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Efficient numerical methods for solving the Schrödinger equation with a potential varying sinusoidally with time
Author(s) -
Hagmann Mark J.
Publication year - 1995
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560560832
Subject(s) - resonance (particle physics) , floquet theory , physics , schrödinger equation , modulation (music) , wave packet , dimensionless quantity , quantum mechanics , quantum tunnelling , amplitude , mathematical analysis , mathematics , nonlinear system , acoustics
Abstract We have used several different methods to solve the one‐dimensional, time‐dependent Schrödinger equation for a sinusiodally modulated barrier. Analytical solutions are given for the case in which the time‐dependent part of the potential has several different forms, but is independent of position. For more general fields, Floquet's theorem is used to write the wavefunction as a summation of components which have different energies as a result of the absorption and emission of modulation quanta. A system of coupled ordinary differential equations is obtained, which is then solved numerically using shooting methods. The examples show a resonance in which the tunneling current is markedly increased. For square barriers, this resonance occurs when the particles absorbing modulation quanta are above the barrier, and the length of the barrier is an integer multiple of one‐half the de Broglie wavelength. The existence of the resonance is confirmed by asymptotic solutions for large and small frequencies. Examples suggest that it may be possible to make a microwave power amplifier by illuminating a field emitter array with an amplitude‐modulated laser operating near the new resonance. © 1995 John Wiley & Sons, Inc.