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Physical Review

Quantum Theory of Fluctuations

H. Ekstein, N. Rostoker

Published Nov 15, 1955• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.100.1023

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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.
The dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and discussed.
The Callen-Welton formula (fluctuation-dissipation theorem) of voltage and current noise of a resistance are the sum of Nyquist's classical Johnson noise equations and a (quantum) zero-point term with power density spectrum proportional to frequency and independent of temperature. At zero temperature, the classical Nyquist term vanishes however the zero-point term produces non-zero noise voltage and current. We show that the claim of zero-point noise directly contradicts to the Fermi-Dirac distribution, which defines the thermodynamics of electrons according to quantum-statistical physics. As a consequence, the Johnson noise must be zero at zero temperature, which is in accordance with Nyquist's original formula. Further investigation shows that the Callen-Welton derivation has conceptual errors such as neglecting phonon scattering, disregarding the Pauli principle during calculating the transition probabilities and using bosonic (linear oscillator) energies leading to the zero-point noise artifact. Following Kleen's proposal, the possible origin of the heterodyne (Koch - van Harlingen - Clark) experimental results are also discussed in terms of Heffner theory of quantum noise of frequency/phase-sensitive linear amplifiers. Experiments that failed to see the zero-point noise term are also mentioned.
The goal of this work is, the utilization of a method recently developed by the author, in quantum dynamical problems. Main idea is based on the relation between quantum and classical dynamics such that quantum dynamical problems become their classical dynamical counterparts when they are formulated via expectation values of certain operators and the probability density tends to be sharply localized. Expectation values can be expanded into a series in ascending appearence multiplicity of the complement of the projection operator which projects the space spanned by the wave function. This expansion contains fluctuation terms as unknowns besides the expectation values. We construct an infinite set of differential equations. The finite truncations of this set enables us to approximate the quantum dynamics of the system under consideration.
A formal apparatus is developed for dealing with quantum fluctuations in a driven (nonequilibrium) state. Phenomenological equations for the decay of the system from an arbitrary state to the driven state are presumed known. By assuming that at some initial time the system-reservoir density matrix can be factored, we establish that the spectrum of fluctuations is given by the Fourier transform of the solution of the phenomenological equations, i.e., the regression of fluctuations obey the "macroscopic" equations of motion even in the nonequilibrium case.
The fluctuation-dissipation relations (FDR) in physical systems are studied at all levels of the statistical description. The most general FDR are the relations for the fluctuations of many-body distribution functions. It is pointed out the problem of formulation of FDR is related to the problem of deriving irreversible equations based on the reversible equations of classical and quantum mechanics. The FDR are divided into two classes: 1) FDR for fluctuations with infinite correlation times (collisionless approximation), which correspond to infinitely narrow resonances, and 2) FDR for fluctuations with finite correlation times (collisional approximation). The corresponding spectral densities have finite widths, determined by the collision integrals. The fundamental questions about which different viewpoints have been published in the literature are critically analyzed: 1) the limits of applicability of the Callen-Welton formula and 2) the quantum generalization of Nyquist's formula for the intensity of a Langevin source of oscillatory systems. It is shown that the traditional form of the quantum Nyquist formula, is not well-founded and leads to unphysical consequences. A different expression, used in the literature, for the quantum Nyquist formula is examined. It is not universal, but holds in many important cases. Its region of applicability is determined by the corresponding quantum kinetic equations. The consequences of the two forms of the quantum Nyquist formula, which can be checked experimentally, are studied. The question of the formulation of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source and the corresponding diffusion coefficient in quantum systems. It arises, in particular, also in quantum electrodynamics in the calculation of the Lamb shift (Sec. 12). In this connection two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the subtraction formalism of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula. The exposition is illustrated with many specific examples.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.