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European Journal of Physics

Numerical modelling of explicitly time-dependent quantum systems with δ-potentials

P. S. Antsiferov

Published Oct 26, 2001• 0 citations
Publisher

Abstract

This paper can serve as an illustration for a course on quantum mechanics. It gives a non-trivial example of the time evolution of a system, which includes a particle in the potential of two δ-traps: one of which is in the fixed position, and the other moves with constant velocity. The numerical solution of the Schrödinger equation represents the behaviour of the vector of state in quite a transparent way; one can see rough pictures of the processes of charge transfer and ionization. The numerical procedure is explained, and an exercise can be carried out in constructing other time-dependent systems, containing δ-potentials.

DOI
10.1088/0143-0807/22/6/307

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This paper can serve as an illustration for a course on quantum mechanics. It gives a non-trivial example of the time evolution of a system, which includes a particle in the potential of two δ-traps: one of which is in the fixed position, and the other moves with constant velocity. The numerical solution of the Schrödinger equation represents the behaviour of the vector of state in quite a transparent way; one can see rough pictures of the processes of charge transfer and ionization. The numerical procedure is explained, and an exercise can be carried out in constructing other time-dependent systems, containing δ-potentials.
We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.
The time-dependent Schrodinger equation for a particle in one dimension in the field of two colliding delta potentials can be solved exactly. Assuming an initial state which is bound about the left delta potential for large negative times, the solution has been used to calculate the probabilities of finding a state bound about one of the potentials long after the collision. These probabilities are oscillating functions of the reciprocal velocity. The ionisation probability goes to zero for high collision velocities and reaches 1 / 2 for small velocities.
In this paper we numerically solve the time dependent Schrödinger equation for scenarios using wave packets. These examples include the free wave packet, which we use to show the difference between group and phase velocities, the packet in a harmonic oscillator potential with non-trivial initial conditions in one and two dimensions, which is compared with their classical analogs to show how Ehrenfest theorem holds. We also include simulations of the diffraction through the single and double slit potentials, the refraction with a step potential and the dispersion by a central potential. The aim of this paper is to illustrate with simulations, nowadays easy to implement, scenarios that can help explaining the basics of the wave-particle duality.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
A quantum one-particle model for electron delocalization and transfer induced by a time-dependent potential is studied with a formalism based on the Lippman-Schwinger approach and fractional as well as operational calculus methods. The potential examined consists of a line of an arbitrary number of short-range (\ensuremath{\delta} function) potentials whose amplitudes have an arbitrary functional dependence on time. The temporal evolution of the ionization probability, the energy spectrum, and the time of transfer are obtained analytically and calculated numerically through a stable and accuracy-controllable technique without any assumption regarding the strength of the perturbing potential.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.