Follow the Bouncing Neutron

By Sean Carroll | June 2, 2007 11:35 am

Stefan at Backreaction has a great post up about measuring the quantum state of a bouncing neutron. If you drop a basketball, it falls freely along a geodesic in the curved spacetime around the Earth, until it comes in contact with the floor; at that point it bounces back up and falls freely again. The cycle repeats, although basketballs come with dissipation (otherwise you wouldn’t hear them bounce), so the bounces gradually lose altitude, unless you impart some force to the ball by dribbling.

Well, the same goes for neutrons, except that there isn’t any appreciable dissipation, so the neutrons just keep bouncing. And neutrons are subatomic particles, so we can imagine observing not just their classical position, but their quantum wavefunction! And that’s what people like Valery Nesvizhevsky have been able to do, using interferometry. I won’t explain the details, since Stefan has already done it better than I could, and you should read it there.

Quantized Neutrons

So there’s both “quantum” and “gravity” involved here, although not “quantum gravity.” The neutron is quantized, but the effects are just those of a classical background gravitational field. (Quantum gravity would become involved if you measured the gravitational field caused by the neutron, and that’s a bit harder.) But still, you’re observing the effects of spacetime curvature on the wavefunction of a subatomic particle, which is pretty neat. And it’s plausible that someday measurements could improve enough that you’re measuring Newton’s inverse-square law for gravity at very small scales, which is relevant for constraining all sorts of theoretical models. And I’m guessing that you could even test the Equivalence Principle, if you could do the same exact experiment with some other kind of neutral particle (a hydrogen atom, maybe?). But really, it’s just cool, and that’s its own reward.

Also from Backreaction I learned that the current mood of the internet is:

The current mood of the Internet at

It’s good to be updated on these things.

  • Quasar9

    “So there’s both quantum and gravity involved here, although not quantum gravity (Quantum gravity would become involved if you measured the gravitational field caused by the neutron, and that’s a bit harder). But still you are observing the spacetime curvature on the wavefunction of a subatomic particle.

    The Earth’s Quantum Gravity or Quantum Gravity on Earth?

    Great post Sean, thanks for the link to the interesting post on “the bouncing neutron” by Stefan @ Backreaction.

  • PK

    It is too bad that neutrons are the only particles that are massive, electrically neutral, and stable enough to do this. Otherwise they could redo Galileo’s experiment!

  • Neil B.

    The quantum aspects of gravity are interesting, but so is the relationship to electromagnetism. I thought of this paradox many years ago; not claiming it was original, but I never saw it posed just so: Let a charged body fall back and forth (SHM) in a tunnel through the earth. It should radiate because there is a changing local field that must propagate (even if the EM radiation is distorted some by the gravity field.) However, the charge is in free fall (albeit surrounded by a bit of tidal field…) and hence should not feel the wrongfully obscure radiative self force f_rad = 2kq^2 v dot dot/3c^3. If so, there’s a problem with conservation of energy because the radiation “isn’t paid for” by the self-force opposing the velocity (look at the derivatives for SHM and see how it works out.)

    Later, I found an article saying that the self-force on a freely-falling charge is just the same as it would be if someone were accelerating it that way in the absence of gravity. OK, no more conservation problem it seems. But if so, how does the charge “know” it is being accelerated that way? (Furthermore, the v dot dot depends on (dg/dr)(dr/dt), so the charge has to detect its velocity, relative presumably to what is causing the gravity – that doesn’t sound right. What if the planet’s material moved around like the protoplasm in a cell…)

    Whatever the answer to that particular conundrum, getting a handle on gravity-EM is a first step before gravity can be integrated with QM, correct?

  • Shantanu

    Sean, I haven’t read stefan’s post in detail. But isn’t this experiment similar to the COW experiment (which is discussed in Sakurai’s book
    on quantum mechanics)?

  • Joseph Smidt

    That’s really interesting. I would have never thought you could measure something like this since, if I remember correctly, the Washington group was able to measure classical gravity down to ~10^{-6} m and yet a neutron is so much smaller than that.

    I hope the day comes when we can measure the inverse square law down to the 10^{-15} level. I hope we see some new physics in the process. :)

  • Sean

    I don’t know anything about the COW experiment, I’m afraid.

  • B

    Hi Sean:
    Thanks for the mentioning :-) I was reminded of the paper when searching for topics that would fit into our discussion group about experimental tests of quantum gravity, but decided against it – for exactly the reason that there’s not actually a quantization of gravity. So it ended up on the blog instead.

    (Also, I am happy to report that my planned workshop Experimental Search for Quantum Gravity was approved and will take place in November.)

  • Stefan

    Hi Sean,

    thank you that you like the post – I’m fascinated by this experiment, because it contains so great, and elementary physics, and wanted to share that!

    I’m not an expert in this stuff – I just know it from reading the papers. But let me add the following:

    This Nesvizhevsky et al. “bouncing neutron” experiment is different from the interferometric experiments with neutron beams of Colella, Overhauser and Werner mentioned in the Sakurai text.

    In the COW experiments (see Rev. Mod. Phys. 51 (1979) 43 for a review) the phase shift between two neutron paths propagating at different gravitational potentials is measured. As far as I know, this has been the first demonstration that the gravitational potential has an influence on the wave function of massive particles.

    In contrast, the “bouncing neutron” experiment shows for the first time that there are bound states of neutrons in the gravitational field of the Earth. It shows that it makes perfect sense to plug in the elementary expression mgh for the potential energy of a particle in the gravitational field into the Schrödinger equation, and that the resulting wave function, the Airy functionn, indeed describes the behaviour of neutrons.

    As for checks of modifications of Newtonian gravity, or for Yukawa-type corrections to the Newtonian force, it seems from what I gotr from the papers that unfortunately, the “bouncing neutron” technique has already reached its limits – mostly because of unavoidable imperfections of the neutron reflectors and absorbers involved in the experiment.

    For checks of the equivalence principle, there is another class of experiments using ultracold beams of atoms (PRL 93.240404). From my understanding this seems to be some COW type of interferometric experiment, but with atoms instead of neutrons. Maybe such experiments can even be made with molecular beams. The atomic beams are used for tests of the equivalence principle, and for measurements of the Newtonian constant G (see e.g. the MAGIA experiment).

    Best regards, Stefan

  • Stefan

    Hi Joseph,

    as far as I know, the “bouncing neutron” experiment could not be used, unfortunately, to derive stronger limits on modifications of the Newtonian force by extra dimensions or Yukawa-type corrections or whatever than the EOTWASH experiment – but I am not really an expert in this.

    One should keep in mind that what is measured is essentially, in an indirect way, the wave function of the neutron in the gravitational potential gh of the Earth near it’s surface. The ground state wave function has an extension of about 30 micrometer. Any short-range modification of the Newtonian force would be relevant only for the gravitational interaction of the neutron with the reflector, or the absorber, and this would imply only a small modification of the linear potential, and thus of the wave function. That’s why it is very difficult to extract stronger limits for the Newtonian force law from this experiment.

    Best, Stefan

  • Count Iblis

    #3 Neil B,

    The boundary conditions at infinity you have to impose on the EM-fields are important.

    If the charge is in an inertial frame and you look at it from an accelerated frame, then the asymptotic behavior of EM-fields at large distances is different from that of a uniformly accelerated charge.

  • Neil B.

    Count Iblis:

    Thanks for getting me started about charges in gravitational fields. I still wonder: does the charge oscillating in the tunnel show the effect of the self-force as given above, or not? That would actually change its rate of acceleration (slowing it down), and is independent of how you are considering distant fields.

  • nigel

    Neil, isn’t that additional inertia for an accelerating charge really just the normal relativistic inertial mass increase, m’ = m[1 – (v/c)^2]^{1/2} ?

    Unless there is a cyclical acceleration, you don’t get actual electromagnetic waves being produced. Although your electron is oscillating through the earth, the frequency will be very low, because the maximum acceleration the electron experiences is that at the Earth’s surface, a = 9.8 ms^{-2}, so the maximum radiating power is merely

    P = (e^2)(a^2)/(6*Pi*Permittivity*c^3) = 5*10^{-52} W.

  • nigel

    I mean m’ = m[1 – (v/c)^2]^{-1/2} if m’ is mass at velocity v and m is rest mass.

  • Neil B.


    No, I am not talking about the relativistic mass increase, but rather the special Abraham-Lorentz force that specifically compensates for radiated power. The body falling through the earth is indeed making the classic sine wave SHM. BTW, the fact that the power would be so weak from an elementary charge is beside the point of the principle of the thing; also note that we could easily use a highly charged macroscopic body, and that the radiated power is proportional to q^2. (Also, what if we used a neutron type star instead! Billions of g’s…. Well, coring it is problematical….)

    (BTW, could we please get sub and superscripts for this posting, it would be so helpful?)

  • Jonathan Vos Post

    Do it also with an antineutron. Compare. Any difference is “new physics.”

  • Ellipsis

    Jonathan V.P.: an antineutron would immediately annihilate with the “floor”. Very different — but not new physics. It isn’t clear how to do this with antineutrons (even capturing antineutrons is not generally possible, because they like to annihilate with basically everything).

  • nigel

    Neil B: so you’re suggesting there is a net reaction (recoil) force to the radiation emitted by an electron accelerated in a straight line?

    Surely, there isn’t any such force in this case because the radiation is emitted perpendicular to the direction of acceleration of charge. The charge accelerates along a radial line to the earth’s core. Hence the radiation in the transverse direction to that line. Think about the radiation from electrons accelerating along the surface of a radio transmitter antenna. The radio waves come off with in all directions perpendicular to the direction of acceleration of the electrons. Ie, the radio beam goes out in a horizontal direction is the aerial is vertically orientated. There’s no recoil force because there is no preferred transverse direction for the radiated waves: they are emitted in all directions on the horizontal plane so the recoil forces cancel.

  • Neil B.


    You have confused the recoil reaction to the momentum of the emitted radiation (which does depend on direction of radiation) with the acceleration-resistance force which normally does indeed act parallel to acceleration and velocity of the charges. That force acts for example in ordinary omni-directional antennas as an impedance, an antenna resistance. Otherwise, you could violate conservation of energy, gravity or not, by just running charges back and force without effort (like the energy recovery of classical masses). and collecting the radiation energy. Look up radiation resistance, Lorentz-Abraham force etc. in Wikipedia or etc. and see the difference.

    However, your point is interesting in quantum mechanics: Well, atoms supposedly emit photons in a radially-symmetric wave (maybe not as strong along a given bidirectional axis as another, but with no net directional vector.) How then can absorption later of a photon somewhere (with its momentum pushing the absorber in a given direction) work back to give conservation of linear momentum to that atom and its environment? Does it depend on which one is measured first? etc.

  • Aaron Sheldon


    Try looking up radiative cooling in gravitational collapse, particularly references to the evolution of white dwarfs and neutron stars, although the process is important for nearly all forms of stellar and astronomical evolution.

    The classical phenomology to relate Abraham-Lorentz radition to Black Body radition is to consider the collision of two atoms in a hot gas of differing velocities as an acceleration of dipoles. Unfortunately this classical approach leads to a UV divergence. Hence Plancks need to invoke quantization.

    Even with the tunnel through the earth example, we would still get integer steps in the radiation of the electron, because it is effectively bound by an inverse square field, to which we know the solutions very well.

    Now the really tricky question is to ask what the gravitational force is between to photons (and if that is the same question as asking if space is hyperbolic, flat, or parabolic)

  • nigel

    Niel B: OK, I’ve got it. Some kinetic energy of the electron is converted into the radiation released due to the acceleration. The loss of the kinetic energy of the charge constitutes the radiation resistance.

    This is nothing to do with the Abraham-Lorentz force you referred me to in comment 14:

    “Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. The Abraham-Lorentz force is the average force on an accelerating charge due to the emission of radiation.”

    From what you now say in comment 18, you’re not concerned with such a recoil at all (there isn’t any such recoil, as shown in comment 17). Instead, you are now making quite a different point: the electron experiences a decelerative force due to losing forward kinetic energy while accelerating, due to the transverse emission of radiation.

    Your problem with conservation of momentum is that the electron is losing momentum (by being decelerated by kinetic energy loss due to emission of radiation), but that radiation isn’t going in the right direction to conserve momentum. But the electron is gaining momentum from the gravitational field which is causing it to accelerate in the first place. Whenever the electron accelerates, it flattens due to Lorentz contraction.

    While it is at constant velocity, the electric isofield strength lines around the electron form an oblate spheroid shape, with propagation along the axis of symmetry. But in order to get into that shape for a constant velocity, the electron must first, during its acceleration, be distorted so that the field at the front moves slower than the field at the rear: this allows the field at the rear of the electron to catch up, squashing or flattening the electron in shape.

    Hence, the electric isofield strength lines are not a perfect oblate spheroid while the electron is accelerating, and this distortion is physically necessary to explain the Lorentz contraction effect. This front-rear distortion during accelerations means that the emitted radiation is not emitted perpendicularly to the acceleration, but at a slight angle (hopefully satisfying conservation of momentum for the electron’s momentum loss which results from the deceleration of the electron due to loss of forward kinetic energy via radiation).

    … atoms supposedly emit photons in a radially-symmetric wave (maybe not as strong along a given bidirectional axis as another, but with no net directional vector.) How then can absorption later of a photon somewhere (with its momentum pushing the absorber in a given direction) work back to give conservation of linear momentum to that atom and its environment?

    The nature of a transverse electromagnetic wave like light is that it can’t propagate outward in spherical symmetry. You need a charge acceleration in a specific direction in order to emit radiation, so a spherical radio antenna emits nothing: you need some asymmetry in the direction of charge acceleration in order to transmit radiation. An electron doesn’t radiate a radially-symmetrical photon: such a thing can’t be detected, so it’s not a physical concept, really.


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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] .


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