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American Journal of Physics

Initial phase and free-particle wave packet evolution

T. L. Beach

Published May 13, 2009• 6 citations
Publisher

Abstract

The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.

DOI
10.1119/1.3089531

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The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.
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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.
Abstract The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.
The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase-space metric---a phenomenon having no analog in Hermitian theories.
We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.