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The Journal of Chemical Physics

Expansion of the Time-Dependent Wavefunction in Quasistationary Energy Eigenfunctions

Robert B. Frey, Everett Thiele

Published Apr 1, 1968• 7 citations
Publisher

Abstract

The motion of a wave packet moving in a potential well with a penetrable barrier is considered. A formally exact expression for the evolution operator is given as a sum over quasistationary energy eigenfunctions plus a remainder term. Numerical calculations are carried out for the square barrier potential. The approximation in which one neglects the remainder term is shown to be a useful and accurate one in computing properties of the particle which depend on the value of the wavefunction inside the potential well.

DOI
10.1063/1.1669597

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The motion of a wave packet moving in a potential well with a penetrable barrier is considered. A formally exact expression for the evolution operator is given as a sum over quasistationary energy eigenfunctions plus a remainder term. Numerical calculations are carried out for the square barrier potential. The approximation in which one neglects the remainder term is shown to be a useful and accurate one in computing properties of the particle which depend on the value of the wavefunction inside the potential well.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
The sum of a one-dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half-space, serves as a convenient and instructive example for which the Schrödinger equation may have both bound-state and continuum solutions. A superposition of these solutions is used in a study of barrier penetration by a wave packet representing a particle with an initial position in the region of the potential well. The presence or absence of a bound state in the superposition is shown to be the key factor determining the evolution of the wave packet. If no bound state exists, the probability of the particle having a position within the potential well is a monotonically decreasing function of time. If the superposition includes a bound state, however, this probability oscillates slightly because of an interference between the bound-state and continuum components of the wavefunction.
To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.
To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
A simple exact solution for the spacetime particle propagator in the presence of a one-dimensional rectangular potential is obtained. This propagator solves the Cauchy problem for the corresponding time-dependent Schrodinger equation in terms of the forward-moving (positive momentum) and backward-moving (negative momentum) components of the wavefunction. The exact formulae for the time-dependent wavefunction in different spatial regions (before, inside and after the barrier) are found. These results demonstrate that generally the forward- and backward-moving components of the wavefunction as well as their interference contribute to the probability density of finding a particle in different spacetime regions. The time of arrival to and the dwell time in the above-mentioned spatial regions are also obtained. The relative contribution of the positive- and negative-momentum components of the wavefunction as well as those of the below- and above-barrier energy regions are discussed for the case of an initial Gaussian wave packet. Numerical visualization of the probability density to find a particle in the different spacetime regions is performed and discussed in the tunneling regime.