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Physical Review A

Invariant theory and exact solutions for the quantum Dirac field in a time-dependent spatially homogeneous electric field

Xiao-Chun Gao, Jian Bo Fu, Jinbo Xu, Xu‐Bo Zou

Published Jan 1, 1999• 7 citations
Publisher

Abstract

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of the quantum Dirac field in a time-dependent spatially homogeneous electric field. We solve the functional Schr\"odinger equation for the Dirac field and obtain the exact solutions and corresponding total phase. The total phase includes both the dynamical phase and geometric phase (Aharonov-Anandan phase).

DOI
10.1103/physreva.59.55

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On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of the quantum Dirac field in a time-dependent spatially homogeneous electric field. We solve the functional Schr"odinger equation for the Dirac field and obtain the exact solutions and corresponding total phase. The total phase includes both the dynamical phase and geometric phase (Aharonov-Anandan phase).
On the basis of the invariant formulation, we find the quantum and classical exact solutions and corresponding total phase for the relativistic charged Klein-Gordon (KG) field in a time-dependent spatially homogeneous electric field. The total phase includes both the dynamical and geometric phases (Aharonov-Anandan phases). The connection between the quantum and classical solutions is then obtained. From this connection we discuss the condition under which the geometric phase for the KG field can be defined.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms.They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture.These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality.For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states.This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations.These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman's work on path integrals).They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier "paradoxes" (such as the van Kampen thought-experiment), and to also complete Peshkin's discussion of the electric Aharonov-Bohm effect in a causal manner.The present formulation offers a direct way to address time-dependent singlevs double-slit experiments and the associated causal issues-issues that have recently attracted attention, with respect to the inability of current theories to address them.