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Computers in Physics

Time-dependent quantum-mechanical scattering in two dimensions

Christopher J. Sweeney, Paul L. De Vries

Published Jan 1, 1989• 4 citations

Abstract

While scattering phenomena are discussed in most quantum mechanics textbooks, the analysis is usually performed using time-independent techniques. These techniques have certainly been successful in predicting the results of scattering events but fail to present an intuitive picture of what happens to the physical system during a collision. A remedy to this situation could be affected by the introduction of time-dependent techniques, utilizing the conceptually simple idea of a wavefunction evolving over time, but these methods have traditionally been computationally unmanageable. This paper describes a novel numerical technique, the split-operator Fourier transform method, which greatly reduces the amount of computational effort required to accurately solve the time-dependent Schrödinger equation. This method is applied to the Rutherford scattering problem in two dimensions and it is shown that the method is not only efficient enough to yield results in less than 15 min when implemented on a standard personal computer, but is also simple enough to be employed by an advanced undergraduate. The method also has the advantage of reinforcing the quantum-mechanical formalism in the student’s mind; both the evolution operator and the complementary relationship between operators and wavefunctions in coordinate and momentum space are exploited. The results of the calculations are presented as color graphs portraying the magnitude and phase of wave packets as they evolve through time.

DOI: 10.1063/1.168336

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While scattering phenomena are discussed in most quantum mechanics textbooks, the analysis is usually performed using time-independent techniques. These techniques have certainly been successful in predicting the results of scattering events but fail to present an intuitive picture of what happens to the physical system during a collision. A remedy to this situation could be affected by the introduction of time-dependent techniques, utilizing the conceptually simple idea of a wavefunction evolving over time, but these methods have traditionally been computationally unmanageable. This paper describes a novel numerical technique, the split-operator Fourier transform method, which greatly reduces the amount of computational effort required to accurately solve the time-dependent Schrödinger equation. This method is applied to the Rutherford scattering problem in two dimensions and it is shown that the method is not only efficient enough to yield results in less than 15 min when implemented on a standard personal computer, but is also simple enough to be employed by an advanced undergraduate. The method also has the advantage of reinforcing the quantum-mechanical formalism in the student’s mind; both the evolution operator and the complementary relationship between operators and wavefunctions in coordinate and momentum space are exploited. The results of the calculations are presented as color graphs portraying the magnitude and phase of wave packets as they evolve through time.

We present a numerical Fourier-transform method for solving the time-dependent Schrödinger equation in spherical coordinates. The time propagation is accomplished with a novel implementation of a Crank–Nicholson scheme combined with a Feit–Fleck split-operator method. Calculations on model problems give results that are in excellent agreement with those obtained by other methods. The procedure avoids the numerical instabilities associated with the division by the sine of the polar angle. The propagator is unitary, does not require the calculation of any matrix elements, and uses no matrix multiplications other than fast Fourier transforms.

A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.

A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed.

Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it against two well-known numerical methods: the Runge-Kutta method and the Split-Step Fourier method. Quantum tunneling is selected as the case study due to its relevance in quantum physics applications. Comparative analysis is based on numerical error metrics. The proposed method can be applied to other physical phenomena, expanding its potential application in physics research and technology industries.

Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.

Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.