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Looking for: Schrödinger equation in terms of phase and modulus 
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#1
Mar112, 12:36 PM

P: 1,412

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 = \psie^{i S}[/itex]. Can anyone show me the way? 


#2
Mar112, 03:11 PM

P: 1,583

What you're looking for is called the de BroglieBohm 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.



#3
Mar112, 03:14 PM

P: 1,412

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. 


#4
Mar112, 03:17 PM

P: 1,583

Looking for: Schrödinger equation in terms of phase and modulus



#5
Mar112, 03:20 PM

P: 1,412

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. 


#6
Mar112, 03:27 PM

P: 1,583




#7
Mar112, 05:15 PM

Sci Advisor
P: 1,626

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 GrossPitaevskii 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. 


#8
Mar112, 06:19 PM

P: 1,412

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



#9
Mar212, 12:28 AM

Sci Advisor
P: 5,366

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


#10
Mar212, 01:34 AM

P: 1,412




#11
Mar212, 04:25 AM

Sci Advisor
P: 5,366

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



#12
Mar212, 06:26 AM

Sci Advisor
P: 4,575

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 nondeterministic interpretation of classical mechanics: http://xxx.lanl.gov/abs/quantph/0505143 [Found.Phys.Lett. 19 (2006) 553566] http://xxx.lanl.gov/abs/0707.2319 [AIPConf.Proc.962:162167,2007] 


#13
Mar212, 08:35 AM

PF Gold
P: 3,080

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.



#14
Mar212, 08:08 PM

P: 1,583




#15
Mar212, 09:10 PM

PF Gold
P: 3,080

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.



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