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arXiv:1208.0845

Study the dynamics of the nonspreading Airy packets from the time evolution operator

Chyi-Lung Lin, Te-Chih Hsiung, Meng-Jie Huang

Published Aug 3, 2012• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1208.0845

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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 present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.
We show that for Hamiltonian of the form, H = c ( t )∂ x ∂ x + V ( x , t ), the general potential V ( x , t ) for which Airy packets remain nonspreading under time evolution can only be of the form V ( x , t )= p ( t ) x + q ( t ), where p ( t ) and q ( t ) are real functions. We derive rules for constructing nonspreading Airy packets from this general potential. We also consider normalizable packets obtained from Airy packet times a converging factor, such as e β x , with β> 0. We show that there is no potential which sustains nonspreading wave packets of this kind.
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).
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.