Classical and Quantum Mechanical Hypervirial Theorems

TLDR

In classical mechanics the time‑average of the Poisson bracket (H,W) vanishes, and in quantum mechanics the expectation value of (WH−HW) is zero for stationary states, yielding a dynamical relationship for any choice of W, of which the Clausius virial theorem is a special case. The authors aim to apply the classical hypervirial relations to determine liquid equations of state and to use quantum‑mechanical hypervirial relations to estimate constants in approximate wave functions. They analyze two classes of W: coordinate‑only functions and coordinate‑times‑momentum functions, and discuss applying the quantum relations to fit wave‑function parameters. For coordinate‑only W, neither classical nor quantum hypervirial results yield useful information.

Abstract

If W is a function of the coordinates and momenta, then in classical mechanics the time average of the Poisson bracket (H, W) is zero. In quantum mechanics it follows from the Heisenberg equation of motion that the expectation value of (WH — HW) for any wave function, corresponding to a stationary energy state of the system, is zero. For each selection of W there is a dynamical relationship, in a time-average or space-average sense, which the system must obey. Two classes of W's are considered in detail: W as a function only of the coordinates, and W as a function of the coordinates times the first power of a momentum. For the first class, neither the classical mechanical nor the quantum mechanical results provide useful information. The usual virial theorem of Clausius is a special case of the second class of relations. The classical treatment should be useful for the determination of the equation of state of liquids. The use of the quantum-mechanical relations for determining the constants in an approximate wave function is discussed.