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Let's say I have a probability amplitude $\psi:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ (so, $\psi$ satisfies $\int_\Sigma |\psi|^2=1$). Is there a way to use $\psi$ as initial conditions for diffusion of probability values?

In particular, is there a PDE for computing a function $\psi_t$ with $\psi_0\equiv\psi$ such that the induced probability distribution $\rho_t=|\psi_t|^2:\Sigma\rightarrow\mathbb{R}^+$ satisfies the heat equation $\Delta_\Sigma\rho=\frac{\partial\rho_t}{\partial t}$ for Laplacian $\Delta_\Sigma$ (or any other diffusive equation)?

This document seems to think that it might be an open mathematical question, at least for the "Anderson model" of diffusion (see e.g. conjecture (iii) on page 30), but I'm not sure if I am reading it properly. My hope was that it would be as easy as using the Dirac operator $D$ since it looks like the "square root" of the Laplacian in the right way, but apparently things aren't so easy!

[This website has been incredibly useful for me to get my bearings in this area of physics despite my total lack of background. I sincerely apologize for posting so many questions and really appreciate everyone's support!]

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What exactly are you taking $\Sigma$ to be? and relatedly, what do you mean by $\Delta_\Sigma$? – Emilio Pisanty Oct 10 '12 at 18:26
I'll actually be taking $\Sigma$ to be a surface (consider a sphere, and writing $\psi$ in the basis of spherical harmonics). But as a math guy I can translate into this case -- an example with $\Sigma\subseteq\mathbb{R}^n$ is perfectly fine. I'm using $\Delta_\Sigma$ to denote the Laplacian associated with domain $\Sigma$ -- e.g. on $\mathbb{R}^n$ it would be $\Delta=\sum_i\frac{\partial^2}{\partial x_i^2}$. – Justin Solomon Oct 11 '12 at 0:52

This is not a answer to your question but a close cousin to it, perhaps you will find it of interest. The Schrodinger equation can be analytically continued to give the Heat Diffusion equation.


Google can point you further elaborations and references.

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Yes, this property is of interest, although I'm hoping to find something closer to diffusion of probability values. – Justin Solomon Oct 10 '12 at 3:37

Nice question. It looks like you are asking about the basic question of how probability amplitudes change over time, ie, what is the mechanics of the probabilities in quantum mechanics. As you know, the schrodinger equation is based on the amplitudes and, in the end, we concern ourselves with the probabilities $p = |\psi|^2$. We do the work on the amplitudes and then compute the probabilities. You have asked if we can just do the work on the probabilities. In fact, the equation you wrote is the schrodinger with no potential and except on the probabilities, not the wavefunction. How about you use the schrodinger and the amplitudes? Afterall, you get probabilities in the end anyway.

Another thing I might mention is about Liouville mechanics. That is definitely a mechanics of pure probabilities. Here is a snippet from Wikipedia:

 the density of system points in the vicinity of a given system point travelling through phase-space is constant with time.

This might suggest that probabilities don't diffuse.

Also, have a look at the Hamiltonian form of the Liouville equation:

$ \frac {\partial \rho }{\partial t} = \{ \rho, H \} $.

Now this is really starting to look like a nice mechanics of probability densities.

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Thanks -- I'll take a look in Liouville mechanics. In my application I'm actually working with objects that look like probability amplitudes $\psi$, so I'm hoping to find analogs from QM that can help me. In particular, I'm hoping rather than switching to doing all my math on $\rho$ to be able to do math on $\psi$ in a way that has the intended effect (in this case diffusion) on $\rho$. Does this make sense? – Justin Solomon Oct 11 '12 at 0:50

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