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International Journal of Quantum Chemistry

Calculation of the coherent memory functions from the eigenvalues and eigenvectors of the corresponding stationary Schrödinger equation

Lubomír Skála

Published Feb 1, 1983• 8 citations
Publisher

Abstract

Abstract Using some properties of the Bessel functions the algebraic representation of the Nakajima–Zwanzig generalized master equation was derived. This representation makes possible the calculation of the memory functions in terms of the probabilities. Using this approach, the formulas for the memory functions giving their dependence on the eigenvalues and eigenvectors of the corresponding stationary Schrödinger equation are given and discussed in detail.

DOI
10.1002/qua.560230220

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Abstract Using some properties of the Bessel functions the algebraic representation of the Nakajima–Zwanzig generalized master equation was derived. This representation makes possible the calculation of the memory functions in terms of the probabilities. Using this approach, the formulas for the memory functions giving their dependence on the eigenvalues and eigenvectors of the corresponding stationary Schrödinger equation are given and discussed in detail.
The Nakajima–Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system’s dynamics. The standard approach is based on using a perturbative treatment of the system–bath coupling for calculating this kernel, and is therefore restricted to systems weakly coupled to the bath. In this paper, we propose a new approach for calculating the memory kernel for an arbitrary system–bath coupling. The memory kernel is obtained by solving a set of two coupled integral equations that relate it to a new type of two-time system-dependent bath correlation functions. The feasibility of the method is demonstrated in the case of an asymetrical two-level system linearly coupled to a harmonic bath.
The Nakajima-Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system's dynamics, and the inhomogeneous term accounts for initial system-bath correlations. In this paper, we propose a new approach for calculating the memory kernel and inhomogeneous term for arbitrary initial state and system-bath coupling. The memory kernel and inhomogeneous term are obtained by numerically solving a single inhomogeneous Volterra equation of the second kind for each. The new approach can accommodate a very wide range of projection operators, and requires projection-free two-time correlation functions as input. An application to the case of a two-state system with diagonal coupling to an arbitrary bath is described in detail. Finally, the utility and self-consistency of the formalism are demonstrated by an explicit calculation on a spin-boson model.
Using the recently discussed quantum dynamics in phase space, we derive a master equation, starting from the phase-space equivalent to the Schr\"odinger equation of motion for the density operator. Use is made of Zwanzig's projection-operator techniques and some explicit realizations of the projection operators are given. The master equation is then applied to show that the time-correlation functions, as defined in the text, satisfy an integral equation of the Volterra type. Next, a master equation for a system interacting with a large system is derived. As an illustration, we determine the lowest-order Born approximation and carry out a short-memory-approximation calculation for an oscillator coupled to a reservoir and for a two-level system interacting with an oscillator heat bath; we obtain equations of the Fokker-Planck type. Some physical implications of these equations are also discussed.
Abstract The stationary Schrödinger equation can be cast in the form , where H is the system’s Hamiltonian and is the system’s density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as SSE , applied to many-body systems with symmetries. For a nondegenerate energy level, the solution of the SSE is merely a projection on the corresponding eigenvector. However, in the case of degeneracy is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE for a general system of spins 1/2 with Heisenberg interactions, and address simple systems of spins 1. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem , where is the total spin of N spins 1/2, and is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of .
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. By employing the exact solution for the model, analytical expressions are determined for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows an explicit comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, the structure of widely used phenomenological master equations with a memory kernel may be incompatible with a nonperturbative treatment of the underlying microscopic model. Several physical implications of the results on the microscopic analysis and the phenomenological modeling of non-Markovian quantum dynamics of open systems are discussed.
A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.