Feynman and Bohmian mechanics (at the macroscopic level)

[Demystifier]

I have just found out that Feynman also (re)discovered (some essential aspects of) Bohmian mechanics a long time ago, in his "Feynman Lectures on Physics" part III. Namely, in the last chapter devoted to superconductivity as macroscopic manifestation of quantum mechanics, he derives equations that are exactly equal to the Bohmian equations of motion (with classical electromagnetic field). Yet, he does not mention Bohm in this context, so I presume that he was not aware that these equations have already been discovered earlier by Bohm (and much more earlier by de Broglie). Indeed, Feynman's equations (21.19) and (21.31) are the Bohm equations for velocities. Likewise, Feynman's equation (21.38) is the Bohm equation for the acceleration, including the term with the quantum potential. In fact, Feynman even calls it "mystical quantum mechanical potential".

Another interesting feature of the Feynman discussion is the fact that he interprets these velocities as MACROSCOPIC velocities of an electron current in a superconductor. Since such a current is macroscopic, it suggests that it might be MEASURABLE. In fact, Feynman points out that the quantum correction to the classical acceleration is not very big EXCEPT AT THE JUNCTION BETWEEN TWO SUPERCONDUCTORS. I don't know if a measurement of that kind has already been performed (the Feynman's book is quite old), but it would be truly remarkable to look for an experimental evidence of a macroscopic effect directly related to Bohmian mechanics.

Does anybody know more about measurements of electric currents in superconductors? If yes, are they compatible with Eqs. (21.19) and (21.31), or with (21.38)?

 

[Demystifier]

Nobody?

Perhaps I should have asked it on the "Solid state" physics forum?

 

[Demystifier]

I continue my research in superconductivity literature and find surprising claims.

Take, for example, C. Kittel, which is a STANDARD textbook for solid state physics.
Chapter 12 is "Superconductivity". Eq. (20) is essentially the same equation as an equation written by Feynman. But this equation gives particle flux calculated from the wave function, which gives LOCAL VELOCITY of electrons at given position. This is, indeed, the Bohmian velocity, but according to orthodox QM, such a statement should not make any sense. Namely, according to orthodox QM, you cannot associate both velocity and position to a particle. Yet, this equation does precisely this. And yet, it is a STANDARD result in the theory of superconductivity. Does it mean that standard theory of superconductivity does not make sense in orthodox QM, but only in Bohmian QM? I am aware that it would be a very strong claim, but I don't see how to avoid such a conclusion.

Or let me rephrase my point in a form of a "naive" question (that could be asked by someone who never heard about Bohmian mechanics): Is Eq. (20) of Chapter 12 in Kittel compatible with the Heisenberg uncertainty principle?

 

[DrDu]

Again I have the problem that I don't have my books here at hand, but the "wavefunction" you are referring to is not a true wavefunction but more precisely the complex Landau-Ginzburg parameter. I think it was Gorkov who showed how to derive these equations starting from the Greens function formalism. If you want so, it is a ground state wavefunction of the Cooper pairs, but due to the macroscopic occupancy, it is nevertheless a classical observable which does not fulfill a Heisenberg relation.

 

[Demystifier]

Again I have the problem that I don't have my books here at hand, but the "wavefunction" you are referring to is not a true wavefunction but more precisely the complex Landau-Ginzburg parameter. I think it was Gorkov who showed how to derive these equations starting from the Greens function formalism. If you want so, it is a ground state wavefunction of the Cooper pairs, but due to the macroscopic occupancy, it is nevertheless a classical observable which does not fulfill a Heisenberg relation.

That sounds very very illuminating. Do you know a reference where I can find more details about that? I mean, about the idea of a wave function which represents a classical observable and does not fulfill a Heisenberg relation?

 

[f95toli]

It is true that it is not a "normal" wavefunction. Note however that the macroscopic phase that is associated with it is an observable that DOES fulfil an uncertainty relation (charge and phase are conjugate variables).

The standard textbook for this would be Tinkhams's book(but e.g. the book by Waldram contains more of less the same information).

 

[Demystifier]

The standard textbook for this would be Tinkhams's book(but e.g. the book by Waldram contains more of less the same information).

Unfortunately, neither of these books is available at my institution.

But I will search for the books elsewhere. In the meantime, I would be VERY GRATEFUL if someone could point to an appropriate journal reference. (A modern review, NOT an original paper!)

 

[Demystifier]

Wikipedia is also quite informal:
http://en.wikipedia.org/wiki/Landau-Ginzburg_theory

In particular, the "Bohmian" equation for j is presented. What is even more remarkable, the "Schrödinger" equation is non-linear. It seems to fit nice with my argument (presented elsewhere) that non-linear quantum mechanics is consistent ONLY with the Bohmian interpretation:
http://xxx.lanl.gov/abs/0707.2319

 

[DrDu]

L. P. Gor'kov, Sov. Phys. JETP, 9, 1364(1959).
Now this is sometimes calle the Gorkov Landau Ginzburg Abrikosov (GLAG) theory.

 

[DrDu]

You can download it here:

http://www.w2agz.com/Library/Classi...ns from BCS, Sov-Phys JETP 9, 1364 (1959).pdf

 

[Demystifier]

You can download it here:

http://www.w2agz.com/Library/Classi...ns from BCS, Sov-Phys JETP 9, 1364 (1959).pdf

Thanks!

 

[xepma]

The Psi in superconductivity is an order parameter of the system -- it's not the wavefunction of the electrons itself, even though it's frequently called that way. It's pretty easy to see that it can't be a wavefunction, since it has only one coordinate (hence it's not the wavefunction of a many-body system). The order parameter can be interpreted as the object responsible for breaking the U(1) symmetry. It therefore represents the charge density of the system (the absolute value of Psi) and the velocity field of the current (the phase term). The fact that charge is conserved automatically leads to the current equation you mentioned.

There is no reference to individual electrons.

The Ginzburg-Landau equation is an expression for the free energy of the system. The order parameter is treated as a classical object, and the non-linear Schrödinger equation follows from minimizing the free energy -- it is not based on the quantum-mechanical Schrödinger equation for the wavefunction (although you can derive it, strating from BCS theory).

In a Ginzburg-Landau approach there is absolute no reference to the underlying degrees of freedom carried by the electrons, i.e. the microscopics. It is solely phenomenological. You can derive this effective theory by starting from a quantum-mechanical treatment. A book such as Tinkham has treatments on that.

 

[Demystifier]

Thanks, xepma. It starts to make sense to me.

 

[Demystifier]

After a repeated reading of Feynman, now I have a better understanding of what is going on physically. If I have a LOT of particles with the same wave function, then rho(x,t) is not only a probability density of individual particles, but also an approximate ACTUAL macroscopic density of a pseudo-continuous charged fluid. Then this "fluid" also has a natural local velocity, which mathematically turns out to be identical to the Bohmian velocity. It is important to stress that this fluid velocity is a correct macroscopic description even if individual particles have totally different velocities; their average velocity coincides with the Bohmian one. Thus, even though all this does not prove that the Bohmian interpretation is correct, at least it demonstrates that it is VERY NATURAL (not artificial, as some people try to argue). In fact, it is very similar to an emergence of Bohmian velocities from standard QM with "weak" measurements:
https://www.physicsforums.com/showthread.php?t=252491

This is also related to the general physical interpretation of the probability current in orthodox QM. In an orthodox context, it seems that it makes physical sense only when a lot of particles in the same state is present, in which case it is much more than something that has only to do with probability.

 

[DrDu]

Yes, I remember having seen this Bohmian description for the flux of many particles in quantum molecular dynamics simulations.

 

[Demystifier]

I have found another paper on the relation between Ginzburg-Landau theory of superconductivity and Bohmian mechanics:
http://xxx.lanl.gov/abs/0812.4118