As an active participant on the Physics Stack Exchange I have, on several separate occasions, run into some vague remarks about the 'intimate relation' between the famous Black-Scholes model in economics and the Schrödinger equation that appears in quantum mechanics. In particular, people mention that path integrals (a certain technique that is very useful in quantum mechanical calculations) find many applications in economic settings.
I'd love if someone could make this connection clearer by
Explicitly (i.e. in equations) showing how the Schrödinger equation and the Black-Scholes equation are related
Discussing the use of path integrals in economics