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

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

recategorized Dec 18, 2016

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.

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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 (775 points)
edited Dec 18, 2016

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