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Journal of Physics A Mathematical and General

A general procedure for solving the time-dependent Schr$ouml$dinger equation

Chuyi Chen

Published Jul 25, 2002• 1 citation
Publisher

Abstract

A nonperturbative procedure for solving the time-dependent Schrödinger equation, called the multi-projection approach or phase dynamics of quantum mechanics, is derived and illustrated. With the help of this new procedure, time-dependent quantum systems become generally solvable (if corresponding stationary systems are solvable) and the phase character of quantum mechanics reveals itself very clearly.

DOI
10.1088/0305-4470/35/31/305

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A general procedure for solving the time-dependent Schroumloumloumldinger equation
A nonperturbative procedure for solving the time-dependent Schrödinger equation, called the multi-projection approach or phase dynamics of quantum mechanics, is derived and illustrated. With the help of this new procedure, time-dependent quantum systems become generally solvable (if corresponding stationary systems are solvable) and the phase character of quantum mechanics reveals itself very clearly.
Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.