Looking for: Schrödinger equation in terms of phase and modulus

In summary, the conversation revolves around the mathematical reformulation of the Schrödinger equation in terms of modulus and phase, and the potential use of this formulation in Bohmian mechanics. The de Broglie-Bohm pilot wave theory is mentioned, along with the concept of rewriting classical mechanics as a single complex nonlinear Schrödinger equation. The possibility of a deterministic interpretation for both classical and quantum mechanics is also discussed. References for further reading are provided.
  • #1
nonequilibrium
1,439
2
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?
 
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  • #2
What you're looking for is called the de Broglie-Bohm pilot wave theory. "plato.stanford.edu/entries/qm-bohm" 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.
 
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  • #3
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
mr. vodka said:
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.
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.
 
  • #5
And I'm telling you, that is exactly what Bohmian mechanics is.
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.
 
  • #6
mr. vodka said:
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.
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.
 
  • #7
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.
 
  • #8
Ah, as I'm Belgian German should be in my range of "pretty understandable", thank you
 
  • #9
lugita15 said:
What you're looking for is called the de Broglie-Bohm pilot wave theory.
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
 
  • #10
@ mr. vodka, it shouldn't be too hard to derive the equations on yor own

I know, but a reference can be nice when writing a paper; besides it's a good check for your calculations.
 
  • #11
I am not sure but perhaps the QM textbook of Sakurai contains this calculation
 
  • #12
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]
 
  • #13
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
Ken G said:
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.
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.
 
  • #15
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.
 

Related to Looking for: Schrödinger equation in terms of phase and modulus

1. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time.

2. What is the significance of representing the Schrödinger equation in terms of phase and modulus?

Representing the Schrödinger equation in terms of phase and modulus allows for a more intuitive understanding of the quantum state of a physical system. Phase and modulus are related to the amplitude and frequency of a wave, which are important concepts in quantum mechanics.

3. How is the Schrödinger equation in terms of phase and modulus different from the traditional form?

The traditional form of the Schrödinger equation uses the wavefunction, which describes the probability amplitude of a particle at a given position and time. The phase and modulus form uses the complex representation of the wavefunction, which separates the amplitude and phase components.

4. What are the benefits of using the Schrödinger equation in terms of phase and modulus?

Using the phase and modulus form of the Schrödinger equation allows for a more efficient and accurate description of quantum systems, particularly for systems with multiple particles. It also simplifies certain calculations and makes it easier to visualize the behavior of quantum systems.

5. How is the Schrödinger equation in terms of phase and modulus used in practical applications?

The phase and modulus form of the Schrödinger equation is used in a variety of practical applications, including quantum computing, quantum information theory, and quantum optics. It is also used in the development of new technologies such as quantum sensors and quantum cryptography.

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