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Journal of Physics A Mathematical and General

A simple model of a decaying quantum mechanical state

A Ludviksson

Published Oct 1, 1987• 30 citations
Publisher

Abstract

A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.

DOI
10.1088/0305-4470/20/14/017

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A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.
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.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.
Abstract An exact analytical solution to the non-stationary Schrödinger equation is found for a particle confined in a short-range potential in a spatially uniform, time-dependent field. The solution is used to calculate the decay probability of an unstable quantum state initially created in the potential well of a double-barrier system. An exact analytical expression for the decay probability is obtained without assuming weak laser field intensity. Calculations show that, for certain sets of parameters, the entire decay curve oscillates significantly. The physical meaning of the oscillations is revealed, and an estimate of the oscillation period is given. The behavior of the decay probability at short and long times is discussed. If the barrier height tends to zero, then at long times, the decay law is inversely proportional to time. The number of non-decaying particles can be varied over a wide range by changing the length from the emission point to the detector that records the particles.
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.
Total Absorption in Finite Time in an iδiδiδ Potential
We consider the evolution of Green's function of the one-dimensional Schrödinger equation in the presence of the complex potential −ikδ(x)-ikδ(x)−ikδ(x). Our result is the construction of an explicit time-dependent solution which we use to calculate the time-dependent survival probability of a quantum particle. The survival probability decays to zero in finite time, which means that the complex delta potential well is a total absorber for quantum particles. This potential can be interpreted as a killing measure with infinite killing rate concentrated at the origin.
We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.