Classical and Quantum Mechanical Hypervirial Theorems

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.