Single quantum particle entropy

Question

Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}) \, ,$$

where $V$ is a potential, and $\underline{x}\in \mathbb {R}^d$ with $d=1,2,3$.

My question: Does there exist a notion of entropy for the solution $\psi$? This might be vague, but I'm looking for an integral functional of $\psi$ which increases as time increases. Specifically, I'm interested in the free-space case, i.e., $V\equiv 0$.

What I know: First, for an entropy to be defined we may need some sort of a probabilistic structure which I did not specify, mostly because I don't know how.

Second, I know that in quantum field theory there is a sense of light-entropy, but that's a totally different model equation than that of the linear Schrodinger equation. See e.g., Loudon's Quantum Theory of Light.

(cross posted from physics.se)

Answer 1

A quantity that might well satisfy the OP is the socalled "entropy of position", $$S(t)=-\int |\psi(r,t)|^2\log|\psi(r,t)|^2 dr.$$ As derived in The entropy of position and the spreading of wave packets (1986), provided that $S(0)<\infty$ this entropy grows faster than $\log t$. One can also define the analogous quantity in momentum space. It has found applications in a variety of quantum mechanical problems, here is a list of references from Google Scholar.

Answer 2

Quantum dynamics of pure states, as described by the Schrodinger equation, is unitary and reversible, so there can be no entropy at that level. In order to formulate entropy you need mixed states. All this is standard material in Quantum Statistical Mechanics.

Answer 3

I think this question might have been misunderstood. Of course the Von Neumann entropy of a pure state (like the state you described) is 0. But I take it you aren't asking about Von Neumann energy, else there would be nothing to ask.

A notion of entropy that suits your requirements would have to be basis dependent (unlike Von Neumann entropy) otherwise it won't distinguish any state from any other. So the first step might be to choose a specific basis relative to which your entropy will be defined. A reasonable choice would be the position basis. Then, the Born rule gives a probability distribution on this chosen basis, and the entropy of this probability distribution can be calculated. Generically (in an imprecise sense), this entropy should be nondecreasing in time.

Your question is a reasonable one to ask, there is intuition that dispersion causes wave packets to spread, not to contract. This implies that there really is a sort of entropy here that's worth caring about. The question is ultimately a physics one though.