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International Journal of Quantum Chemistry

Efficient numerical methods for solving the Schr�dinger equation with a potential varying sinusoidally with time

Mark J. Hagmann

Published Feb 25, 1995• 9 citations
Publisher

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.

DOI
10.1002/qua.560560832

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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.
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.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the tunneling current. For square barriers the resonance occurs when the tunneling particles absorb modulation quanta and the length of the barrier is a multiple of one-half de Broglie wavelengths. The resonance has a similar mechanism with triangular barriers. However, the relationship is more complex because the absorption and emission of modulation quanta takes place throughout the full length of the barrier, whereas this exchange only occurs at the ends of a square barrier.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
The authors present a variety of theoretical methods to treat the general problem of a quantum particle interacting with a time-dependent barrier. Assuming a sinusoidal form for the time dependence, they discuss the classical, semiclassical and quantum solutions for two specific problems: a barrier whose amplitude is modulated, and one whose mean position changes. While the classical solutions to these problems are quite different, the time-independent quantum solutions appear almost identical. It is only when the particle is modelled as a wavepacket that the quantum solutions behave like their classical counterparts.
Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.