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Chinese Journal of Physics

The Effective Hamiltonian and Ehrenfest Theorem in the Propagation Dynamics of Nonspreading Wave Packets

Chyi-Lung Lin

Published Oct 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.6122/cjp.20140504

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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 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 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.
Berry and Balazs showed that an initial Airy packet Ai(b x) under time evolution is nonspreading in free space and also in a homogeneous time-varying linear potential V(x,t)=-F(t) x. We find both results can be derived from the time evolution operator U(t). We show that U(t) can be decomposed into ordered product of operators and is essentially a shift operator in x; hence, Airy packets evolve without distortion. By writing the Hamiltonian H as H=H_b+H_i, where H_b is the Hamiltonian such that Ai(b x) is its eigenfunction. Then, H_i is shown to be as an interacting Hamiltonian that causes the Airy packet into an accelerated motion of which the acceleration a=(-H_i/( x))/m. Nonspreading Airy packet then acts as a classical particle of mass m, and the motion of it can be described classically by H_i.
We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrödinger’s equation for general F (t) and discuss the motion for some special forms of F (t).
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.