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International Journal of Quantum Chemistry

Exact solution to a general quantum mechanical problem with time-dependent boundary conditions

Rafael Almeida, Daniel A. Morales

Published Jan 1, 1997• 4 citations
Publisher

Abstract

Exact analytical solutions are found for the quantum mechanical problem of a particle subject to a time-dependent potential with time-dependent boundary conditions. The method of solution employs time-dependent invariants, rescaling of space and time variables along with an unitary transformation. Several applications are discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 827–833, 1997

DOI
10.1002/(sici)1097-461x(1997)63:4<827::aid-qua3>3.0.co;2-y

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Exact analytical solutions are found for the quantum mechanical problem of a particle subject to a time-dependent potential with time-dependent boundary conditions. The method of solution employs time-dependent invariants, rescaling of space and time variables along with an unitary transformation. Several applications are discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 827–833, 1997
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.