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American Journal of Physics

Illustration of resonances and the law of exponential decay in a simple quantum-mechanical problem

Herbert Massmann

Published Jul 1, 1985• 15 citations
Publisher

Abstract

The exact quantum-mechanical solution of a particle moving in a one-dimensional well formed by a delta barrier in front of an impenetrable well is used to illustrate the relation between scattering resonances and exponential decay from the well.
DOI
10.1119/1.14284

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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.
A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential V(x)V(x)V(x) and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.
Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert WWW function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation ΓT=TtransΓT = T_{\text{trans}}ΓT=Ttrans​ between the decay rate ΓΓΓ, oscillation period TTT, and transmission probability TtransT_{\text{trans}}Ttrans​, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.