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arXiv:1306.1311

The effective Hamiltonian which governs the propagation dynamics of nonspreading wave packets

Chyi-Lung Lin

Published Jun 6, 2013• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1306.1311

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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.
We analyze the propagation dynamics of nonspreading wave packets by decomposing the Hamiltonian H into two parts: H = □(t) + H_c(t). The first part □(t) is such that Ψ(x, t) is its instantaneous eigenstate, and is therefore irrelevant to the propagation of the packet. The second part H_c(t) is shown to be the effective Hamiltonian governing the propagation dynamics of nonspreading wave packets both quantum mechanically and classically. The Ehrenfest theorem is thus based on the Hamiltonian H_c(t) connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets, and explains why Airy packets can self-accelerate in free space.
We show the method for constructing nonspreading wave packets whose shape and motion can be general. We analyze the time evolution of nonspreading wave packets by decomposing the Hamiltonian into two parts. Of the two, one changes the instantaneous state, the other does not. Through this decomposition, the time evolution operator is shown to be effectively a spatial shifting operator. This explains why nonspreading wave packets can be nonspreading. And we show that the part of the Hamiltonian which changes the instantaneous state governs the motion of the nonspreading wave packets.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.
The spreading or nonspreading of wave packets in one-dimensional linear and quadratic potentials can readily be determined by solving Ehrenfest’s equations. The results for quadratic potentials can be understood in simple intuitive terms by using quantum distribution functions in phase space which, like classical distributions, rotate clocklike in time once per motional cycle. For more complicated situations, such as potentials with quantum energy spacings which are integral multiples of a basic unit, much can be learned instead by studying the quantum recurrence of their time-dependent wave functions or probability densities.
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.