Schrödinger equation derivation and Diffusion equation

Question

I am aware of the debate on whether Schrödinger equation was derived or motivated. However, I have not seen this one that I describe below. Wonder if it could be relevant. If not historically but for educational purposes when introducing the equation.

Suppose that we have the time dependent Schrödinger equation for a free particle, $V=0$.

$$-\frac {\hbar i}{2m} \nabla^2 \Psi_\beta = \frac {\partial \Psi_{\beta}}{\partial t} $$

As the particle moves its heat is diffused throughout space. Now consider that we consider Heat equation or in general Diffusion equation:

$$\alpha\nabla^2 u= \frac {\partial u}{\partial t} $$

Where $u$ is temperature.

Also we have particle diffusion equation due to Fick's second law.

$$D \frac {\partial^2 \phi}{\partial x^2}= \frac {\partial \phi}{\partial t} $$

Where $\phi$ is concentration.

Furthermore, probability density function obeys Diffusion equation. So as the free particle moves, the heat, the temperature, or the density is diffused.

Now we can motivate Schrödinger equation in an intuitive way. Mathematically it is describing the same diffusion. Am I right? Have you seen more like this motivation elsewhere?

Answer 1

I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics".

However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary. That tells us that we have to separate the Schrödinger equation into two, one equating the real parts of the two sides, and one equating the imaginary parts. The meaning of this imaginary diffusion coefficient is therefore that the wave-function is complex, or, in other words, it has an absolute value and a phase, like the electromagnetic wave.

Answer 2

The Schrödinger equation is a wave equation, not a diffusion equation. While the equations look similar, the $i$ in Schrodinger equation differentiates them; that allows non-decaying oscillatory solutions, which diffusion equations do not allow.

That said there are certainly relations between the two.

The Schrödinger equation is analogous to the Fokker-Planck equation, which is the evolution of a classical probability distribution subject to random noise. That can result in diffusion.

There is also the stochastic interpretation of quantum mechanics, which relates the Schrödinger equation to a kind of quantum Brownian motion. (Truthfully, I don't understand it; the original paper is here.) Classical Brownian motion leads to diffusion.

Answer 3

I wanted to write a comment but my quote is too long to fit as a comment.

I think I found a relevant quote from James Gleck who said the following in page 175 of his book Genius

"The traditional diffusion equation bore a family resemblance to the standard Schrödinger equation; the crucial difference lay in a single exponent where the quantum mechanical version was an imaginary factor, i. Lacking that i, diffusion was motion without inertia, motion without momentum. Individual molecules of perfume carry inertia, but their aggregate wafting through air, the sum of innumerable random collisions, does not. With the i, quantum mechanics could incorporate inertia, a particles memory of its past velocity. The imaginary factor in the exponent mingled velocity and time in the necessary way. In a sense, quantum mechanics was diffusion in imaginary time."

Answer 4

I'm not familiar with quantum mechanics yet, but I have taken a course on partial differential equations where we did look at Ficke's law.

The form of the equations do seem to be quite similar - the first time derivative is proportional to the second spatial derivative. This implies solutions that end up settling down over time (i.e. steady-state solutions). However, the complex term is a bit of a wild card, because it can turn exponential factors into periodic ones via Euler's formula. So I'd be careful in trying to compare the two.

Answer 5

I think we are missing a very important point. In SE the time it imaginary, where as in diffusion equation it is real. And the consequence of imaginary time is that it gives a phase freedom in wave function, leads to oscillatory solution. Whereas in diffusion equation, real time leads to decaying solution as mentioned already.

Answer 6

The Schrodinger equation (SE) is already in the form of a diffusion equation, but there is an imaginary number in front of the time derivative (or the diffusion coefficient is imaginary), as other respondents here already noted. I think the best way to motivate the SE is still via higher classical mechanics (e.g. Hamilton-Jacobi).

However the analogy between the diffusion equation and SE is still interesting. So let us examine the analogy in more detail and see where the differences lie to get more insight.

Because of the imaginary unit $i$ in SE, $\psi$ is likely complex. To get a real number, the founders interpret its modulus squared as a probability density, analogous to the number density or concentration. Where the density is high, you are likely to find more particles (or that one particle described by the Schrodinger equation). A bit of a stretch, but still ok.

The classical diffusion equation can also be derived from the equation of continuity. This needs the current in the form of Fick's law, which is phenomenological. It says the current is proportional to the gradient of the density or concentration. Particles tend to flow from higher to lower concentrations.

QM and SE also have an equation of continuity for the probability density. But as far as I know, the corresponding current cannot be derived from a Fick's law form. It is not the gradient of the density. The probability current is some sort of average of the quantum mechanical particle velocity operator. Also, while this continuity equation can be derived from the SE, the argument cannot be reversed, as far as I know. The SE cannot be derived from this continuity equation. Another way to look at this is that the continuity equation for the density emerges from the invariance of the lagrangian of the Schrodinger Equation to a phase change in psi (gauge invariance -> Noether's first theorem -> conserved current).

So formally the main difference, aside from the imaginary number in SE, is that the SE/QM current is not gradient of the density of anything.