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Physical Review A

Invariant formulation and exact solutions for the relativistic charged Klein-Gordon field in a time-dependent spatially homogeneous electric field

Xiao-Chun Gao, Jian Bo Fu, Xin‐Hua Li, Jun Gao

Published Feb 1, 1998• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.57.753

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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.
The structure of additional electromagnetic fields to the Aharonov–Bohm field, for which the Schrödinger, Klein-Gordon, and Dirac equations can be solved exactly are described and the corresponding exact solutions are found. It is demonstrated that aside from the known cases (a constant and uniform magnetic field that is parallel to the Aharonov–Bohm solenoid, a static spherically symmetrical electric field, and the field of a magnetic monopole), there are broad classes of additional fields. Among these new additional fields we have physically interesting electric fields acting during a finite time or localized in a restricted region of space. There are additional time-dependent uniform and isotropic electric fields that allow exact solutions of the Schrödinger equation. In the relativistic case there are additional electric fields propagating along the Aharonov–Bohm solenoid with arbitrary electric pulse shape.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.