Semi-classical work and quantum work identities in Weyl representation

TL;DR

This paper derives a semi-classical quantum work identity using Wigner-Weyl quantization, providing a geometric interpretation and addressing the challenge of defining work in quantum mechanics.

Contribution

It introduces a semi-classical work identity in quantum mechanics based on phase space trajectories, bridging classical and quantum descriptions.

Findings

Derived a semi-classical nonequilibrium work identity.

Provided a geometric interpretation in complex phase space.

Validated with the quantum harmonic oscillator example.

Abstract

We derive a semi-classical nonequilibrium work identity by applying the Wigner-Weyl quantization scheme to the Jarzynski identity for a classical Hamiltonian. This allows us, to the leading order in $\hbar$, to overcome the problem of defining the concept of work in quantum mechanics. We propose a geometric interpretation of this semi-classical relation in terms of trajectories in a complex phase space and illustrate it with the exactly solvable case of the quantum harmonic oscillator.