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Physica Scripta

Exact solution of Schrödinger equation for time-dependent ultra-short barrier

Swati Mudra, Aniruddha Chakraborty

Published Jul 9, 2019• 1 citation
Publisher

Abstract

Abstract In this paper, the time-dependent Schrödinger equation for an ultra-short barrier has been solved. Here, the height of the barrier has been taken as time-dependent with different possible cases. A new mathematical technique using Green’s function method is presented. These mathematical calculations are purely analytical and give exact results.

DOI
10.1088/1402-4896/ab307d

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Abstract In this paper, the time-dependent Schrödinger equation for an ultra-short barrier has been solved. Here, the height of the barrier has been taken as time-dependent with different possible cases. A new mathematical technique using Green’s function method is presented. These mathematical calculations are purely analytical and give exact results.
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A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.