Abstract
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 Schrodinger 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.
Highlights
The law of exponential decay is typically discussed in association with atomic transitions or resonances in scattering amplitudes
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
On Khalfin [1] used dispersion relations to show that even quasi-stationary states with spectral functions that have a lower bound in their energy spectrum must decay non-exponentially at large times
Summary
The law of exponential decay is typically discussed in association with atomic transitions or resonances in scattering amplitudes. Winter [2] examined the infinite wall plus repulsive delta function potential and obtained a single implicit solution in the form of an integral for the special case in which the initial wavefunction is an eigenfunction of the infinite square well of the same width and as a result it is a near-resonance (quasi-stationary) state of the actual potential His analytic approximation to the integral in the limit of low barrier transmittance (large strength of the delta function) proved that the survival probability exhibits exponential decay in the (intermediate) time interval-when the dominant quasi-stationary resonance prevails inside the well-while at very large times it decays following the power law t 3. Numerical solutions are developed for finite-range potentials and shown to exhibit a rich, non-exponential decay behavior, including oscillations
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