The theory of quantum mechanical systems with a stochastic Hamiltonian which are of importance in the theory of dissipative systems and in experimental investigations of the response of physical systems by means of electronic devices is discussed. A new formal development of quantum mechanical density matrices is given that is valid even for strong stochastic perturbations. If the stochastic part of the Hamilionian has a Gaussian distribution and an almost constant spectral density the given solution reduces to an expansion in terms of Hermite functionals which are othonormal respective to the Wiener measure (Cameron-Martin development). This expansion is operationally meaningful and characterized by good convergence and simple properties. As an example of the application of the theory a new foundation of Bloch's relaxation theory is sketched. (auth)
The theory of quantum mechanical systems with a stochastic Hamiltonian which are of importance in the theory of dissipative systems and in experimental investigations of the response of physical systems by means of electronic devices is discussed. A new formal development of quantum mechanical density matrices is given that is valid even for strong stochastic perturbations. If the stochastic part of the Hamilionian has a Gaussian distribution and an almost constant spectral density the given solution reduces to an expansion in terms of Hermite functionals which are othonormal respective to the Wiener measure (Cameron-Martin development). This expansion is operationally meaningful and characterized by good convergence and simple properties. As an example of the application of the theory a new foundation of Bloch's relaxation theory is sketched. (auth)
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.
A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrödinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.
When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
In 1946 Cameron and Martin concluded a paper [‘‘The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals,’’ Ann. Math. 48, 385–392 (1947)] by saying: ‘‘This paper opens up the whole set of techniques of Orthogonal Developments...for use...in the study of non-linear functionals... .’’ Here, the Cameron–Martin–Wiener expansion is developed with mathematics familiar to physicists, and its use is illustrated and applied to a quantum theory of electron transport.
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.