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International Journal of Modern Physics B

TIME-DEPENDENT EVOLUTION OF A WAVE PACKET IN QUANTUM SYSTEMS

Chi‐Shung Tang, Pi‐Gang Luan

Published Sep 30, 2008• 0 citations
Publisher

Abstract

We consider wave packet propagation in mesoscopic quantum systems. A number of approaches are compared to look at the general solution of a time-dependent Schrödinger equation and the validity of the Ehrenfest theorem. Detailed calculations are presented to illustrate the results of a charged particle motion in the time-dependent systems, and show that the Ehrenfest theorem is not directly applicable in topologically nontrivial systems.

DOI
10.1142/s0217979208048887

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We consider wave packet propagation in mesoscopic quantum systems. A number of approaches are compared to look at the general solution of a time-dependent Schrödinger equation and the validity of the Ehrenfest theorem. Detailed calculations are presented to illustrate the results of a charged particle motion in the time-dependent systems, and show that the Ehrenfest theorem is not directly applicable in topologically nontrivial systems.
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.
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.
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.
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.