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Physical Review C

Analytical approach to the wave function of a decaying quantum system

W. van Dijk, Yukihisa Nogami

Published Jan 16, 2002• 40 citations
Publisher

Abstract

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 $\mathbf{S}$ 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.

DOI
10.1103/physrevc.65.024608

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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 \ensuremathα\ensuremath{\alpha}\ensuremathα-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.
We report on a novel, practical method of determining the nonrelativistic wave function of a decaying quantum system at all positions and times. The system consists of a particle which is initially confined around the origin and leaks out tunneling through a potential barrier. We show that the wave function $\ensuremath{\psi}(r,t)$ of the particle can be expressed as a linear combination of Moshinsky functions, each of which is associated with a pole of the scattering S\mathbf{S}S matrix of the system. The $\ensuremath{\psi}(r,t)$ so obtained is normalizable. In the light of this approach we reexamine the Gamow wave function, which is not normalizable. We elucidate the source of this difficulty.
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.
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.
In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.