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Journal of Physics A Mathematical and General

The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator

C M Cheng, P. C. W. Fung

Published Nov 21, 1988• 97 citations
Publisher

Abstract

The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a 1 (t)J 4 +a 2 (t)J 0 +a 3 (t) J - where J + , J 0 and J - are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.

DOI
10.1088/0305-4470/21/22/015

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The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a 1 (t)J 4 +a 2 (t)J 0 +a 3 (t) J - where J + , J 0 and J - are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.
In a previous paper the authors have employed a group theoretic method to find the evolution operator for a quantum system having a SU(2) Hamiltonian. In this paper, they consider a more complicated system whose Hamiltonian consists of SU(2) and h(4) group generators. A transformation method will be introduced, in conjunction with the above group theoretic method, to tackle this quantum problem. The result thus obtained will be applied to the problem of a mass-varying harmonic oscillator under an external force. Their result shows that the equation of motion of this oscillator is identical to a damped harmonic oscillator under an external force. In addition, it is shown that an initial coherent state will evolve as a squeezed state under the above Hamiltonian.
Recent papers by Cheng and Fung (1988) and Lo (1990) have studied the generation of squeezed states out of coherent states in systems described by some time-dependent Hamiltonians. They have employed the evolution operator method for the Schrodinger equation with a Hamiltonian expressible as H(t)=a l (t)J + +a 2 (t)J 0 + +a 3 (t)J - , where J +- ,J 0 are the SU(2) group generators. Here, for the sake of comparison of methods and results, the present authors employ the normal ordered Hamiltonian H(t)= Sigma h l,m (t)(a) l a m and investigate squeezing through an alternative approach using the normal ordering technique to solve the evolution operator for the same system.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.
We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For simplicity, we consider a free particle with a variable mass M(t), a particle with a variable mass M(t) in an alternating homogeneous field, and a harmonic oscillator with a variable mass M(t) and frequency ω(t) subject to a variable force F(t). To construct the evolution operators for these systems in an explicit disentangled form, we use a simple technique to find the general solution of a certain class of differential and finite-difference nonstationary Schrödinger-type equations of motion and also the operator identities of the Baker–Campbell–Hausdorff type. With known evolution operators, we can easily find the most general form of the propagators, invariants of any order, and wave functions and establish a unitary relation between systems. Results known in the literature follow from the obtained general results as particular cases.