Looking for: Schrödinger equation in terms of phase and modulus

nonequilibrium

Hello,

I can't seem to find a reference formulating the Schrödinger equation as a set of two differential equations in terms of the modulus [itex]|\psi|[/itex] and the phase [itex]S[/itex] instead as one diff. eq. in terms of the complex wavefunction [itex]\psi = |\psi|e^{i S}[/itex].

Can anyone show me the way?

lugita15

What you're looking for is called the de Broglie-Bohm pilot wave theory. Here's an explanation of it. It is the basis of the Bohmian interpretation of quantum mechanics, a realist interpretation which escapes Bell's theorem by being nonlocal.

nonequilibrium

I wasn't looking for that, why would you assume that?

What I'm looking for is what I said: the mathematical reformulation of the Schrödinger equation in terms of the modulus and the phase.

lugita15

And I'm telling you, that is exactly what Bohmian mechanics is. You have two real differential equations, one for the quantum potential, which represents the modulus of the wave function, and one for the pilot wave, which represents the phase of the wave function.

nonequilibrium

That doesn't make sense, how can a theory be a mathematical rewriting of an equation.

The two equations might be used in Bohmian mechanics, but anyway, I'm just looking for a reference that shows the equations for the modulus and phase of the wave function equivalent to the Schrödinger equation.

lugita15

Those equations are pretty much only used in the context of Bohmian mechanics. Bohmians rewrite the Schrodinger equation in terms of two real equations, and then they give their own interpretation for what the modulus and phase really represent. But the rewriting of the equation itself has nothing to do with their philosophical interpretation, it's just a mathematical procedure.

Cthugha

That formulation goes back to Madelung who noted in the 1920es that one can interpret the Schrödinger equation in polar form [tex]\psi=a e^{i \phi}[/tex]as the equation describing the hydrodynamics of a compressible fluid with density [tex]\rho=a^2[/tex] and velocity [tex]v=\nabla \phi[/tex]. That form is often used when dealing with the Gross-Pitaevskii equation and quantum fluids in general.

The original publications were:
Madelung, E. (1926). "Eine anschauliche Deutung der Gleichung von Schrödinger". Naturwissenschaften 14 (45): 1004–1004.
Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Z. Phys. 40 (3–4): 322–326,
but I doubt these are of much use to you as they are in German.

However, performing a google search for "Madelung equations" might already point you in the right direction.

These equations have indeed also been used by Bohmians (although using different terminolgy), but they have not invented them and are certainly not the only ones who use them. They are more heavily used in quantum hydrodynamics and in dealing with quantum fluids.

nonequilibrium

Ah, as I'm Belgian German should be in my range of "pretty understandable", thank you

tom.stoer

It's correct that Broglie-Bohm use the Schrödinger equation in terms of phase and modulus. But of course you need not agree to their interpretation only b/c you rewrite the equation.

@ mr. vodka, it shouldn't be too hard to derive the equations on yor own

nonequilibrium

I know, but a reference can be nice when writing a paper; besides it's a good check for your calculations.

tom.stoer

I am not sure but perhaps the QM textbook of Sakurai contains this calculation

Demystifier

As is well known now, single complex Schrodinger equation can be rewritten as two real equations for probability density and phase of the wave function, which suggests (but in no way proves) a deterministic Bohmian interpretation of QM.

However, it is much less known that a sort of reverse is also possible. Certain equations for CLASSICAL mechanics can be rewritten as a single complex nonlinear Schrodinger equation, which suggests a non-deterministic interpretation of classical mechanics:
http://xxx.lanl.gov/abs/quant-ph/0505143 [Found.Phys.Lett. 19 (2006) 553-566]
http://xxx.lanl.gov/abs/0707.2319 [AIPConf.Proc.962:162-167,2007]

Ken G

That's an interesting paper. I've often made the argument that there is no real reason to regard classical mechanics as deterministic, or dealing in exact positions and velocities, but it's nice to see a formal presentation of why one can hold that position.

lugita15

I think Demystifier was referencing the paper for the opposite reason, to argue that, if we're willing to consider classical mechanics deterministic even though it also has a nondeterministic interpretation, we should be willing to consider quantum mechanics deterministic as well.

Ken G

Actually, I think he is quite clear that that blade cuts both ways. He leaves it up to each person to decide if they will choose a deterministic or nondeterministic interpretation-- his point is that the same choice can be made for both classical and quantum mechanics. Indeed, I think he has a point there-- there is little reason to adopt one interpretation for one of them, and the other for the other, even though that is what is most often done.