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)