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arXiv:2205.14670 · Physical Review C

Time evolution of a decaying quantum state: Evaluation and onset of nonexponential decay

Markus Nöth

Published Feb 15, 2023• 0 citations
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Abstract

The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.

DOI
10.1103/physrevc.107.l021602

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The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.
An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the sss-wave $\ensuremath{\delta}$ shell potential model.
We examine a one-dimensional model of a decaying quantum system, e.g., one that could simulate decay. A particle is initially confined in a region and leaks out, tunnelling through a potential barrier. The same model has been examined before, primarily to study decay properties which are determined by the wavefunction of the particle within the potential barrier. However, the wavefunction of this model (and of any similar models) outside the potential barrier has not been obtained explicitly as a function of position and time. We obtain the wavefunction outside as well as inside the potential barrier by solving the time-dependent Schrödinger equation, and explore various features of the space-time development of the system. In an early study by Winter it was pointed out that the probability current just outside the barrier, after a long time, may fluctuate and can be negative, i.e., inward. We find that under certain circumstances the amplitude of the fluctuations increases significantly as the distance from the barrier increases. We propose a simple approximate wavefunction that works well when the decay is very slow. The Gamow wavefunction, commonly used to describe the -decay process, is not appropriate at large distances. Our wavefunction can be used anywhere and is normalized in the usual manner.
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Recently we outlined an analytical method of solving the time-dependent Schr\"odinger equation for a model which simulates a decaying quantum system such as an $\ensuremath{\alpha}$-decaying nucleus. A particle in the model is initially confined around the origin and leaks out, tunneling through a potential barrier. The solution can be expressed as a linear combination of the Moshinsky functions, each of which is associated with a pole of the scattering S\mathbf{S}S matrix of the model. In this paper we give a full account of the method with a few explicit examples. We examine deviations from the exponential decay law at very large times. We comment on a recent controversy regarding the t dependence of the survival and nonescape probabilities when t is very large.
It is shown that a representation of the decaying wave function as a resonant sum plus a nonexponential integral term may be written as a purely discrete resonant sum by evaluating at long times the integral term by the steepest descents method, and then expanding the resulting expression in terms of resonant states. This leads to a representation that is valid along the exponential and the inverse power in time regimes. A model calculation using the $\ensuremath{\delta}$ potential allows us to make a comparison of the expansion with numerical integrations in terms of continuum wave functions and, in the long time regime, with an exact analytic expression of the integral term obtained using the steepest descents method. The results demonstrate that resonant states give a correct description of the long-time behavior of decay.
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.