Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Is there a set of variables over which the Schrodinger eq. can be interpreted as a diffusion eq.?

+ 2 like - 0 dislike
959 views

Is there a set of variables over which the Schrodinger eq. can be interpreted as a diffusion eq.?

asked Dec 18, 2016 in Theoretical Physics by anonymous [ revision history ]
recategorized Dec 18, 2016 by dimension10

Yes, with imaginary time and using the ground state as a guiding function.

Substituting $t=i\tau$ (maginary time) one gets a diffusion equation. But the properties of the equation change completely. In particularly, probability is no longer conserved.

What is a "guiding function"?

Diffusion may have a non trivial stationary profile function $\psi(x)$, for example, between two plates, with a constant flux $\propto\nabla\psi$. For that there should be a stationary gradient of $\psi(x)$. If the initial state is far away from the stationary state, then the general solution contains all eigenfunctions. After a while (or in so called regular regime), the diffusion function reduces to a sum of two functions: the final stationary profile plus decaying exponentially the ground eigenstate. Maybe this was meant by "guiding function".

After another while nothing but the final stationary state remains. It may be non zero function.

In QM we may have non stationary populating different eigenstates due to time-dependent perturbation. One may think of "spreading" the initial eigenstate over the others, but it may happen instantly, - it is not like a diffusion in the state space. The coefficients may reach their stationary values instantly if the perturbation is sudden.

See, E. Nelson https://web.math.princeton.edu/~nelson/books/qf.pdf, p. 76ff, on stochastic quantization. 

 

1 Answer

+ 3 like - 0 dislike

No, except in a generalized sense.

As partial differential equations, the diffusion equation is of parabolic type, whereas the Schrödinger equation can be considered to be of wave equation (hyperbolic) type. The type does not change under reparametrization of spacetime coordinates $(x,t)$. But also a field redefinition would not work, because the solutions to these PDE have different properties. For instance, parabolic equations tend to smear out singularities of the initial data, while wave equations propagate them. Another important restriction is that the Schrödinger equation has several conserved quantities: Total probability  $1 = \int dx\, |\psi(x,t)|^2$ and total energy $E=\int dx\, \psi(x,t)^*H\psi(x,t)$, whereas a diffusion equation typically has only the total probability as conserved quantity. Any attempt at writing the Schrödinger equation as a parabolic equation would have to make up for that.

That said, people often introduce a concept called imaginary time and write $\tau = it$ and use it to write the Schrödinger equation as

$$ \frac{\partial}{\partial \tau} \psi(x,\tau) = -H \psi(x,\tau) $$

However, the wave function in real time, $\psi(x,t)$, is only defined for well, real time $t$, and is, at best, only related to the wave function $\psi(x,\tau)$ by means of an analytic continuation (which need not be unique or possible).

That said, imaginary time is quite useful. After all, one way to solve the Schrödinger equation is to construct the evolution operator $\exp(-itH)$, for which we need to know the spectrum of the Hamiltonian $H$. The imaginary time variant gives use the evolution operator $\exp(-\tau H)$, which is not directly useful for solving the Schrödinger equation, but still contains a lot of information about the spectrum of $H$. For instance, if we know that $H \geq 0$ and send $\tau \to \infty$, then this operator will converge to the projector onto the ground state, $\exp(-\tau H) \to P_{E=0}$ in some operator topology.

Summary: There is no direct way to write the Schrödinger equation as a diffusive equation, but going to imaginary time $\tau = it$ is still a useful way to gain information about its solutions.

answered Dec 18, 2016 by Greg Graviton (775 points) [ revision history ]
edited Dec 18, 2016 by Arnold Neumaier

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ys$\varnothing$csOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...