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Physical Review

Evolution of a Quasi-Stationary State

R. G. Winter

Published Aug 15, 1961• 182 citations
Publisher

Abstract

To elucidate the time development of quasi-stationary states, a simple barrier penetration problem has been studied. Both approximate expressions and numerical results for some parameters were obtained for the decay rate. First, irregular oscillations occur for a short time. Second, the exponential region follows. Third, further oscillations occur during which the decay rate dips to negative values, so that the probability of finding the undecayed system increases briefly at several times. Fourth and finally, the decay rate decreases like an inverse power of the time.

DOI
10.1103/physrev.123.1503

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To elucidate the time development of quasi-stationary states, a simple barrier penetration problem has been studied. Both approximate expressions and numerical results for some parameters were obtained for the decay rate. First, irregular oscillations occur for a short time. Second, the exponential region follows. Third, further oscillations occur during which the decay rate dips to negative values, so that the probability of finding the undecayed system increases briefly at several times. Fourth and finally, the decay rate decreases like an inverse power of the time.
Approximately 40 years ago it was realized that the time development of decaying systems might not be precisely exponential. Winter [Phys. Rev. 123, 1503 (1961)] analyzed the simplest nontrivial system---a particle tunneling out of a well formed by a wall and $\ensuremath{\delta}$ function. He calculated the probability current just outside the well and found irregular oscillations on a short-time scale followed by an exponential decrease followed by more oscillations and finally by a decrease as a power of the time. We have reanalyzed this system, concentrating on the survival probability of the particle in the well rather than the probability current, and find a different short-time behavior.
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.
The tunneling dynamics of the spin-boson problem has been computed using discretized path-integral simulations for temperatures T and couplings, i.e., the Kondo parameter \ensuremath{\alpha}, spanning the entire T-\ensuremath{\alpha} plane. The inherent problem of alternating weights has been solved using a combination of the stationary-phase Monte Carlo method and contour-distortion techniques. A transition from coherent to purely incoherent relaxation was observed for the spin correlation function. The time correlation functions and the location of the coherent-incoherent boundary on the T-\ensuremath{\alpha} plane are well described by the noninteracting-blip approximation. In the deep-tunneling limit of large \ensuremath{\alpha}, low T, and high bath frequency, the system relaxes exponentially, with its relaxation time constant following a power-law temperature dependence, in accord with perturbation theory. At higher T and low bath frequency, the relaxation time crosses over to a classical Arrhenius temperature dependence, reflecting the onset of activated processes. For a narrow region within 1/21, numerical results suggest that the system undergoes incoherent relaxation, with a short-time exponential decay, followed by a long-time tail of the power-law type. The short-time exponential relaxation time follows a peculiar power-law temperature dependence, with the relaxation rate increasing as a function of decreasing temperature.
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.