A Statistical-Averages Theorem for Pauli Operators

Ian Duck, E. C. G. Sudarshan

Ian Duck, E. C. G. Sudarshan, Arthur S. Wightman

Peter T. Landsberg

Time averages are important in physics and in statistical mechanics. An ergodic theorem is a way of justifying the replacement of time averages by averages in phase space. A derivation of this theorem from quantum mechanics was given by von Neumann in 1929. Pauli and Fierz found a shorter argument in 1937. They agreed that for ergodicity to hold there should be no energy degeneracy in the Hamiltonian. I trace the circumstances and consequences of a disproof of these arguments.

Donald H. Kobe

The virial theorem is derived in quantum statistical mechanics for a time-dependent Hamiltonian that describes a charged particle in a time-dependent electromagnetic field. The time derivative of the expectation value of the Clausius operator satisfies an equation which is the same form as the classical equation for the Clausius function (r⋅mv). By taking a time average, the virial theorem of quantum mechanics for time-dependent states is obtained. By taking a statistical average and a time average, the virial theorem of quantum statistical mechanics is obtained.