QM (section 17) shows that the classical limit of Schrod is the Hamilton-Jacobi eqn.

    * Schrodinger’s paper of 1926 “Quantization as a problem of proper values” (reproduced in the book below) explains how he come up with time-independent version. He starts with H-J eqn.

    * The book “Probability and Schrodinger’s Mechanics” by David Cook promise a “variational derivation” of Schrod Eqn in 8.1. I think it’s an expansion of schrod’s original argument, but haven’t read it, so not sure.

    Anyway, my view is that the Schrod eqn is not something that can be derived – it simply is.

    (*) I guess you can derive schrod eqn from quantum field theory, but to me that just begs the question of how do you derive quantum field theory.

  • ira

    1 What does the many-worlds interpretation really mean ? And why do so many physicists believe it’s correct ?

    2 How is many-worlds different from parallel universes or the multiverse ?

    3 Given that quantum mechanics is the basic theory that explains how the world works, why don’t we see its bizarre aspects in everyday life ? (I think this gets at the heart of many laypeople’s interest in QM) Could you discuss decoherence vs the theory of Zeillinger’s group (referenced in the link in comment # 50 above)

    4 The work of Vedral and Zeilinger’s group on entanglement in many-body systems or bulk materials (this gets at beginning to see one of QM bizarre aspects
    in (closer to) everyday life situations)

    Why do you have to limit yourself to one episode ? Sean, you do a great service by attempting to explain physics to non-professionals, and in the age of the Internet the ability (bandwidth) to communicate has been immensely increased.
    After all, you did inherit Feynman’s desk :-)

  • Jason Dick

    I have a beef against “decoherence” as a supposed solution for semi-solving (even at best) the problem of the collapse of the wave function. Maybe or not I truly appreciate the concept, but in any case: regardless of whether the wave superpositions are coherent or incoherent, waves would just stay waves unless there was some additional influence or principle forcing the sudden localizations that we call “collapses.” There is nothing intrinsic to the math of waves that enables or even describes “collapses” (not to be confused with the coming and going of e.g. thick spots due to shuffling of frequencies and the resultant Fourier composition of peaks, etc. – that’s still just a pure wave effect per se.) The collapse thing has to be “put in by hand.”

    This is the beauty of decoherence: it doesn’t have to be put in by hand at all. A really simplistic way of looking at it is as follows. Imagine that we start with a two-state system:
    $latex midpsirangle = mid 1rangle + mid 2rangle$
    (normalization ignored for clarity)

    Now, as we progress this state forward in time, the two states will oscillate between one another dependent upon the difference in energy. But what happens when we observe the state? Observation of the state is provided by an interaction with some other system, a system that happens to be vastly more complex (let’s call the other system $latex midphirangle$:
    $latex midpsiranglemidphirangle = left(mid 1rangle + mid 2rangleright)midphirangle$

    After this interaction, what happens is that the oscillation time between states 1 and 2 becomes huge. If the state $latex |midphirangle$ is complex enough, the time to oscillate can become effectively infinite. Due to the fact that we are complex, then, every time we interact with such systems we end up preventing further interference: the wave function decoheres, and only one result is ever visible. Nothing put in by hand, just simple wave function evolution.

    P.S. I hope the tex came out okay.

  • Jason Dick

    From my attempts to understand the topic, it seems particles and waves of light are treated as two descriptions of the same state, but are they? Energy and matter are different but interchangeable states of energy, but while matter is gravitationally collapsing, energy, as radiation, is expanding. Since the ignition of matter is where matter turns to energy, where is it that energy turns to matter? Is it when radiation as a wave becomes a photon? Say the process of measuring this wave amounts to grounding it to the measuring device. So the energy wave collapses to this point of contact, like a miniature lightning bolt.

    I think you’re misunderstanding what is meant by energy. Energy and matter are not different sots of thing. You don’t convert from one to another. Instead, energy is a property of matter. Now, you can convert from one sort of energy into another. For example, if I collide an electron and a positron together, the two can annihilate to form a pair of photons. This isn’t a conversion between matter and energy: photons are still a form of matter. What happens, however, is that the mass-energy of the electrons gets converted into kinetic energy of the photons.

    So far as we know, these interactions are point interactions. But this will probably be amended once we understand more about high energy physics.

  • http://pantheon.yale.edu/~pwm22/ Peter Morgan

    Sean, your

    Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t.

    compares quantum mechanics with classical particle physics, which is a straw man. Compare quantum field theory with “probability densities over classical field states” (aka random fields) — which, given that all our best theories are field theories, and given that probability is ever-present in Physics, seems less likely to be a straw man — we find that superposition is present in both. Indeed, a random field can be formulated as a commutative algebra of operators, making differences between quantum and classical only of the algebraic structure and interpretation of their measurement algebras. Bell inequalities become an essentially negligible issue (I argue that the conspiracy assumption is natural for a random field, whereas I agree with the conventional argument that it is unreasonable for classical particle models).

    With apologies for grinding my axe, for “Bell inequalities for random fields”, see cond-mat/0403692, J. Phys. A: Math. Gen. 39 (2006) 7441-7455; for an algebraic formulation of interacting random fields, “Lie fields revisited”, see Arxiv:0704.3420, J. Math. Phys. 48, 122302(2007). These will not convince you of much, I have not yet developed my approach enough for it to be much more than speculative, but you might perhaps take away that superposition is not a good separator between classical and quantum. If you understand and start to implement in your thinking that classical particle physics is a straw man you will be ahead of the curve. Asking how much of this can be put into a discussion directed at the layman is an awkward question of course.

  • http://albatross.org Albatross

    Here’s one for today

    Lately I find myself feeling less contemptuous of Creationists than sorry for them. The creation myths of religion were only primitive efforts to understand the origin of the world and the nature of being in the absence of any information. Now, for the first time in history, we’re actually able to get at the answers–the actual, correct answers–and most people are too blinded by their emotional attachment to the old bedtime stories to look at the truth now that it’s staring them in the face. It’s especially depressing because the truth, insofar as we’re able to apprehend it, is so much more elegant, complex and beautiful

  • Fermi-Walker Public Transport

    Interesting discussion everyone. A question for the experts. I recently came across Ballentine’s quantum mechanics book and he makes a fairly convincing case to me that this “collapse of the wave function” business is not needed in the measurement problem if one uses the statistical point of view for the wave function. How generally accepted is this viewpoint ?

  • weichi

    Thinking more about the question of “deriving” the schrod eqn, maybe an analogy would be to the “derivation” of maxwell’s eqns in, e.g., Schwartz’s Principles of Electrodynamics, where he takes coulomb’s law, invariance of charge under lorentz boosts, and lorentz invariance, and argues not only that a magnetic field is required, but gets the full maxwell equations (if i recall correctly). So he is starting with three very well-established experimental facts and shows that consistency requires this extra structure (loosely speaking).

    (I’m not sure whether his argument can really be viewed as an air-tight logical deduction, but it certainly provides motivation and insight).

    It would be nice if there were a similar set of well-established experimental facts that could similarly lead you to schrod eqn. My guess is that there *isn’t* such an argument, because it seems like the experimental facts would have to involve the wavefunction, but we dont’ really *have* any experimental facts about wavefunctions. The fact that the wavefunction – the very quantity whose evolution schrod eqn describes – is not an observable gives it a very different flavor than the electromagnetic field.

  • http://tyrannogenius.blogspot.com Neil B.

    Jason Dick, thanks for the helpful attempt. However, I still don’t think you get the deep objection, which is that even that one resulting wave “that we observe” still has no reason to suddenly shrink into a tiny space, it should still be an extended wave anyway, etc. You and others are still taking the observation regime for granted and can’t seem to “get above it”, you are IMHO like fish who can’t appreciate what their being in water does.

    In any case, the multiple transmission thought experiment I have been describing above doesn’t rely on any particular interpretation of decoherence, etc., it is based on the logical implications of what we already know about those particular interactions between photons and wave plates.

  • TimG

    weichi, I think you’re missing andy s.‘s point a bit. It’s all fine and good to say the Schrodinger eqn. “just is”, and we may not have some sort of derivation from first principles, but nevertheless Schrodinger himself must have got it from somewhere, right?

    That is to say, Schrodinger didn’t simply write down random symbols until he found an equation that predicted the spectrum of the Hydrogen atom, did he?

    Most intro-level text books do seem to just state Schrodinger’s equation without any explanation of where it comes from. I’ll go check my graduate QM book and see if it does any better.

  • TimG

    Well, according to Merzbacher’s Quantum Mechanics textbook, Schrodinger wanted an equation that agreed with classical mechanics in the classical limit.

    Taking Schrodinger’s equation and using psi = exp(i*S/hbar), we are led to the Hamilton-Jacobi equation (which is one formulation of classical mechanics, equivalent to Newton’s laws, Lagrangian mechanics, etc.) Except, this equation has one extra term proportionate to hbar.

    In other words, it agrees with classical mechanics in the limit where hbar is negligibly small.

    That’s a reverse derivation — what Schrodinger really did was probably used his knowledge of classical mechanics to constrain the form of the quantum mechanical equation, and then played around with it a bit until he got one that also reproduced known experimental data. At least, that’s my best guess without actually looking up Schrodinger’s original papers (or rather, translations of them to English).

  • TimG

    To clarify the above post, I don’t mean that the Hamilton-Jacobi equation has an extra term compared to classical mechanics. I mean that the Schrodinger equation leads to the Hamilton-Jacobi equation plus an extra term. Hamilton-Jacobi by itself is exactly equivalent to classical mechanics.

  • Fermi-Walker Public Transport

    I heard the story that Schrodinger got his idea from attending a lecture given by De Broglie on his thesis work
    on wave-particle duality. Schrodinger and Kramers were sitting next to each other and at the end of the talk, Kramers said to Schrodinger that if matter had wave proporties, then there must be a wave equation. Supposingly this comment started Schrodinger thinking about the topic though of course it does not answer andy s.’s question.

  • John Merryman

    Jason,

    Thank you for responding to my handwaving.

    Energy and matter are not different sots of thing. You don’t convert from one to another. Instead, energy is a property of matter.

    Given my mental dyslexia I think QM makes more sense if this statement is reversed, that matter is a property of energy. We assume there is some underlaying monolithic property, but by all evidence so far, reality seems to be a function of relative interactions of opposing forces, with matter as a fairly stable manifestation of this. The search for this underlaying property leads to smaller and smaller points of observation, while the macrocosmic reality continues to balance itself between polarities. What if they don’t find the Higgs? Are they just going to keep looking? String theory is seemingly about the strings, but it’s only their fluctuation that really matters. What if they are simply vortices?

  • weichi

    TimG,

    Well, I did provide a pointer to Schrod’s original paper, so I don’t think I *totally* missed his point ;-) Agreed that it would be nice if textbooks explained why schrod choose particular eqn.

  • Aaron

    Probably not historical, but I think the clearest way to look at Schroedinger’s equation is that energy is the generator of time-translation in exactly the same way that momentum is the generator of space-translation. Noether’s theorem, etc.

  • http://pantheon.yale.edu/~pwm22/ Peter Morgan

    Agreeing with Aaron: once we represent the results of experiments by operators acting on a Hilbert space, time evolution gives a one-parameter group, so by Stone’s theorem there is a self-adjoint operator, which in the case of time evolution we call the Hamiltonian, that generates the group. It’s convenient to represent statistical measurement by operators, and to represent the impossibility of joint probability distributions, for certain combinations of measurements, by using noncommuting operators.

  • TimG

    More on “deriving” Schrodinger’s equation:

    Consider a plane wave: psi = e^[i (k x – w t)]

    This has
    k psi = -i d/dx psi
    and
    w psi = i d/dt psi

    From the de Broglie relations we have p = hbar k and E = hbar w

    For this reason, we define:
    p = -i hbar d/dx
    and
    E = i hbar d/dt

    Now take the classical equation E = p^2 / 2m + V
    and replace with our expressions for E and p in terms of differential operators.

    This (acting on psi) is the Schrodinger equation.

    Now, starting from plane wave solutions doesn’t make sense if V varies over space. But let’s suppose the Schrodinger equation is still correct. From this we can see:
    (1) The equation has the right classical limit (as I mentioned above)
    (2) It correctly predicts the spectrum of the hydrogen atom.

    So, it looks like the equation is right even for the case where V depends on x.

  • weichi

    TimG,

    I like it. So we can use electron diffraction experiments to justify both 1) treating electron as a wave (so we have wavefunction) and 2) p = h-bar k deBroglie relation. For E = h-bar omega I guess you can refer to planck blackbody, and make a leap that it will hold for matter waves as well (just like deBroglie did!)

    So treating these as your the experimental “facts”, your simple argument leads to Schrodinger (and also, I believe, Klein-Gordon if you take the relativistic case). It’s not a derivation from first principles, but better than pulling things out of thin air.

  • TimG

    Yep, it gives the Klein-Gordon equation if you start from:
    E^2 = p^2 c^2 + m^2 c^4

    I’ve read somewhere that Schrodinger actually first discovered the Klein-Gordon equation, but rejected it because it gave incorrect predictions for the hydrogen atom. (Of course we now know the equation is appropriate for spinless particles, not electrons.)

  • http://celsetialmechanician.org Celestial mechanician

    1. Why can’t QM predict, i.e. have an analytic solution, the line spectra of He or any multielectron atom?

    2. How does QM account for the continuous spectra of molecules?

  • Lawrence B. Crowell

    ndy.s on Jul 7th, 2008 at 3:50 pm

    Question 3: In a delayed choice two-slit experiment, …. How does it GOD DAMN KNOW HOW … ahem excuse me again.
    —————–

    It should be possible to use an Einstein lens as a great beam splitter. EM radiation emitted by a distant source, then lensed around a large elliptical galaxy (Abell cluster), will be have split paths through the Schwarzschild metric. For EM radiation in the radio band, bandpass filters with a very narrow frequency range can exploit the Heisenberg uncertainty principle to get around the path length difference take by either side. So if a photon strikes our radio telescope it has a quantum amplitude for traversing one path or the other. We may then be able to perform a Wheeler Delayed Choice experiment across hundreds of millions of light years.

    How does the photon know which path to take? It doesn’t, in fact it knows nothing. Think of there being two types of relationships in the universe, one which involves invariant quantities such as distances, intervals, the momentum interval

    $latex
    m^2c^4~=~E^2~-~p^2c^2,
    $

    and the like and covariant elements such as curvatures. The other relationship involves quantum nonlocality, entanglements, superpositions and the like. This relationship system is independent of the geometric type of relationship. There happens to be a representation of quantum waves or states in spacetime, where

    $latex
    psi({vec r},~t)~=~langle{vec r}|psi(t)rangle,
    $

    here nonrelativistic, which obeys wave equations. In this way we can identify oscillator modes on each ADM relativity spatial slice of spacetime. Yet the wave function is nonlocally defined on each time slice (spatial surface) and that relationship is nonlocal. The Wheeler Delay Choice experiment indicates this nonlocality is not just across spatial distances, but temporal ones as well.

    Lawrence B. Crowell

  • John R Ramsden

    Albatross (#77) quoting from http://www.thepaincomics.com:
    >
    > Lately I find myself feeling less contemptuous of Creationists than sorry for them.

    Although agnostic, I am contemptous of Creationists. Nowhere in the Bible does it detail or even indicate *how* the Universe was created or, explicitly, state that creation was ever completed. Nor, aside from the seven day business, where the word “day” was probably meant more as “epoch”, can a reliable timescale be deduced.

    So dogmatic creationists are presumptuous and foolish to insist on their interpretation when the facts clearly indicate otherwise in countless ways. Maybe God if he chose to comment, far from approving of their blind faith, would give them the same bollocking he gave Job (Job Chap 38):

    Then the Lord answered Job out of the whirlwind, and said, “Who is this that darkeneth counsel by words without knowledge? Gird up now thy loins like a man; for I will demand of thee, and answer thou me. Where wast thou when I laid the foundations of the earth? Declare, if thou hast understanding. Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it? Whereupon are the foundations thereof fastened? or who laid the corner stone thereof; When the morning stars sang together ..

  • TimG

    Celestial mechanician wrote:

    Why can’t QM predict, i.e. have an analytic solution, the line spectra of He or any multielectron atom?

    Some differential equations can’t be solved analytically. Hydrogen works out nicely because of the spherical symmetry of the problem. The electron feels a potential due to the nucleus — and if you put the nucleus at the origin of your coordinate system and rotate around the origin, that potential doesn’t change.

    In helium, an electron feels a potential due to the nucleus and due to the other electron, so the spherical symmetry is broken. You can’t put both the nucleus and the second electron at the point of rotation.

  • James Robson

    I see that this list of responses has been growing rapidly since the time that I first looked just a day or so ago. I think this is good news on the one hand (as I believe that general interest in scientific thinking and ideas to be a essential thing. Of course, experts have contributed but it’s the general audience that matters here). However I think it also reflects a real problem with the understanding of quantum mechanics and the consequent frustration held by the general public (well, unfortunately not quite general since in many countries most are more concerned with finding food and avoiding wars and oppression, and elsewhere a lot of people can’t be bothered…however…)

    Time! That seems to me to be the key. It is one of the most natural, obvious, and simple things, yet I think it is the biggest enigma remaining in the understanding of the world. In GR the whole thing seems to be fixed and “cast in stone”. The Wheeler-de-Witt eqn (just an example of one bit of QM thinking applied to GR) doesn’t even mention it at all. But, time seems to keep plodding on regardless. There is, as well, the entropy question that Sean has commented on before, and which is also related to time.

    And what is the “quantum measurement problem” if it is not a question about time – at least as we humans experience it? The unfortunate cat is observed first to be alive, and later to be dead (either as the result of the malevolent Mr. Schroedinger or later from natural causes). But it all seems to depend on who you ask and what they may know from their particular involvement with the whole sad business at any particular time. Thinking about this I have some enthusiasm for (but no real understanding of) Carlo Rovelli’s ideas of relational QM – but who knows?

    The fact remains that IMHO nobody has got a clue. ALL modern theories are (or aspire to be) quantum mechanical. They increase in their sophistication and mathematical complexity. But, right now, and for the foreseeable future, nobody knows what is going on – despite the fact that the thing seems to work. The vast majority of physicists/chemists/engineers, etc. who use it day-in-day-out just get on with the calculations, having long-since exhausted their enthusiasm for the associated philosophy after too many nights spent arguing with friends while doing their degrees.

    So – “Quantum mechanics (and GR if deemed appropriate) and the question of time” would be my request for your discussion, and my also my”Clay” $1000000 prize” (if I had any mony) for your solution.

    -James

  • Lawrence B. Crowell

    James Robson on Jul 9th, 2008 at 5:24 pm

    In GR the whole thing seems to be fixed and “cast in stone”. The Wheeler-de-Witt eqn (just an example of one bit of QM thinking applied to GR) doesn’t even mention it at all.

    ————

    The absence of time in the WD equation stems from its classical roots in the ADM “space plus time” approach to general relativity. It leads to a Hamiltonian constraint NH = 0, where the Hamiltonian is due to the Gaussian second fundamental form, and N is the lapse function. There is also a momentum contraint as well N^iH_i = 0, where N^i is a shift function that tells how points are slid around from one spatial surface to the next. The lapse and shift functions are not “God given,” but depend upon how one fixes a coordinate condition on a problem. So time is not something which has any “cast in stone” quality to it.

    A similar thing happens with quantum mechanics. The momentum and position have commutator relationships [x, p] = hbar/2, which under canonical quantization come from the Poisson bracket in classical Hamiltonian mechanics. QM also has an uncertainty relationship between energy and time, but there is no Poisson bracket for the Hamiltonian and time, and there is neither a time operator. The most you can work up is a sort of periodicity operator. So even in QM time has a curious property and does not really have the same status as position.

    I could go on about this at considerable length, but I might find myself accused of “theory mongering.”

    Lawrence B. Crowell

  • Diocletian

    Please discuss the paradox that arises when one tries to attribute expansion of the universe to the energy of quantum vacuum fluctuations. The fact that a discrepancy of 10^120 is found when making this attribution should I think first suggest that the two ideas have in fact nothing to do with each other and that there is really no difficulty at all. Nevertheless, physicists have expended a great effort in trying to reconcile this argument. What am I missing here? Why should I insist that vacuum energy causes the universe to accelerate?

    The vacuum energy that physicists burden themselves with as an attempted explanation of cosmological acceleration is calculated the same way as the Casimir force, which has been measured and verified. So the discrepancy when one applies this idea to the cosmological expansion can only be a misapplication, somehow, of the theory and not the deeply troubling deficiency or inconsistency of theory that is always presented. But obviously I’m missing one or more ideas here.

    Would it be possible to perversely attribute the Casimir force to the measured cosmological constant and calculate an inverse discrepancy of 10^-120? Why would this problem be improperly motivated?

    Thank you to one and all with opinions.

  • Mark N

    Can undergraduates really run through walls, or is that just a ploy to get them more interested in the topic of QM?

  • http://celsetialmechanician.org Celestial mechanician

    TimG,

    I’m pretty rusty and never was interested in it much but isn’t it the case that all differential equations have numerical solutions to arbitrary precision. Then it is not even possible to write down the differential equations that describe multi-electron atoms, not only that they have no analytic solution?

  • http://celsetialmechanician.org Celestial mechanician

    The inner product in 3-d is simply the dot product we know so well from freshman calculus with its simple geometric meaning. QM is in 3-d. The Born interpretation is simply that the complex conjugate of psi is a 3-d probability density function, no different than the pdf’s used in probability. What could be simpler? We have to pick solutions to the Schrödinger equation that are normalizable, drop off as e^-x. But then atoms do not have surfaces, they extend to infinity in all directions. Sean, my question about QM that I was never afraid to ask, is how can atoms not have surfaces when all macroscopic systems do have surfaces?

    The arena: the discrete, the dynamic, + and -, space and time.

  • John Merryman

    James,

    To make a brief observation about time, the question is whether it is the basis of motion, or a consequence of it. If physical reality moves along a dimension of time, then it proceeds from past to future, but if time is being produced by motion, than one event is replaced by the next and the linear process of time is going from future potential to past circumstance.
    My opinion is that distance and volume are features of the vacuum, while time and temperature are effects of fluctuation.

  • weichi

    Celestial mechanician,

    TimG didn’t mean that some differential equations don’t have solutions, he meant that they don’t have solutions that can be written down as a nice simple formula – that’s what’s meant by the term “analytic solution”. Although I’m not sure how “analytic solution” is defined in a precise way.

  • James Robson

    Hi,

    Mainly to Lawrence B. Crowell,

    Apologies – I should not have mentioned WD as I am not an expert “a little knowlege is a dangerous thing” but it occurred to me when typing and it was too hard to resist – but still I think it is odd.

    I did not think of the QM parallel at the time i.e. E/t (despite its somewhat delicate nature – you have probably seen the John Baez discussion (too lazy to find link)) this is never taught at uni (well, not in the lectures I went to) and seems to just be left be taken for granted by people (A similar complaint might be levelled at E=mc^2 which many people seem to think is a consequence of Lorentzian-relativity without any conservation laws)

    Still, I think QM has to be re-interpreted – not with math(s – UK!) necessarily but with something that allows intuition (I’m no philosopher but I’m tempted to call that “understanding”) to explain how you can solve the “unanswered question” while taking a walk in “Central Park in the dark”.

    -James

  • Lawrence B. Crowell

    Diocletian on Jul 9th, 2008 at 8:00 pm

    Please discuss the paradox that arises when one tries to attribute expansion of the universe to the energy of quantum vacuum fluctuations. The fact that a discrepancy of 10^120 is found … physicists have expended a great effort in trying to reconcile this argument. What am I missing here? Why should I insist that vacuum energy causes the universe to accelerate?

    ————–

    The equation between the vacuum energy density and the cosmological constant (CC), or negative pressure assigned as responsible for the expansion of the universe, is a model. It does not necessarily hold as a fundamental theory in my opinion, but as some effective theory or approximation. There are a number of things which bring questions to this whole enterprise. I have sat through talks where people on slides have pictures of virtual loops and the rest. The one problem is that these things are an artifact of quantization. These objects which couple to real particles (on shell etc) are really pertubative terms that are summed over. Just as in elementary calculus a summation variable is a dummy variable, in many ways these terms are much the same. Yet there is a tendency to see these things as “real.”

    These quantum loops in space are the result of quantization. Yet as an exercise one can with the classical variables add the negative of the commutator which happens in quantization to get the a^*a + 1/2. There is nothing wrong with this since classically all you are adding is zero. If you then quantize the ZPE term is removed. This is one reason so called normal ordering works. The whole ZPE term can be removed.

    The Casimir effect might be seen as an interaction of plates with the vacuum. It could also be seen as a quantum response or nonlocal effect due to the presence of the plates in some proximity with respect to each other.

    Assigning the cosmological constant to the vacuum energy is a convenient thing to do. It is tempting to say that there exists a symmetry between positive and negative energy terms that is slightly broken to give the CC we observe that is small. The problem is that all of these ideas are increasingly to my mind fundamentally wrong. My reason for saying this is that the cosmological constant is the result of pure spacetime curvature. It is a curvature term R_{ab} = /g_{ab} and defines an Einstein space. We can of course cast about and say that / = 8piG(e + 3p) and assume the CC is determined by a “fluid” that has certain properties. Yet this muddies up the picture of general relativity.

    There is a more basic reason why these ideas may diverge from reality. A cosmology is a Petrov type-O solution that has no Killing vector system. In effect cosmologies in their lack of such isometry permits the creation of momentum and energy — aka the big bang. We have been through this some on this list, with how cosmologies fail to admit a global conservation of energy. Assume that the spacetime of the universe has a source by the Einstein field equation

    R_{ab} – (1/2)Rg_{ab} = 8pi G,

    here I have written this in a semi-classical form, then the symmetries of the spacetime (the left hand stuff) is given by the symmetries of the source by its momentum energy tensor. If we assume that the source is given by a gauge field then

    T_{ab} = F_{ac}F^c_b – (1/4)g_{ab}F^{cd}F_{cd},

    has a set of symmetries which determine the symmetry of the spacetime. This is a very difficult sort of analysis to enter into, but the punchline is that the spacetime symmetries are type-N. Due to the nature gauge symmetries if they define a large enough of a source (say a vacuum state etc) then they couple to spacetime in the form of gravity waves or related structures. You can’t build cosmologies from this source, not without adulterating general relativity in some subtle ways.

    It is for this reason and others that I have had growing doubts about these types of models. In spite of having upheld these ideas in years past I have called these models into question — in spite of their population growth on the arXiv like bunnies. In order to do these things right we have to be honest about what we are talking about, and on a fundamental level these models appear to have some inconsistencies.

    Lawrence B. Crowell

  • http://www.geocities.com/aletawcox/ Sam Cox

    LC said…and this paragraph is so conceptually central to cosmology it is worth repeating and studying carefully…

    “Assigning the cosmological constant to the vacuum energy is a convenient thing to do. It is tempting to say that there exists a symmetry between positive and negative energy terms that is slightly broken to give the CC we observe that is small. The problem is that all of these ideas are increasingly to my mind fundamentally wrong. My reason for saying this is that the cosmological constant is the result of pure spacetime curvature. It is a curvature term R_{ab} = /g_{ab} and defines an Einstein space. We can of course cast about and say that / = 8piG(e + 3p) and assume the CC is determined by a “fluid” that has certain properties. Yet this muddies up the picture of general relativity.”

    To repeat the critical point one final time…LC points out…

    “My reason for saying this is that the cosmological constant is the result of pure spacetime curvature”….

  • http://commonsensequantum.blogspot.com Arjen Dijksman

    Hello all,

    About the Schrödinger equation, I also had much difficulty to understand it. I didn’t find any textbook that could explain it intuitively. It took me tens of times re-reading Feynmans presentation on how states change with time or due to external forces before coming to the conclusion that it is a common sense law for any arrow-like object. Feynman taught us that all we do in QM is drawing little arrows on a piece of paper… that’s all. The differential of a 3-D arrow is always perpendicular to the arrow itself and the proportionality factor is the differential of the angle. If the arrow represents |psi>, then it’s common sense to have:

    d|psi>=i.d(angle).|psi>

    which is a generalized form of the Schrodinger equation. I present this at my blog and youtube channel.

  • Eric

    I think that if you’re looking to explain quantum mechanics to the masses, an explanation of just what a wave function means would be a useful precursor to talking about “the wave function of the universe”. Having recently TAd introductory quantum mechanics (i.e. end of a freshman sequence), it seems that interpreting the wave function is a huge stumbling block for a lot of people. If you have a good way of explaining it to the laiety in ten minutes, it might be the most worthwhile thing you could do in that context.

  • http://commonsensequantum.blogspot.com/ Arjen Dijksman

    Eric said “I think that if you’re looking to explain quantum mechanics to the masses, an explanation of just what a wave function means would be a useful precursor to talking about “the wave function of the universe”.”

    Yes, that’s a good precursor, I’ll going to explain the wave function in one of my next sequences. But I needed first to explain the ket (arrow) and operations on kets (arrows). The wavefunction may than be seen as the projection of the ket on the base kets. Feynman explained that very well. That projection is an ondulatory function when you vary time or position (or any other observable).

  • John Merryman

    Lawrence, (Sam,)

    “My reason for saying this is that the cosmological constant is the result of pure spacetime curvature”….

    Since the rate of expansion, with the added dark energy assumption, seems to mirror a cosmological constant, if it is pure curvature, what does this do for the Big Bang assumption that redshift is due to recessional velocity?

    The gravitational curvature causes lensed sources of light to shift apparent location, but it is an optical effect. So is redshift an optical effect?

  • James Robson

    Just one final gripe…

    I have always disliked the (Heisenberg) uncertainty principles (to some extant this refers back to earlier comments from Lawrence B. Crowell where the energy/time relation one was alluded to).

    These “principles” often seem to be taken as somehow basic in QM. However, as far as I can see (and please correct me since this is the reason for sending this in the first place) I thought that all the uncertainty relations say are:

    1. The Universe is quantum mechanical
    2. It is NOT classical mechanical
    3. If you will insist on trying to force a classical picture on the universe due to deficiencies in your cognition resulting from evolution in a non-QM (or non-GR for that matter) environment, then, well, OK, – this will be the best that you can do… Here are the (uncertainty limits) that are possible.
    4. This doesn’t strike me as fundamental to QM – perhaps a bit too human
    5. Of course, this relates to lack of any understanding of the measurement problem.

    -James

  • chemicalscum

    @Celestial mechanician

    In celestial mechanics there is no simple analytical solution to the many body problem for the differential equations of three or more massive objects. Approximate solutions have to be obtained.

    In quantum mechanics the same many body problem applies to systems of particles such as multielectron atoms and molecules. Only two body systems such as the hydrogen atom and the He+ ion have analytic solutions. Considerable effort has been expended in attempting to get ever more accurate solutions for the electonic structure of atoms and molecules over the past four decades. Of particular importance here is the application of the variational principle to the QM of such systems.

    If you want to get a good overview of the approaches taken here I would recommend Szabo and Ostlund’s book Modern Quantum Chemistry: Introduction to Advanced Electronic Structure.

  • http://www.geocities.com/aletawcox/ Sam Cox

    Hi John,

    Lets go back to Einsteins original work. He found that in his 4D model of GR, cosmologically there was a left-over tendency toward expansion, which with GR is ridiculous, because by definition the GR universe is conceptually everywhere (quasi-static) and there is nothing outside it to expand into.

    The CC was introduced to keep the concept logical…to insure the math made sense, NOT just to “make the universe static”.

    This fact, that the CC is not really a vestigial artifact, but an essential element of the concept has been kind of “glossed over” or just plain ignored by many.

    LC’s point is that gravity and the CC are related conceptually. The CC is not some material ingredient of the universe, it is a characteristic of the structure of a GR universe.

    The “force” which causes the universe to (seem to) accelerate outward is is a result of the way we observe the structure of the universe.

    The universe overall is not going anywhere, certainly not by the GR definition…we just observe it to seem to expand and accelerate outward from our frame of reference in 4D, which also by the GR defintion is “real”…to us as observers.

    We are familiar with the basics, the concept of free fall, space-time curvature, invariant frames and so forth, and understand the possibility of an extra three-space in the structure. The universe (according to Einstein, at least) just “is”.

    Best Wishes, Sam

  • layman_42

    @Sean

    May the landscape problem be related to the manywords interpretation of QM?

  • John Merryman

    Sam,

    One of the big problems with understanding physics in the first place is that there are any number of versions. When I first heard of the CC, say early ’80’s, it was described something Einstein added because according to his calculation, gravity would cause the universe to collapse to a point, so he added it to maintain a static universe. Since then, I’ve heard a fair number of other descriptions. That he did reject it upon learning of the redshift didn’t correspond to my original understanding, as the redshift would seem to be reasonable evidence of something balancing the contraction of gravity, so it would seem to be evidence of a cosmological constant! Then again that the most recent measurements of the rate of expansion attributed to dark energy have been said to match a CC would seem to imply this original understanding was accurate.

    So, yes, I do see gravity and a CC as intimately related. So far as I can understand, while gravity is the contraction of space, the CC does seems to be a corresponding expansion of space in those areas not dominated by gravity. That’s pretty much been my point all along; How, if space is expanding between gravitational wells, at approximately the same rate it is collapsing into these wells, how is it that the universe as a whole is expanding? These two effects are happening simultaneously, not sequentially, so the idea that the universe is expanding and then collapse into a Big Crunch doesn’t make any sense. To use your(I think it was yours.) analogy of the merry-go-round, it’s like saying that because the horses close to us are moving to the left, the entire merry-go-round moves to the left, but since those further away are moving to the right, eventually it might move back to the right, if it doesn’t move too far to the left(fadeout scenario).
    That’s why I describe it as a convective cycle of expanding energy and collapsing mass.

    The universe overall is not going anywhere, certainly not by the GR definition…we just observe it to seem to expand and accelerate outward from our frame of reference in 4D, which also by the GR defintion is “real”…to us as observers.

    I agree it does seem to expand outward from our frame of reference, but our only measure is of light that has managed to traverse the most distances and thus the paths least blocked by gravitational obstacles, therefore the space that is expanding the most. The effect of this expansion is eventually absorbed by those gravity wells, including the telescopes collecting the light we measure.

  • Lawrence B. Crowell

    The cosmological constant came about from the continuity equation on the momentum-energy tensor

    $latex
    nabla_a T^{ab}~=~0,
    $

    where the differential is covariant. An integration of this recovers the momentum-energy tensor plus an ingration constant

    $latex
    frac{8pi G}{c^4}T^{ab}~+~Lambda g^{ab}~=~R^{ab}~-~frac{1}{2}Rg^{ab},
    $

    where the second term on the left hand side is the integration constant. For the observable universe the density of matter and radiation is very small and so the momentum-energy tensor can approximately be set to zero. The result is an equation that says, “Ricci = constant x metric.” This suggests that the structure of the universe is entirely due to spacetime (classical) vacuum physics with no sources.

    Historically Einstein included this integration constant in order to model a static universe. This was a sort of bias that existed then, and to keep the universe from collapsing in this constant was inserted. Hubble’s results on redshift called this into question and the term was abandoned as Einstein’s greatest blunder. Then in 1997 the SNI results by Perlmutter caused the term to be reconsidered.

    Many models regard the cosmological constant / as due to the quantum vacuum, or the vev. This certainly sounds reasonable. The problem is two fold in my opinion. The GR problem is that the vacuum fields will result in a spacetime of type N, due to some grand summation over all these virtual waves etc. It is hard to know how one gets a solution with no Killing vector structure, cosmologies of type O, built up from type N solutions with a 4-fold null direction or Killing direction. The other problem frankly are the idea of all these virtual waves as a source of anything. The argument is similar to the Casimir result. Yet this result is usually taken as due to a difference in vacua — that between the plates and that outside. With a cosmology there is only one space or spacetime.

    Now, it might be argued that since cosmologies have no Killing structure that globally we can’t define a constant vev. As such the vacuum here might be different from the vacuum “there.” If so there might be a difference in vacua structure globally across the universe. These different vacua are likely to be unitarily inequivalent, similar to what happens with Hawking radiation and the Unruh effect. Remember, it is a difference in vacuum energies in separated regions that can give rise to dynamics!

    However, this leads to some rude questions. This physics is constructed by a difference between Killing fields. Wald spells this out in his little book on spacetime thermodynamics and BH QM, where the Unruh effect is due to a deviation between the coordinate time and a Killing time. But with cosmologies we have no Killing symmetries to work with! But then again maybe we do have one thread to grab onto, the anisotropy of the universe. If we were to consider some spacetime with Killing vector fields K_a we might consider some series

    $latex
    K_a~=~xi_a~+~sum_pC^p_{abcd}[xi_p^b,~xi_p^c]xi_p^d~+~O(xi^3).
    $

    For a type O solution we will have

    $latex
    xi_a~=~0.
    $

    I write the first order term with a “C,” for I suspect this would be related in some ways to the Weyl tensor. The index “p” means a summation over the other Petrov-Pirani types. This means a cosmology is being perturbed by a series term built up from other solution types, type D for black holes, N for gravity waves and I, II, III types etc. This of course could be “jazzed up” to include in the first order some harmonic sum of C’s and so forth which can be made to fit with the WMAP data. The anisotropy could then be modelled this way. This then might provide the Killing structure required to understand how the universe might have different vacua, and how these perturbation terms are related to the quantum vacuum.

    Lawrence B. Crowell

  • Lawrence B. Crowell

    erratum,

    I just realised I make a typo on the order of indices in the perturbation series with the “C.” It is easy to make the correction.

    L C.

  • http://www.geocities.com/aletawcox/ Sam Cox

    LC, Awesome and very interesting…a very nice, brief and understandable review and technical evaluation. I appreciated your look at the anisotrophy of the universe. Your approach makes possible a mathematical treatment of a subject which may cosmologically be quite a bit more complex overall, with smooth cosmic variations rather than sets of contant ones. Computer analysis and modeling might be possible?

    John, You touch on the “tightrope” aspect… that the universe “rides the line” between a tendency to either collape or expand. (Astronomically we observe the universe to expand and accelerate outward). The merry-go-round analogy in which the different parts of the mery-go-round are observed to be heading in different directions at the same time, when in fact the overall structure stays in the same place (in GR, everywhere, all the time) is appropriate.

    Sean was discussing the single Wave Function of one Electron, two Electrons- and the universe. I am reminded of possible matter/antimatter occillations at the quark level of scale, in the Planck Realm, everywhere, all the time….that all order, information, complexity and observation in the universe is everywhere and all the time (eternally) entangled.

    I think the not so subtle reality that at the farthest extent of our (theoretical)astronomical observations, 13.7BLY distant, the universe is everywhere geometrically singular (white hole…big bang) at 360 degrees is important to keep in mind. Any observer, anywhere in the universe has this common horizon of existence and experience.

    Very interesting!

  • Lawrence B. Crowell

    Sam Cox:

    Computer analysis and modeling might be possible?

    —————

    This approach, if it really works, might allow this problem to be converted into a perturbation problem with Lie algebras. This is similar to what is done with astro-mechanics, where pertubation theory is worked according to symmetries in a problem. Jacobi found that the gravitational three body problem has some remarkable symmetries which permit it to be reduced to four integrals of motion. This removes some of the mystery of the problem. Such techniques have been used in other perturbative problems. If this can extend to cosmology then the problem can be considerably simplified.

    Before I continue, the O(xi^3) should be O(xi^5).

    If this works then the perturbative Killing fields are on comoving frames, and the difference between them from region to region might then lead to inequivalent vacua from which some of these types of analyses might be performed. This might be particularly important in the early universe. Rather than the cosmological constant being of the form

    $latex
    Lambda~=~Lambda(rho),
    $

    it would be of the form

    $latex
    Lambda~=~Lambda(deltarho)
    $

    where this variation is across two overlapping regions. This variation would be computed according to differences between Killing vectors across these regions. There are clearly some scaling or conformal issues here, but … .
    Yet if the CC depends upon small variations in the vev this might at least ameliorate some of these 10^{123} orders of magnitude problems.

    There is one problem with this — it is wrong on a most fundamental basis. For various reason I think this can only be at best a better approximation or effective theory.

    Lawrence B. Crowell

  • John Merryman

    Lawrence,

    Historically Einstein included this integration constant in order to model a static universe. This was a sort of bias that existed then, and to keep the universe from collapsing in this constant was inserted. Hubble’s results on redshift called this into question and the term was abandoned as Einstein’s greatest blunder. Then in 1997 the SNI results by Perlmutter caused the term to be reconsidered.

    Wouldn’t this raise the question of whether the universe actually is static?

    Sam,

    I think the not so subtle reality that at the farthest extent of our (theoretical)astronomical observations, 13.7BLY distant, the universe is everywhere geometrically singular (white hole…big bang) at 360 degrees is important to keep in mind. Any observer, anywhere in the universe has this common horizon of existence and experience.

    I take it to mean that at 13.7 billion light years, everything appears to be receding at the speed of light? So if there were anything radiating from beyond this distance, we would only record black body radiation and not be able to pinpoint the source. For me, a horizon line effect remains a reasonable possibility, especially in light of such proposals as Inflation that are required to make the BB work.

  • Lawrence B. Crowell

    John Merryman: Wouldn’t this raise the question of whether the universe actually is static?

    —————–

    Remember, the CC came about as an integration constant. As such one can set this to what ever you want, ignoring data and … . With the so called eternal inflation the / is set so that the equation of state w = -1 holds and the CC pushes the universe out. If cosmology has demonstrated anything it is that the universe is not static.

    Lawrence B. Crowell

  • Lawrence B. Crowell

    John Merryman: I take it to mean that at 13.7 billion light years, everything appears to be receding at the speed of light? So if there were anything radiating from beyond this distance, we would only record black body radiation and not be able to pinpoint the source.

    ————–

    This ain’t necessarily so. The spatial manifold is itself stretching out, so the light cones at distant points are pointed in different directions. As a result the universe up to the CMB opaque boundary is about 80 billion light years out. This sounds funny, since this is further out than the time for the universe.

    Google Ned Wright, he has a good cosmology site which gives some really good pictures and explanations of these things.

    Lawrence B. Crowell

  • http://www.geocities.com/aletawcox/ Sam Cox

    “There is one problem with this — it is wrong on a most fundamental basis. For various reason I think this can only be at best a better approximation or effective theory…”

    Lawrence B. Crowell

    Yes, but you may be on the right track. After all, the universe IS anisotropic. We don’t have to look through a telescope or use WMAP to see that! The very existence of order, energy density information and complexity in the cosmos belies anisotrophy.

    Does the Planck Realm exist at the same level of local scale in the center of the Sun as it does on the surface of the Earth? Is any place in the universe so remote as to have no Planck Realm at the local scale? Black holes ARE the Planck Realm, and the space nearby them has a measured Planck Realm much higher in local scale than, say here on the surface of the Earth.

    The Planck realm, however, regardless of its observed location in local scale, is the same thing cosmologically and is universally quantum entangled, just as the photonic realm is quantum entangled. Whether we observe photons or the singular is but a matter of the direction we observe within the manifold, and the “size” and “mass” of the Planck Realm, but a matter of our relative observing location in the manifold…our coordinates.

    John,

    Of course you are correct. The fact that galaxies close to the big bang are observed and measured to be accelerating and receding at almost the speed of light, however relates to the fact that at those coordinates, near the singular realm (with respect to us), light experiences the effects of gravitational time dilation and red-shifts. The closer we observe things to be, relative to the big bang, the more rapidly they accelerate. In spite of the time gravitational time dilation involved, the acceleration outward can (obviously) still be measured.

    As we have discussed, all the implications of these phenomena and the geometry for the universe which they imply, are profound, and relate directly to the quantum nature of reality as espressed within a two-sphere GR manifold, with energy densities as observed from our frame, behaving in an SR manner- according to the grand proportion.

    I’m sorry for the very much and quite over-simplified perspective of a retired high school math and physics and college research paper writing teacher! I really appreciate Lawrence as a teacher. It is a privilege to have these opportunities to share ideas with him and you as well, John. I’m looking forward to the upcoming discussion of Quantum Mechanics.

    Sam Cox

  • John Merryman

    Lawrence,

    This ain’t necessarily so. The spatial manifold is itself stretching out, so the light cones at distant points are pointed in different directions. As a result the universe up to the CMB opaque boundary is about 80 billion light years out. This sounds funny, since this is further out than the time for the universe.

    This would be implicit to the concept of a horizon line, as it is a function of perspective.

    My reason for saying this is that the cosmological constant is the result of pure spacetime curvature.

    If the CC is pure spacetime curvature, then the further light travels, the more this effect is compounded as it crosses more space, so that distant sources appear to recede at a greater rate than closer sources, but if we were closer to them, they wouldn’t appear to be traveling faster, as it is only because those distant sources are at the edge of what is our horizon line that they do appear to be traveling away from us at the speed of light. This makes much more sense as an lensing effect than actual recessional velocity. Since it would be a very gradual effect that only applies outside the gravitational field of the galaxy and across enormous distances, it would seem at least worth considering.

  • http://www.geocities.com/aletawcox/ Sam Cox

    “This ain’t necessarily so. The spatial manifold is itself stretching out, so the light cones at distant points are pointed in different directions. As a result the universe up to the CMB opaque boundary is about 80 billion light years out. This sounds funny, since this is further out than the time for the universe.”

    I have a profound respect for Ned Wright, and have carefully studied his tutorial. Of course he has carefully thought out conceptual reasons for making the above statement. However 13.7 billion Earth years ago at the “Big Bang” everything in the universe was singular. To state that anything in the universe can, at any time, be farther from anything else than the age of the universe itself requires considerable explanation. If folks in a distant galaxy 10 BLY out measured the distance to us, they would get the same result we do, when we measure in their direction.

    Yes, space has expanded since the “Big Bang”…of course, to the extent that a tiny quantum entity now has an observed radius of 13.7 Billion light years.

    I like the way Lawrence takes an idea, works it carefully and then, after all that work, disgards the idea as conceptually inconsistent, or lacking in some way. He does not feel his reputation as a scientist and teacher is in any way encumbered by engaging in the scientific process. He and Ned Wright… in fact John and Sam are searching. Our search may be mathematical and emperical. Our search may be conceptual and of course our search may be philosophical.

    However, in science we expect to make mistakes. I told my students I would give them exta credit if they followed their instructor carefully and were able to point out errors. I gave a lot of extra credit! However, the students also paid closer attention to what was going on, and most of all, I taught them the spirit in which scientific investigation should proceed.

  • Lawrence B. Crowell

    When it comes to Planck scale physics I think that there is a need for a totally different manner of thinking. The Simon & Garfunkel song modified to “When I think back on grad school, it’s a wonder I can think at all,” almost comes to mind. In many ways it requires some form of “unlearning,” or acknowledging that our theories are basically symbolic forms which as best have a tentative sort of truth.

    Lawrence B. Crowell

  • http://tyrannogenius.blogspot.com Neil B.

    James Robson, the uncertainty principle (at least, the position/momentum version) isn’t just an ad hoc rule without basis in something even more fundamental. I’ve seen it explained in terms of the Fourier analysis of the wave function of a particle. If the momentum is well defined, the frequency spread is tiny and the wave “packet” created by interference of all those waves is very long. If the position is well-defined (small packet) then the Fourier analysis requires a large spread in the frequency composition. Hence the momentum is wide spread and thus uncertain (since frequency is related to kinetic energy and thus velocity, given a mass.) This is an unavoidable consequence of the fundamental maths of waves; I suppose it “couldn’t be otherwise” given the wave nature of matter/energy to begin with. Contrast this with the projection postulate, that “measurement” must convert the uncertainties in the WF into a basis state. AFAIK the PP doesn’t have to be true in the same sense as the UP, and I offered a possible way around it in #s 20 and 22.

    However there is one oddity, a sort of paradox, I thought of about the whole idea of giving a wavelength to “an” object using lambda = h/mv [m is relativistic mass here.] That seems OK at first glance, but consider an objection Galileo made to Aristotelian mechanics: A-mech said that the speed of an “object’s” fall was proportional to its mass. Galileo objected, that it is ambiguous what constitutes “an” object or several, etc. For what if we tie two rocks together with a string. Now are they “a” more massive object or still “two”? What if the connection was more substantial? It is ambiguous “how many objects” there are, but the total mass is a coherent concept. This lent weight (pardon the pun) to Galileo’s notion that rate of fall was independent of “object” properties, ultimately enshrined in GR as the Equivalence Principle.

    Hmmm… But the matter wavelength rule (related to the UP) specifies a mass of a specific object! So, what if we “tied two neutrons together” with a piece of string, etc? What de Broglie wavelength and thus uncertainties should apply? Sure, fundamental particles are not like rocks and have some special identity, etc. But the problem is real as a matter of principle, and QM is supposed to apply “to anything.” So, what about molecules tied together with little strands of CH2 polymer, etc? What wavelength applies to this complex, does it matter how well connected they are? I already suspect the dynamical coordination is crucial to the answer, which may not be a perfectly neat answer either. It probably hashes out in the interference of the different possible waves, or such, given the interactive situation. I still wonder how measurement and “collapse” is affected by collective connections.

  • John Merryman

    Sam,

    To state that anything in the universe can, at any time, be farther from anything else than the age of the universe itself requires considerable explanation.

    Which would be where inflation theory comes in and that seems to get more nebulous by the day

    Yes, space has expanded since the “Big Bang”…of course, to the extent that a tiny quantum entity now has an observed radius of 13.7 Billion light years.

    Erk! I can’t resist… Wouldn’t it have to take light 13.7 bly’s to cross that ‘tiny quantum entity,’ otherwise it would seem to be expanding volume in a stable dimension of space, as measured by lightspeed? If space expands, should the normal measure of it increase proportionally?

  • Lawrence B. Crowell

    Neil B.: Yes, space has expanded since the “Big Bang”…of course, to the extent that a tiny quantum entity now has an observed radius of 13.7 Billion light years.

    —————–

    The SNI data and WMAP tells us otherwise. The spatial manifold on the Hubble frame was rapidly expanded. As such the distance to far galaxies begins to increase then the further out they do so faster than expected by simple motion. As one looks further out these galaxies are on different local frames. The light cones are “turned” away from the direction of our local light cone. Of course if one were to transform to this distant frame the light cone coincident with our galaxy would appear turned away. So the universe is homogeneous and isotropic. This also means that while the spatial surface on the Hubble frame might be a three dimensional R^3 flat space it is also embedded in a spacetime that is not flat. This means that the spatial surface evolves into successive spatial surfaces by being stretched, or where spatial points slide away from each other. As a result the CMB exists somewhere out to 70-80 billion light years away.

    Cosmology is a strange science. The one thing which makes it fundamentally different is that there is only one system we have to work with. This gets really strange if one considers quantum cosmology. The outcome of the quantum wave function, proximally thinking of this as similar to the so called wave function collapse, is the one system we observe. Yet QM is a statistical theory — but we have no experimental statistics! There is nothing else out there to compare things to and no way to obtain a “scatter plot” of quantum outcomes. It is a standard of science that experiments and observations must consider an ensemble — observations of many quasars, many particle scattering events, many ant colonies of the same species, and so forth.

    Lawrence B. Crowell

  • http://tyrannogenius.blogspot.com Neil B.

    Lawrence, to avoid confusion note that ‘Yes, space has expanded since the “Big Bang”…’ is from Sam Cox at #125. As for the WF of the universe, what indeed is “collapse” supposed to mean for it as a whole? I have to wonder, about Sean’s initial statement that if we “consider” two electrons, we have one WF for two particles, not two WFs. But there can be “two WFs” existing because we can write up two (?) separate Schrödinger equations to evolve under the rules, it just can’t be done that way if we “consider” them together – ?! Just what is this “considering”, especially transitioned from “thought” to “reality”?

    BTW, have you tried walking through the concept about measuring photon polarization I’ve been talking about?

  • John R Ramsden

    Re #41, exactly, and I’d go further and suggest that in a sense everything of an appropriate nature can (with suitable probability) “observe” everything else that comes its way, like the ultimate police state where each citizen dutifully spies on everyone else!

    Furthermore, if one applies the same principle even to small-scale vacuum fluctuations and states then so-called dark energy could perhaps be explained as a book keeping device to cancel or vitiate extra information/energy that would otherwise be manifested in violation of the “no cloning” QM result.

    This isn’t a hobby horse BTW, little more than a passing fancy. All the same, although much more could be said, with the new rules recently stated for this blog, I daren’t elaborate further ;-)

    BTW, *completely* off topic footnote, but I can’t resist mentioning it for the benefit of all you helpful and interesting contributers, especially those whose native language is not English. There’s a wonderful free pop-up spell-checker and dictionary called WordWeb that works with any Windows app. See http://wordweb.info/. All you do is install it and type ctrl+alt+W to see the spelling and meaning(s) of the word your cursor is on. (Needless to say I’m not associated with them in any way.)

  • John R Ramsden

    lee (#24) wrote:

    As a physicist, I am mostly comfortable with QM, but I can’t shake the overwhelming feeling of guilt that I get whenever I refer to wavefunction collapse! It is the only example I can think of where a fundamental physical process is described by a non-unitary operation.

    Seems like http://arxiv.org/abs/0807.1544 might have a bearing on that.

  • http://celsetialmechanician.org Celestial mechanician

    Are s, px, py and pz orbitals described by a single wave function? What about sp, sp2 and sp3 orbitals, are they also described by one wave function? Is the Periodic Table and bonding in chemistry described by single wave functions?

  • Diocletian

    Lawrence B. Crowell

    most helpful and interesing, thank you.

  • http://tyrannogenius.blogspot.com Neil B.

    I used a “Quantum” model lawnmower to cut my lawn this afternoon! (Really!) You can imagine how it works: You turn it on, then it starts to blur and soon covers the whole yard like a gray mist. Then here and there, “at random”, blades of grass will suddenly be snipped down. With 6.0 HP, quite a few blades go down each second but it takes a while to get 90% cut. (I say, that’s enough and to heck with the stragglers, since the chance of some (but the same chance per any one blade of course) remaining blade of grass being quantum mechanically whacked go down as fewer and fewer are left intact.) You have to get out of the yard of course, since it could slice your ankle if you hang around (I’m not taking any chances!) I love it, but don’t have much clue how it works.

  • http://celsetialmechanician.org Celestial mechanician

    Hey Sean, how do you account for the Periodic Table that is made up of hydrogen like orbitals? How could one wave function describe multi-electron atoms? That is entirely contrary to the structure of chemistry.

  • Lawrence B. Crowell

    It is interesting that this discussion has turned to atomic bonds. Do the electrons in a molecule exist in a single wave function? The unequivocal answer is yes! This is certainly the case for simple molecules, and I suspect also for very complex molecules as well. By this I think that the electronic system for DNA and complex polypeptides may well form a single Fermi-electronic system. At this time the only electronic system which involves molecular biological systems that is well understood is the hydrogen bond between purines and pyramidines in DNA. Outside of that our knowledge literally falls off into a dark age.

    The problem is that this involves some fine tuned quantum physics. With solid state physics the electronic state is defined largely according to a conduction band, usually with respect to a Fermi surface, and atomic levels. With a complex molecule, say a complex saccharide, polypeptide, or DNA the electrons are arranged in what I can only describe in more “artistic” systems. The words I use here betray our complete ignorance of this sort of problem.

    To start to address this problem we might begin to examine some of the symmetires inherent in many body problems, here with Fermi statistics, where they are electromagnetically attracted by “classical” positive charges. The problem really has all the intellectual challenge of quantum cosmology. This might have much to do with the shape and functions of kinases. phophotases, transferases and other molecules involed with biological processes.

    Lawrence B. Crowell

  • TimG

    Celestial mechanician wrote:

    how do you account for the Periodic Table that is made up of hydrogen like orbitals? How could one wave function describe multi-electron atoms? That is entirely contrary to the structure of chemistry.

    The short answer is that this is only the most basic approximation to the true state of the atom, ignoring the Coulomb force between electrons (but taking into account the effects of the Pauli exclusion principle — since otherwise you’d just have all the electrons in the same orbital.) For greater accuracy, there are various ways to approximate the effect of the interaction between electrons.

  • http://tyrannogenius.blogspot.com Neil B.

    TimG, since you seem to know: I am aware that we can show how the electron etc. wave function responds to fields (I think, directly in terms of “potentials” and also affected by weird pseudofields like the vector potential A field as in Aharonov-Bohm effect.) But I wonder what sort of E field etc. comes from an electron wave/s. I intuitively expect, the E field is what you’d expect from a “cloud” of charge spread out according to the probability distribution given by the WF. That ties in with the calculation of upper orbitals being determined by the electrons “orbiting” around a screened nuclear charge, etc. Since the electron waves are spread out, it isn’t then as simple as e.g. the four outer electrons is carbon seeming to orbit a nuclear charge of only +4. But is it even as “simple” as using the entire WF/s to get an effective E field at different distances from the nucleus, did I get the idea right? I am not talking about how hard it is to solve analytically, just to get the physical picture right, tx and this should help the understanding of lots of readers.

  • James Robson

    A few recent comments asked about whether a single wavefuction could (or should, I suppose) be used to describe multi-part systems such as atoms and molecules. This relates (all the long way back in history – over 130+ blog entries!) to Sean’s motivational intro were the point was made that this will be the case according to conventional QM.

    When considering a QM system, the supposedly individual parts are not assigned individual descriptions (wavefunctions if appropriate, but more generally some vector in some linear space where they might live). Instead, this description can only be applied to the whole system as supposedly isolated from us. This is where the real problems start if you (as we all do) approach this with classical intuition. Even if we imagine parts of this system should be regarded as separate (e.g. due to distance/time preventing interactions) then this does not necessarily allow us to split the description up as we might do classically: the EPR “paradox” case is the most famous example.

    Taking this to its logical conclusion we arrive at the “wavefunction of the Universe” as we can’t really draw any dividing lines anywhere.

    Looking forward to the video blog.

    -James.

  • layman_42

    James Robson,

    Wouldn’t be possible, at least in theory, to describe single wavefuctions as [I]intricated[/I] multi-part systems?

  • paul valletta

    In GR there is E=MC^2, which is time dependant. What is the equivelant in QM?..and why must this equation, if triggered in/on, a quantum scale, have instant and far reaching consequnece for conversion process, ie from one local quantum horizon, planck scale?.. to say the edge of the observable Universe?

  • paul valletta

    P.S I meant to ask for/if there exists the above process, then could this be the trigger (instantaneous) mechanicism for Big Bang? in relativity time causes interactions to have discrete locations for events to happen at different times, little bangs!.. but in QM all events are effectivly continueous?

  • James Robson

    Hi Layman_42,

    I was kinda hoping one of the experts would have replied to your question by now and saved me from any embarrassment! My physics training was limited and a long time ago.

    The answer to your question is “yes” as far as I am aware. I was just talking about the model offered by “conventional” quantum mechanics – but if you shop around you might find something more to your liking.

    In fact, you might be able to do better than you suggest in your question, and continue with the idea that your particles have classical positions and momenta at all times – just like in the good old classical days. Of course, everything comes at a price. These “hidden variable” theories, as they are called, seem to be constrained by Bell’s theorem (based on his famous inequalities) and must abandon either locality (things can only affect their neighbours and these effects propagate outwards at finite speed – i.e. something can’t instantaneously affect something else at a distance) and/or “realism” (that a measurement only reveals what was objectively and definitely there in the first place (no casting of dice!)). The most famous of these approaches is due to David Bohm and maintains reality at the cost of locality.

    However, when QM is combined with relativity (only the special theory as yet in established theory) the notion of distinct particles does seem to be become even more unlikely. Particles can now be created and destroyed (in fact this happens all the time), and so their independence is gone. It seems better to think of them as (quantised) vibrations of fields. In fact, when the interactions between these fields are taken into account then things get even worse thanks to Haag’s theorem.

    This is the kind of stuff I personally would like to hear discussed by more knowledgeable people than myself people in the video blog.

    -James

  • James Robson

    Hi Paul Valletta,

    E=MC^2 which, of course, is the most famous equation ever devised, is, as you say, a result of special relativity, not QM, thinking (though you can find debates about the “true” nature of this assertion (SR or QM) on the web if you look). I vote for special relativity here – but still the basic SR geometry needs to be supplanted with something else to get the result (SR only places restrictions on the possible dynamical laws; it does not tell us what they are). Other symmetry and conservation law ideas must brought in to finish it off (these are usually the same thing as Emmy Noether pointed out). See Terrance Tao’s blog for a neat discussion of the derivation of E=MC^2. Also I seem to recall Rindler’s “Special Relativity” book was good for this fundamental stuff such as justifying (well, motivating) the linearity of the Lorentz transformations (this is probably out of print by now).

    As for QM, well as far as I am aware all QM theories in the standard model are based on Lagrangians or Hamiltonians just as classical models usually are. The nature of energy is tied down there. If you take a classical system where the Hamiltonian represents the energy, and put a Newtonian expression in for that enery (just as Schroedinger did on his second attempt) then the QM version of your equations will inherit this same behaviour. Alternatively, if you put in the SR version of energy/momentum instead you will get a quantum version of SR energy/momentum (such as Schroedinger did originally. And, like he, you will find you hit a lot of problems (potentially interesting ones for the video blog. Anyway, of course, it took a Brit – Dirac – to sort this out ;-)

    You mention GR – but as I’m sure you’re aware, energy – particularly the conservation thereof – is a delicate topic in this context. Without some guarantees of particularly good behaviour from the universe it seems to be pretty awkward to work with it. This is largely because GR is “background independent” and the fixed space-time that you might use to organise things and do your energy bookkeeping is not there like it used to be – e.g. it can wave around in an energetically ambiguous way, for example.

    So much for the first part of your enquiry. As for the second, it goes way above my head and I suspect that you may be breaking new ground. As such I can only refer you to the historical experts in these fields:

    http://www.treknation.com/episodes/tng/season6/descent_part_one.shtml

    -James

  • Jason Dick

    Sorry for the delay.

    Neil,

    Jason Dick, thanks for the helpful attempt. However, I still don’t think you get the deep objection, which is that even that one resulting wave “that we observe” still has no reason to suddenly shrink into a tiny space, it should still be an extended wave anyway, etc. You and others are still taking the observation regime for granted and can’t seem to “get above it”, you are IMHO like fish who can’t appreciate what their being in water does.

    No, decoherence explains this just fine.

    Here’s the deal: the essential effect of performing a measurement is to force the state in question into a specific superposition of states. For example, if we pass a particle through a pair of slits, we force the particle into what is basically a two-component wave function: one state that passed through slit 1, the other state which passed through slit 2. When the particle hits the screen, it is forced into a superposition of position states given by the various locations at which we can measure the particle.

    The key, then, is noting that the various states into which the particle is forced can no longer interfere if decoherence occurs. Thus the interaction in question goes as follows:

    Initial superposition of states -> basis transformation to different superposition of states -> decoherence so only one of the latter states is observed.

    Note that in conjunction with Anne’s previous question, this is also why observables are Hermition operators: observing is a process by which we transform the wave function into some particular sort of basis in the Hilbert space. A general invertible basis transformation is given by a Hermitian operator (invertible because we don’t want any components of the wave function to be simply lost).

  • http://tyrannogenius.blogspot.com Neil B.

    Jason, thanks, but I still think you don’t realize, how the collapse-causing imposition from outside on the waves otherwise acting forever as waves, is being taken for granted unwittingly. Classically for example, a wave passing through two slights is just two wave fronts interfering, and stays that way. I mean, the waves could cause charges to jiggle and make other waves, but there wouldn’t be a “hit” from a given photon. It isn’t just about waves and interference, the “h” is put in by hand. Otherwise we’d just have classical physics, where an EM wave would jiggle electrons all around in the path of the wave and it would never zero in on a specific atom etc. if atoms could still exist, and come together right there.

    All the talk of bases, Hilbert space, “observation” as a special event, etc., is only like that because QM imposes the collapse rule. You don’t realize IMHO you are taking the result as if it was an explanation of itself at the end of the chain. This is like pulling up by bootstraps in the bad, logically inappropriate sense (in case anyone thought there was a clever “good” way to do it.) The imposition from who knows what of collapse explains the apparent effectiveness of “decoherence”, not the other way around.

    BTW, if you or anyone has answers to my other questions, I’d appreciate some answers even if brief, tx.

  • Excal

    Funny how the blog conversation on a given topic, even one as inexhustable as this one, eventually collapses. I wonder if QM has any thing to do with it? Does the topic eventually get measured in some way?

    The last QM book I read was “Deep Down” things by Bruce Schumm, in which he says, about quantum spin:

    So the question arises, what exactly is spin and this oddly construed spin space in which it lives?

    On the one hand, it’s quite real, having associated with it the measurable physical quantity of angular momentum [but see Neil B. above]. Furthermore, the angular momentum associated with ordinary orbital angular momentum is the same physical quantity as angular momentum: the total angular momentum of any physical system is just the sum of the various orbital and spin angular momenta of the components of the system. The fact that total angular momentum is observed to be conserved means, according to Noether’s theorem, that physical laws are no less invarient with respect to orientation in spin-space than they are with respect to orientation in normal space.

    On the other hand, a particle with no spatial extent shouldn’t possess angular momentum, and the axis about which it spins shouldn’t have to be rotated through 720 degrees to return the particle to its original state.

    We don’t really have a clue about the physical origin of spin…your guess is truly as good as mine.

    I should have read that before I read Tomonaga’s “The Story of Spin.” I think, as I sit here listening to Coltrane and Garner, the most important point that you can make, Sean, is that, as Einstein put, “It would be enough to understand the electron.” (or something to that effect.)

  • layman_42

    James Robson,

    So if I understand, you say that 1) yes we should be able to describe any single closed system as intricated multi-parts system 2) when QM is combined to relativity it is not obvious that closed systems do exist.

    In other words, if I have a system with n dimensions, I can rewrite it. If this system is only well approximated using n dimension, and “really” needs more than n D, then the cut off causes problem… and I don’t think there is any way to determine for sure the number of dimensions for a system. But maybe I’m wrong?

    Anyway, thanks for your answer ;)

  • TimG

    TimG, since you seem to know: I am aware that we can show how the electron etc. wave function responds to fields (I think, directly in terms of “potentials” and also affected by weird pseudofields like the vector potential A field as in Aharonov-Bohm effect.) But I wonder what sort of E field etc. comes from an electron wave/s. I intuitively expect, the E field is what you’d expect from a “cloud” of charge spread out according to the probability distribution given by the WF. That ties in with the calculation of upper orbitals being determined by the electrons “orbiting” around a screened nuclear charge, etc. Since the electron waves are spread out, it isn’t then as simple as e.g. the four outer electrons is carbon seeming to orbit a nuclear charge of only +4. But is it even as “simple” as using the entire WF/s to get an effective E field at different distances from the nucleus, did I get the idea right? I am not talking about how hard it is to solve analytically, just to get the physical picture right, tx and this should help the understanding of lots of readers.

    I can think of a few ways to try to answer this . . . First I’d say that one has to remember that electromagnetic fields, like atoms, are subject to quantum mechanics. Really you have to quantize the electromagnetic field, and then the interaction between charged particles is governed by the exchange of photons, which are excitations of the EM field. The theory that describes this is quantum electrodynamics.

    (That said, there are still many applications for which a semiclassical picture — treating the electromagnetic field as basically classical, but the atom as quantum mechanical — will suffice. You don’t need to quantize the electromagnetic field to calculate the force on an atom due to a laser beam, for instance.)

    Electromagnetic interactions as described by photons don’t always correspond to classical descriptions of EM fields. In particular, if you have a state with a particular number of photons (a “Fock state”), it basically corresponds to an EM wave with a totally random phase, giving a large uncertainty in the field amplitude. To get something more like classical EM, you can take a certain kind of state (called a “coherent state”) that is a superposition of states with different numbers of photons, so the expected number of photons has a Poisson distribution.

    Let me bring things down to a less technical level and try to give you a simple reason why thinking of things in terms of EM fields doesn’t always make sense. Classically, the EM field at a given position due to one electron is proportional to the force another charged particle (say, another electron) would feel if placed at that position. But in quantum mechanics, you can have a situation where the electrons are correlated — that is, whether we have a second electron (our “test charge”) at a given position effects the probability of having the first electron (our “source charge”) at some position. If source and test charge aren’t independent, then how can we define an electric field due to the first charge?

    Suppose we ignore the question of “What is the electromagnetic field?”, and just try to answer the question of what is the energy due to the interaction of a bunch of electrons orbiting an atomic nucleus. If we knew the full wave function of all the electrons (which includes correlations like I described in the previous paragraph), then we could take the probability (really probability density) of having the electrons at any particular positions, calulate the contribution to the energy (which goes like 1 over the distance between them, as in classical EM), and integrate this over all possible coordinates. Unfortunately we in general don’t know the full many-electron wave function.

    So instead you can (as an approximation) ignore the correlations between the electrons, and treat the full wavefunction as being an antisymmetrized product of single electron wavefunctions. For instance, for problems involving an atom we could initially assume the electron states are just the orbitals that we get from solving the hydrogen atom (except with the nuclear charge adjusted to the actual nuclear charge of the atom we’re interested in.) Then you can construct an operator (called the “Fock operator”), which gives the energy of an electron in any one state due to Coulomb interactions with atoms in the other states, as well as including Coulomb interactions with the nucleus and kinetic energy terms. These Coulomb interactions are calculated basically as you suggest — take the norm squared of the single electron states to find the probability of having two electrons at any two positions, and then the energy is proportional to that times 1/r (with r the distance between those positions). This is integrated over all positions for the full energy.

    The Fock operator (represented as a matrix) also contains some off diagonal elements, corresponding to the fact that you can swap the electrons in any two states without really changing anything (except the overall sign of the wave function), due to the fact that the full multi-electron state must be antisymmetric. So we can diagonalize this matrix and get new single-electron eigenstates, replacing the orbitals we started with. We can then start from the beginning with our new single electron states, and repeat the whole procedure again and again until it converges.

    What I’ve described here is the “Hartree-Fock method” as applied to a single atom. As I said, it’s an approximation that ignores correlations between the electrons, but it still gives good results in many cases. For molecules, the procedure is similar, except you have to first separate out the motion of the nuclei (Born-Oppenheimer approximation), solve the elctron problem for fixed nuclear positions, and then find the nuclear positions that minimize the energy.

    This all sounds very complicated, but this is a well-established technique and there are software packages available that do a lot of the work for you. Much current research is focussed on the case where this sort of approach fails — that is, systems where electron correlations play a crucial role.

  • TimG

    I forgot to add that the above post was a response to Neil B., post 139.

  • TimG

    In response to the discussion between Neil B. and Jason Dick regarding decoherence:

    Neil, I think the key in efforts to use decoherence to answer the measurement problem is assuming there is no collapse of the wave function. That is, all branches of the wave function still exist, but the observer becomes entangled with one of the branches so from his perspective the others are unobservable.

    Let’s focus on the case of a wave (which is spread out in space) turning into a localized particle.

    The wave can be viewed as a superposition of particle states at each different possible position. That is, if S1 is the wave function of a particle at “position 1″, and S2 is the wave function of the particle at “position 2″, then the full wave function of the wave is:

    S = a*S1 + b*S2 + C*S3 + …

    where here a, b, etc. are appropriate coefficients, and * just represents multiplication.

    Now, let’s also consider the wave function of the observer. Let’s say before he observes the wave, he’s in state O, and once in observes the wave in some state Sn (for n an integer) he’s in state On.

    So, before observation, the combined state of wave and observer is:

    (a*S1 + b*S2 + C*S3 + … )*O

    and after observation the combined state of wave and observer is:
    a*S1*O1 + b*S2*O2 + C*S3*O3 + …

    Note that no matter what state the observer is in, he sees the particle as having a particular position (whether it’s S1, S2, etc.) That is, he’s only seeing the particular branch of the wavefunction (meaning the particular term in our sum) that he’s living in. If we could see the wavefunction of the whole system (including the observer), we could see that we still have a non-localized wave, but the observer can’t perform a quantum mechanical mearsurement on this system because he’s part of it. There’s no state where the observer sees everything, only a set of states where he sees the wave function collapsed to various positions.

    To make this point another way, the part of the wave function corresponding to the observer can no longer be factored out of the full wave function — so there’s no way to describe the state as an observer who sees the full superposition (that is, who sees the sum over all the terms).

    So, from this perspective there is no true wave function collapse. When we observe apparent wave function collapse, we just say “I happen to have landed in the branch of the wave function where the particle appears to have this particular definite position. And although I can’t see it, I know that the full wave function contains other copies of me who observe the particle in other particular positions. But each of us sees it as collapsed to some position, because we all measured its position and thus entangled ourselves in this way.”

    This can be seen as fixing the measurement problem, because it makes measurement no different than anything else (while still explaining the apparent collapse of the wave function that we observe). If two electrons interact, they get entangled, but with no collapse of the wave function. Likewise, if a human and an electron interact, they get entangled, but with no collapse of the wave function. Everything is on a level playing field.

    The downside of this is that the other branches still exist, so really this amounts to postulating a virtually infinite number of unobservable copies of myself (one for each possible outcome of the experiment.) Unless these copies have real existence, then we have to admit that the wavefunction collapsed after all, and the measurement problem returns. In this way, it’s kind of like the Many Worlds Interpretation.

    Whether this is a philosophically acceptable solution to the problem is debateable. Some people would say that postulating all that unobservable but still existing stuff is a worse problem than what we started with.

  • http://tyrannogenius.blogspot.com Neil B.

    Hmmm, wave function is a superposition of “particle states” at each position? But look at the expanding wave from an ultraviolet photon emission. It is, typically an expanding spherical shell. I don’t even know what “photon positions” it would or could be considered constructed out of. How wide are those photon positions, about a wavelength wide? Anyway, when this shell encounters a screen, the photon jiggles a single atom, causing say a flash of visible light from a few green photons at that spot. I don’t see how entanglement with an “observer” occurs right there, since the interaction just happens and then we see the other photons later.

    In any case, you still employ the back-door taken for granted, IMHO (innocently of course but I’d like you to scrutinize that) that the observation as a “given” has the power to do something special. For example, you say:

    Now, let’s also consider the wave function of the observer. Let’s say before he observes the wave, he’s in state O, and once in observes the wave in some state Sn (for n an integer) he’s in state On.

    But if there’s really “no collapse” to explain to begin with, even that doesn’t get off the ground and the waves just all interpenetrate and shift each other around like the waves on a ripple tank – it’s still just classical physics, you still can’t get it off the ground.

  • TimG

    In response to Neil B.‘s post, above:

    We can write the wave function as a superposition of basis states for any basis of our Hilbert space.

    For simplicity, let’s talk about the state of a single particle. Even when you shoot particles through a double slit one at a time, you still see interference fringes when you measure the overall pattern that accumulates on your detector screen. However, when you use a detector to determine which slit each particle passes through, the interference fringes go away. This is the standard example to illustrate that individual particles propagate as waves unless you collapse the wave function by measurement.

    Free particles live in an infinite-dimensional space, and the usual basis for this space is plane waves. It’s infinite-dimensional because there are infinitely many plane waves — one for every possible momentum vector.

    The plane waves have a fixed value of momentum and total uncertainty in position. But we could also choose as our basis states which have a fixed value of position and total uncertainty in momentum. These are essentially just the Fourier transform of our original states. They look like a plane wave in momentum space, but in position space they look like a Dirac delta function (a distribution which is non-zero only at a single position).

    So, to answer one of your questions, the particle position states I’m talking about really have no width at all, and the plane wave is equal to a superposition of infinitely many of them. Of course in the above example I was pretending that we had detectors which could measure the exact position of the particle — in real life we’ll at best get a small range of positions with some level of confidence, but I don’t think that distinction really matters for the point I’m trying to make.

    Anyway, when this shell encounters a screen, the photon jiggles a single atom, causing say a flash of visible light from a few green photons at that spot. I don’t see how entanglement with an “observer” occurs right there, since the interaction just happens and then we see the other photons later.

    At that point you are correct that there is no entanglement with the observer. The particle is merely entangled with the apparatus of observation. Then, when the light from the apparatus hits the observer’s eyes, he becomes entangled with it. In my above description I simplified things a bit. I could take P = particle, A = apparatus, O = observer, and then describe the sequence of entanglements as:
    (a*P1 + b*P2)*A*O –> (a*P1*A1 + b*P2*A2)*O –> a*P1*A1*O1 + b*P2*A2*O2
    (where here for brevity I’ve pretended the particle is in a superposition of only two states instead of infinitely many)

    However, this distinction about when what gets entangled with what doesn’t affect my overall point. The point is the observer is always entangled with the system being observed by the time he observes it.

    In any case, you still employ the back-door taken for granted, IMHO (innocently of course but I’d like you to scrutinize that) that the observation as a “given” has the power to do something special. For example, you say:

    “Now, let’s also consider the wave function of the observer. Let’s say before he observes the wave, he’s in state O, and once in observes the wave in some state Sn (for n an integer) he’s in state On.”

    But if there’s really “no collapse” to explain to begin with, even that doesn’t get off the ground and the waves just all interpenetrate and shift each other around like the waves on a ripple tank – it’s still just classical physics, you still can’t get it off the ground.

    I think what you’re misunderstanding is that from the observer’s perspective something special does happen when he makes the measurement — he sees the appearance of wave function collapse. But in this view of things, the wave function collapse isn’t real, it’s an artifact of the fact that the observer has become entangled in the wave function himself.

    Let me restate this point because I think it’s really the crux of what I’m saying

    – In the traditional Copenhagen interpretation of quantum mechanics, measurements changes the wave function in a different way than any other sort of interaction. In technical terms, the way the wave function normally evolves is called “unitary”, and the way it changes when there’s a measurement is “non-unitary”. This gives rise to the measurement problem: “Why should measurement be different than everything else?”

    – In this alternate interpretation, the evolution of the full wave function is always unitary. The wave function never actually collapses. However, because the observer inevitably entangles himself with the observed system in the process of making a measurement, from his perspective there appears to be wave function collapse. The full wave function is now a sum over all possible observers and all possible experimental outcomes, and it is only from that perspective that the lack of collapse is still apparent.

    I should be clear that I personally find this sort of thing philosophically unsatisfying, but I do believe it’s a logically coherent interpretation in which no wave function collapse occurs.

    No pun intended in my use of the word coherent. :)

  • TimG

    To condense even more:

    Before measurement, the wave function is a sum of many terms. After measurement, the wave function appears to the observer to only consist of one term. There are (at least) two possible explanations for this:

    (1) The other terms of the wave function went away.

    (2) We now have a superposition of observers, each observing a different one of the terms of our original wave function. The total number of terms is unchanged.

    With (1), we have the traditional measurement problem.

    With (2), we don’t have the traditional measurement problem, but we are left with a full wave function which contains a copy of ourselves and our experiment for each possible experimental outcome, all of which are unobservable to us. Whether accepting the existence of all this unobservable stuff is an acceptable price to pay to resolve the measurement problem is debatable.

  • http://tyrannogenius.blogspot.com Neil B.

    Is anyone out there not a fan of decoherence and would be willing to take a professional crack at discussing disagreements and problems? I appreciate the effort TimG took to explain it, and at this point I should digest that and read and fiddle more before putting up more generalized conceptual complaints. However I still have a sense of misgiving, and I wonder how good this is even aside from weird multiple universe issues. Also, there must be some critics of what I’d like to call “Art Deco” out there; what are they saying? I remember, was it Penrose not being real impressed, and mentioning perhaps the Renninger issues of null results having consequences of wave function redistribution, etc.
    tx all for your time, forbearance, and patience!

  • Count Iblis

    I don’t have much problems accepting point (2) in TimG’s post above. The superposition you end up with is a unitary transformation of the initial state, so you could just as well interpret as representing the intitial state.

    If you consider the entire mulitiverse, then time evolution become trivial. In the MWI, it is more natural to consider the multiverse static. The wavefunction then satisfies the equation:

    H|psi&gt = 0

    So, the time evolution that we experience is simply an illusion. The multiverse doesn’t change at all. All that happens is that in the same multiverse that I exist in, my “time evolved copies” also exist. All the possible states that I can possible be in exist and they each contain some subjective notion of time and personal history.

  • Count Iblis

    Typo:

    H|psi> = 0

  • John Merryman

    This may not be very well expressed, but if E=mc2 and I strike a match, that turns matter into energy. At what point does energy start to become matter again? Is it when we measure/observe it and the wave becomes a particle? Fundamentally it travels as a wave, yet any effort to measure it requires an interference that results in a particle. Is this what plants do when they photosynthesize light, absorbing it and turning it into mass?

    The idea of C2 doesn’t seem to make much sense as compounding the speed of light, so is it a function of volume, that amount of energy expressed within the x times the y coordinates is squeezed into the volume of the mass? When energy is released, it is as a wave in all directions.

  • John Merryman

    That way, the wave collapses, but the energy isn’t lost and will eventually be released as a wave again. Often by the process of measuring/observing how much energy is in the mass, such as striking the match.

  • TimG

    I feel I should add one more comment on decoherence. In particular I want to clarify that decoherence has a meaning beyond attempts to resolve the measurement problem.

    Basically, decoherence is what happens when a system in a superposition interacts with some other system. (It could be any two systems, not necessarily an atom and an observer like in my above example.) You get a situation like I described above where you can’t “factor out” the original system, so the only way to see the superposition is to perform measurements over the composite system. If we’re talking about an atom interacting with a macroscopic system (such as the surrounding environment), then we can’t possibly measure the quantum state of the combined system, and the superposition is effectively lost.

    Decoherence is an observable effect, and it happens regardless of our interpretation of quantum mechanics. Someone trying to build a quantum computer, for instance, has to worry about decoherence effects. (Quantum computers take advantage of the fact that the bits can be in superpositions of 0 and 1, so destroying these superpositions is a problem for them).

    As for the measurement problem, as I said above it can be stated as:
    “The evolution of the wavefunction is always unitary — except when someone makes a measurement, in which case it’s non-unitary. What makes measurement different than everything else?”

    But really, the evolution is only unitary for closed systems. That is, we expect the quantum state of a system that isn’t interacting with anything else to evolve unitarily. But when we have a situation like above, where the system we’re studying is really a subsystem within some larger closed system, then the evolution of the subsystem need not be unitary.

    Like I said, I don’t think this really resolves the measurement problem, since even though for all practical purposes the superposition is destroyed, at least in principle it still exists within the state of the full system (i.e., the system being studied plus the macroscopic environment it’s entangled with). One way around this is to note that the observer himself inevitably becomes intangled in the wave function, leading to a sort of “many worlds” picture like I discussed above.

    An alternative is to deny the reality of the wave function — to say that it is merely a mathematical device for predicting the results of experiments. If you initially have a quantum system in a coherent superposition of state A or B, then after decoherence if you look at the state of the system alone (ignoring the quantum state of its environment, which is presumably unmeasurable) you can find a certain probability of A and a certain probability of B, but not the superposition. As far as I can see this doesn’t really explain why the system persists in state A after you’ve measured it once — but with this mindset we essentially say “Quantum mechanics allows us to calculate probabilities of different measurement outcomes based on our current knowledge of the system. Once we know the result of the measurement, we no longer need quantum mechanics to know what we’d measure.” (Well, until some time passes, at any rate.)

    I should reiterate that I’m sort of playing devil’s advocate here — I don’t personally find either of these explanations particularly satisfying. If wave functions don’t exist, then the question of what actually exists that explains the predictions of quantum mechanics reamins open. I can’t stomach the extreme positivism in saying “The goal of science is just to predict measurement outcomes.” Someone once made the point that if we discovered a magic oracle that could correctly predict the result of any experiment, no one would consider that the end of science. We’d want to know how the oracle worked, and the reasons why those answers were the right ones. In my opinion the goal of science is to create a conceptual framework that accurately describes the real world and is predictive.

    As I’ve said, I also don’ t care for the sort of “many worlds” interpretation you get from assuming that the wave function has real existence and never collapses. Postulating an infinite number of alternate versions of ourselves which are unobservable even in principle seems to badly violate Occam’s razor, and is in my opinion too high a price to pay to resolve the measurement problem.

    Personally, I don’t think a satisfactory resolution of the measurement problem yet exists — unless it turns out that some as-yet undiscovered physics beyond quantum mechanics actually causes wave function collapse. There are some theories to that effect, but so far none has been successfully tested (although one can always hope).

  • paul valletta

    TMG “The evolution of the wavefunction is always unitary — except when someone makes a measurement, in which case it’s non-unitary. What makes measurement different than everything else?”.

    A “quantum” of anything will always, repeat ALWAYS!, know where you are/exist, long before you can locate a quantum? by default of scale, a quantum has always measured “you” prior to your ability to locate a “quantum”. Think about a quantum needle that exists in a macro haystack, the needle will detect you macro movement, whilst you will not “register” the needle anywhere!

    By default, the evolution of any observation is stacked one-way, from the quantum outwards, in gravitational terms it’s like walking on the surface of the Earth, I am the “quantum” and I know with certainty that the macro Earth is below, but WRT all else that is going on upon the Earth’s surface, does it know with certainty where I am shuffling my feet ?

  • collin237

    Paul, I’m not sure whether you’re being sarcastic.

    If you’re not, what kind of “knowledge” are you referring to? How does a field with only a few components code for the knowledge of everything?

    The way I see it, it goes the opposite way. The macro system “discovers” that it’s linked to the observed quantum when the quantum forces it to separate into distinct possibilities, and the macro “chooses” which to become.

    (I’m not referring to a consciousness or “cat” argument. I’m referring to “dumb luck”. That is, Markov Processes or Martingales or whatever.)

    The decision propagates from the macro to the quantum, starting with how much blurriness the macro can tolerate and “deducing” that of smaller and smaller portions.

    If that doesn’t make sense, what am I missing?

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .

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