Analogous structure of Diffusion and Schrödinger equation and definition of flux?

Question

I came across some analogous structure of diffusion and the quantum mechanical particle (Schrödinger eq.). I have seen that there have been similar questions asked, but the (probablitily flux and the mass/particle conservation was not adressed in those).

In diffusion the particle flux $\vec{j}(\vec{r},t)$ is related to the gradient of the particle density $\vec{\nabla} n(\vec{r},t)$ and the diffusion coeffcient $D$ via Ficks first law

$$\vec{j}(\vec{r},t) = -D \nabla n(\vec{r},t) \tag{1a} $$ When this is combined with the particle conservation condition

$$ \frac{\partial n(\vec{r},t)}{\partial t} = - \nabla\cdot \vec{j}(\vec{r},t), \tag{2a}$$ one obtains the "Diffusion euqtaion" (Ficks second law)

$$ \frac{\partial n(\vec{r},t)}{\partial t} = D \nabla^2 n(\vec{r},t). \tag{3a}$$

Now I find it quite puzzling to compare this with analogous expressions from non-rel. Quantum mechanics.

The probability flux is defined by

$$ \vec{j}(\vec{r},t) = \frac{\hbar}{2m i}\left[\Psi^*\nabla\Psi - \Psi\nabla(\Psi^*)\right]\tag{1b},$$ keeping in mind that the QM particle density $$n(\vec{r},t)=|\Psi\Psi^*|\tag{4}.$$ Thus $\vec{j}$ in (1b) essentially differs from $\nabla n$ in (1a) only by the "-" sign of the second term.

In QM usually the continuity condition (= particle probability conservation):

$$ \frac{\partial n(\vec{r},t)}{\partial t} = - \nabla\cdot \vec{j}\tag{2b},$$ is obtained from (1b) and the time-dependent Schrödinger equation:

$$ i \frac{\partial \Psi(\vec{r},t)}{\partial t} = -\frac{\hbar}{2m} \nabla^2 \Psi(\vec{r},t) \tag{3b}. $$

So in both settings we have two independent equations of close structural similarity form which a third one follows. In both cases (1) defines a flux, (2) a continuity/conservation condition and (3) the time development of a density function.

I am asking myself if there is a theory of a more general structure from which cases (a) and (b) follow as specific cases. I think about something like a Poisson bracket formalism (or the mimisation of action and similar) that contains both cases as special cases. Can anyone hint me to something like that?

In particular I would be interested to understand how in such a formalism the above addressed "-"-sign in the definition of the flux can arise. I am asking this because I suspect some physical interpretation or significance of $\nabla n$ in the QM context of the flux.

I am aware of similar questions like this on PSE about the analogy of the SE and the Diffusion equation, but no one has adressed particle conservation and flux and in addition I have found no comments that would hint at a "common theory" that would unify both in the sense I am asking for.

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Edit: to make the analogy better visible I attach this table $$ \begin{array}{c|c|c} (a) & (b) & \\ \hline \vec{j} = -D \nabla n & \vec{j} = \frac{\hbar}{2m i}(\Psi^*\nabla\Psi - \Psi\nabla \Psi^*) & (1) \\ \frac{\partial n}{\partial t} = - \nabla\cdot \vec{j} & \frac{\partial n}{\partial t} = - \nabla\cdot \vec{j} & (2) \\ \frac{\partial n}{\partial t} = D \nabla^2 n & i \frac{\partial \Psi}{\partial t} = -\frac{\hbar}{2m} \nabla^2 \Psi & (3) \end{array}$$ with $n=|\Psi^*\Psi|$

Answer 1

I'm not sure how to focus on your question your way, but first you must compare apples with apples and use the Hydrodynamic formulation of QM introduced by Madelung in 1926. They key point here is that the Schroedinger equation is complex, so it has two dependent variables, unlike the real diffusion equation, so it is basically two equations, a familiar Euler hydrodynamic one, but also a novel "Hamilton-Jacobi" one.

The idea is to rewrite Schroedinger's wave function in polar coordinates, $$ \Psi=\sqrt{n} e^{iS/\hbar}, $$ when the diffusion equation only has one dependent variable, n. The key point is that probability flow is not driven by just the probability density n, as in Fick's law, but mainly by the phase S, (note $\vec v= {1\over m} \nabla S$),
$$ \vec j= {n\over m}\nabla S, \tag{1b} $$

Thus, (2b), the conservation of probability equation, resembles (2a) conservation of particles in the abstract, but works quite differently, $$ 0=\partial_t n+\nabla \cdot \vec j = \partial_t n+ (n\nabla^2 S + \nabla n \cdot \nabla S)/m. \tag{2b} $$ This Euler equation is only the imaginary part of Schroedinger's equation! (And, as you might have marveled in school, doesn't care a bit about the potential V.)

Nevertheless, the big Kahuna is the real part of that equation (the "Quantum Hamilton-Jacobi" equation), $$ 0=\partial_t S+ ( |\nabla S|^2 /2m+V +Q), \tag{4b} $$ where $$ Q= - {\hbar^2\over 2m}{\nabla^2\sqrt{n}\over \sqrt{n}} $$ is Bohm's celebrated quantum potential. It is amazing what an imaginary unit can do to an equation, but there it is.

(Actually, your (3b) is spurious: You willfully tossed out V by hand, but, as you see here, it influences the flow of S and hence n, after all.)

Looking at the wavepacket might, or might not, help your intuition about quantum flows. Suffice it to say that, in phase space, they are known to exhibit astounding phenomena, thoroughgoingly different than material flows (Steuernagel et al.). But you know QM is weird...

Answer 2

A less known quantum equation of diffusion can also be considered as \begin{equation}\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2}-\nabla^2\psi+\frac{2m}{\hbar}\frac{\partial\psi}{\partial t}+\frac{m^2c^2}{\hbar^2}\psi=0.\end{equation} This is analogous to the Telegraph equation that describes the electric voltage and current in a transmission line. In both cases the electron waves due to their charge and more are coherent.