Quantum Theory of Fluctuations

An operator is constructed for the spectral density of a fluctuating dynamical variable. The formulation is based on an analysis of method of measurement. An essential feature of the operator is the replacement of classical dynamical variables by time-dependent Heisenberg operators. The expectation value of the spectral density includes fluctuations of statistical origin, as well as fluctuations of quantum mechanical origin. It can be computed for any physical system when the initial state has been specified. Two applications are given: Nyquist's law is rigorously deduced for a physical system that is similar to the one discussed by Callen and Welton, and the shot effect is calculated for free uncorrelated electrons. A quantum correction is obtained whose origin lies in the wavelike character of the electron.