Efficient numerical method for finding the initial response of quantum processes to changes in the potential

Abstract

The time-dependent Schrödinger equation is solved for an electron with potentials of the form f1(r) + f2(r, t)U(t), where U(t) is the unit step function. We use a product formulation, solving for F(r, t) where the wave function Ψ(r, t) = F(r, t)Φ(r)e-iEt/h in which Φ(r)e-iEt/h is the solution for t < 0. A simple implementation of the product formulation that does not use absorbing boundary conditions and is explicit, without using split operators, is applied in two examples. The first example pertains to resonances in tunneling with square barriers when the barrier height varies sinusoidally with time. The initial response of quantum tunneling to oscillations in the barrier height shows a buildup for oscillations at the resonance frequency and an off-resonance response that diminishes to approach the steady-state solution after several cycles. The second example is the initial response of a hydrogen atom to an intense applied electric field. In both examples the response is delayed, and for tunneling particles the delay is approximately equal to the semiclassical value for the duration of barrier traversal, defined as the time for traversing the inverted barrier. © 1996 John Wiley & Sons, Inc.