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Communications in Theoretical Physics

Invariant Hermitian Operator and Density Operator for the Adiabatically Time-Dependent System

Yan Feng-Li, Linguang Yang

Published Sep 15, 2001• 1 citation
Publisher

Abstract

The density operator is approximately expressed as a function of the invariant Hermitian operator for the adiabatically time-dependent system. Using this method, the calculation of the density operator for the Heisenberg spin system in a weakly time-dependent magnetic field is exemplified. By virtue of the density operator, we obtain the approximate energy expectation value and spin expectation value of this system starting from the thermal equilibrium.

DOI
10.1088/0253-6102/36/3/263

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The density operator is approximately expressed as a function of the invariant Hermitian operator for the adiabatically time-dependent system. Using this method, the calculation of the density operator for the Heisenberg spin system in a weakly time-dependent magnetic field is exemplified. By virtue of the density operator, we obtain the approximate energy expectation value and spin expectation value of this system starting from the thermal equilibrium.
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.
Introduction to the density operator Motivation and definition It is often inconvenient, for one reason or another, to keep track of all the variables of a complex system. For example, in a many-body system it would be impracticable to consider the co-ordinates of each particle. Furthermore, such information is actually not of much interest. Thus in a spin system the behaviour of the individual magnetic moments is unimportant; the components of the total magnetisation are the variables of primary interest. Very generally, for a system with ∼ 10 23 co-ordinates, one is unlikely to need more than, say, ten variables to describe its observable properties. However, it is quite clear that this reduced amount of information is no longer sufficient to write down a wavefunction and, therefore, it is no longer possible to calculate the evolution of such a system using the usual methods of quantum mechanics. In Chapter 5 we saw that it was possible, by the introduction of probabilistic arguments, to calculate the evolution of the total magnetisation of a spin system. The calculations were performed using the machinery of quantum mechanics and meaningful and useful results were thereby obtained. In this chapter we will look at things from a rather different point of view. We will not, initially, be concerned with the expectation values of certain specified observables, rather we direct our attention to the general description of the ‘state’ of the system and the way it evolves with time – possibly towards thermal equilibrium.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrodinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.