Look here:

http://www.ncnr.nist.gov/summerschool/ss09/pdf/Abele_FP09.pdf

and here:

http://www.ift.uam.es/oldIFT/paginaspersonales/bellido/cuantica/articulos/PhysRevLett.34.1472.pdf

for some examples.

Thanks for the references Stam!

The biggest problem is that since we do not actually have a complete quantum theory of gravitation, we are unable to reliably answer any of your questions to your satisfaction. There are proponents of each of the current schemes for quantum gravity: String/M Theory, Loop Quantum Gravity, Topological Quantum Field Theory, Twistor Theory. But none of them have emerged as The Quantum Gravity Theory. One fundamental problem is the question of whether gravitation is truly unifiable with the other fundamental forces. It behaves in no way like the others, so why must it be unified with the others? One hopes so, because then the Standard Model would be complete with gravitons. But we have never seen a graviton. So, other than esthetics, why should it be unified?

I'm no expert but why would the usual swapping out partial for covariant derivatives for most normal gravitational fields be found lacking when it comes to computing a particle in a free falling box? In this simple view the geodesic deviation equation has or needs no quantum analog. Of course, if you asking about quantum gravity and how all that would work, I have no clue.

Dear Paul, thanks for the answer. Actually I was wondering about solutions of the Schroedinger equation with a potential energy generated by Newtonian tidal forces (static ones if you wish). So, gravity is not geometrized yet and this is precisely the point: the inertial and the gravitational masses do not cancel out from the equation and (as a conequence) it seems that the equivalence principle would fail. What do you think?

Perhaps I'm confused about the equivalence principle since as far as I know it works just fine for Newtonian gravitation. Have you've missed something in the calculation? If I was setting up the calculation you describe mass would be mass. It would help to see the calculation.

Another thought, if tidal forces are of sufficient size to cause detectable changes in energy levels in the box wouldn't they also be detectable by other less exotic means?

Hi,

What is being discussed here is very relevant to experiments that are being performed on freely-falling Bose-Einstein condensates. For example, see:

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.76.063617

One approach to finding the energy levels of a particle in a freely-falling, non-rotating, box would be to start with the Klein-Gordon equation on a general background spacetime, and write it in Fermi normal coordinates adapted to a worldline inside the box. A single-particle Schrodinger equation can be obtained from the Klein-Gordon equation using a phase transformation and then taking the non-relativistic limit. The leading order term in the metric due to the curvature is quadratic in Fermi normal coordinates, so I think that you'll obtain the Hamiltonian for three harmonic oscillators if you make suitable approximations and if the components of the Riemann tensor are independent of time in the Fermi normal coordinate system. You'll need to make an assumption about the positions of the walls of the box. The natural choice would be to assume that the walls of the box are Fermi normal coordinate surfaces.

Best regards,

David

Thanks for your answer David. Amazing experiment, indeed! Yes, it seems that assumptions about the positions of the walls are crucial. The physical dimensions of Einstein's elevators are important for the classical version of the equivalence principle. But, the quantum system automatically "feels" the presence of boundaries. If we make them pretty small in order to avoid tidal effects the spectrum will automatically identify the transformation, do you agree? Regards

The equivalence principle (EP) is indeed no trivial matter in quantum mechanics, even for the simple case of a nonrelativistic particle falling

in a uniform gravitational field (like the Earth's).

You can find some very relevant information here:

http://link.springer.com/article/10.1007/s00340-010-4085-8

The bottom line is that the EP works for quantum mechanics as far as

average values are concerned (ie., the solutions only depend on the ratio of the

gravitational and the inertial masses), but not for the dispersion of the wavefunction or the energy eigenstates!

All the above is for a uniform gravitational field. if you are doing work on higher order corrections, I'm interested.

Another useful reference for theory is this one:

http://link.springer.com/chapter/10.1007/978-3-0348-0043-3_16

As to experimental work, there are many very nice experiments on ultracold

neutrons falling freely in the Earth's gravity field, for instance:

http://www.nature.com/nature/journal/v415/n6869/abs/415297a.html

And more recently, people are trying to measure the free fall of antimatter atoms:

http://iopscience.iop.org/0264-9381/29/18/184008/

Good luck

Giovanni

Many thanks for the enlightening papers Giovanni. And yes, I'm interested in this higher order corrections, but I didn't find many references. It is really interesting that the EP works for quantum mechanics only as far as average values are concerned, but not for the dispersion of the wavefunction or the energy eigenstates! Perhaps the Bohm-de-Broglie version of QM has something interesting to say about this dependence on the ratio of the gravitational and the inertial masses.

Regards

By the way. Is there any attempt to perform the double slit experiment in free fall?

Yes, with BE condensates: http://arxiv.org/pdf/1301.5883.pdf

See also the AEGIS collaboration page: http://aegis.web.cern.ch/aegis/

They plan to do similar things with anti-hydrogen atoms.

Thanks again Giovanni.

In nonrelativistic quantum mechanics isn't this equivalent to simply solving the Schrodinger equation with a gravitational potential added to the box potential? This is a good place to start, even though it only approximates reality in the limit of weak gravitational fields.

Robert, from what I can tell from the linked papers in this thread, yes. If this is a wrong assessment I would benefit from being corrected.

Dear Robert, classically, if the box is in free fall there will be no gravitational forces inside (locally) and consequently no potential. This is the idea behind the equivalence principle. Thus, it seems that solving the Schrodinger equation with a gravitational potential added to the box potential doe not describe a system in free fall. Rather it describes a box at rest in the gravitational field of the Earth (for instance).

Nevertheless there will be some (generally small) tidal effects in the freely-falling box. How does these effects alter the spectrum in the limit of a small box?

Yes, this is right. Obvious when you think of an elevator alternately falling and stationary. Surely the smaller the box the smaller the tidal effect, because delta g across the box will tend to zero?

Thanks Robert, "surely the smaller the box the smaller the tidal effect". The point is that it is not clear to me what a "small" box means in the above context. In classical mechanics you can arbitrarily change the size of the "elevator" without disturbing the actual trajectory under investigation. In QM the size of the box influences directly what is being measured. Formally speaking the equivalence principle is only valid along the world-line of the particle. But of course this does not make sense in the context of QM because the spectrum blows up. That's why I was wondering about the tidal force effects even if they're small.

For the vertical modes in a freely falling box under the influence of an approximately linear tidal force the eigenfunctions are simply Ariy functions. For a finite height box and small tidal forces the spectrum appears well behaved. Why do you feel they blow up?

Correction, the tidal force is linear in, x, the distance from the box center. This makes the tidal potential term quadratic in x. This is similar to a harmonic potential but opposite in sign. The eigenfunctions for an inverted harmonic potential are well known. When the boundary conditions at the box edge are applied the spectrum is well behaved.

Thanks Paul. I'll check the calculations with more details as well. Anyway, it is curious that the spectrum depends both on the inertial and the gravitational masses, isn't it?

Mathematica actually makes short work of this once I purged all my errors. Scaling the energy and coordinate I get,

y''+y^2 + E y = 0

as the eigenvalue equation. The answer pops out as a sum of parabolic Cylinder D functions. To get numerical values for the eigenvalues one has to solve the usual transcendental equation to put nodes at the box boundary. I haven't done this but I have plotted the solutions and they wiggle in the ways one expects with nodes that move with energy. So the spectrum seems like it should be on the up and up to me.

I'm confused by your inertial versus gravitational mass question. I automatically set these equal from the start so the result only depends on the common mass. If I were to assume gravitational and inertial mass were different then there would be a weak dependence on the mass ratio I believe. So, I don't get what's so special about QM in this regard? Isn't assuming a different gravitational and inertial mass a violation of the EP from the get go?

The energy eigenvalues of a particle in a box are inversely proportional to L^2, where L is the box dimension, so the spectrum does indeed "blow up" in the limit of small L. In "real physics" this is reflected in the fact that the smaller the system you are investigating the more energetic the probe required (nucleus vs atom etc). Gravity is a very weak force but because there are no negative gravitational "charges", it accumulates, so to speak, and you feel it in large systems, but in small systems it's effects are dwarfed by the other forces.

Apologize for spelling of its, blame iPad automatic text corrector

If there is a gravitational field, then the space isnt isotropic, and then almost in principle we can conjecture a few experiments capable of differentiating this anisotropy ... but stantis rebus all the equations that we can use to describe particle dynamics are agnostic to this feature; unless of working on twins systems placed in positions symmetrical with respect to a few observable that shows the characteristic of symmetry: in this case I think to a kind of interferometer for those waves oriented in the vertical and horizontal direction (relatively to grav field, in the box of dimension L that must act as a resonant cavity: we now are focused on em waves). Photons with the same horizontal and vertical | P | show a different path, and using the Eiconal equation (that leads to the P = -Nabla U, with U potential) we can argue something...

Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?

http://arxiv.org/pdf/1404.3765v1.pdf

(a quite recent related discussion)

I'm not certain I agree with the implications equation 4 at all. From my own experience simply replacing coordinates followed by the deduction of physical implications is a very error prone procedure. For electrodynamics where the approach is even justified in the strong gravitational limit almost all the conclusions of this naive approach are simply wrong. Virtually all imagined EM effects of gravitational waves are eaten up by subsequent coordinate motion of the problem boundaries which are easy to overlook and very hard to incorporate in general.

The problem I have with Labed's paper is with the use of the unprimed coordinate as an "inertial" system which it is not for the situation described. Non relativistic quantum mechanics is not generally covariant, however, the primed coordinates defined in Equation (3) are proper lengths and times. So Eq 4 is as close to the correct treatment as one can get in the (weak) gravitational field of a point mass. In the lab when making measurements one uses clocks and meter sticks which measure proper lengths and times corresponding to the primed coordinates; not the unprimed ones. The unprimed coordinates don't correspond directly to times and distances in the rest frame of the atom. Once tidal forces are neglected, the results computed in the primed coordinates are the physical ones. The "proof" of this is simple. A frequency of transition of hydrogen is measured on the earth's surface by a collocated spectrometer. The measured frequency is compared to an identical measurement performed in an inertial frame out in space again with a local spectrometer. These measurements will yield the exact same result provided tidal forces are ignored. I believe the results of the paper are in fact specious.

I think question is not posed properly. Also one should be clear about limitations of quantum mechanics as well as gravitation. Quantum mechanics is valid over microscopic length scales where as gravity makes its presence in macroscopic length scales. So, correct thing is, quantum mechanical box has microscopic dimension and gravitational field is constant in this region. So, there can not be any tidal effects. To consider tidal effect, we would need space dependent gravitational field or potential that is quadratic in distances. The potential problem can be solved and one would get energy spectrum which would be different from square well potential problem but that's not very important for this discussion. How should one find tidal effect? For tides you need time dependent potential. So you now have time dependent Schrodinger equation. That is not a problem. You need to have a solution. It can be found if potential varying slowly in time. One can use adiabatic approximation and find time dependent solution. But that will describe time dependence of each state. But the physics says that the length scales are macroscopic. So you need to take the limit of Plank's constant vanishing. Actually, the levels are so dense that one will be averaging over many any quantum states and then one gets classical solutions. So quantum physics is irrelevant.

Speaking of quantum mechanics here will make sense if gravitation field varies over microscopic length scales. We are then getting into domain of general relativity and Schrodinger-like equation in general relativity. Such equation is not known.