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Physica Scripta

Exact solutions of time-dependent Dirac equations and the quantum–classical correspondence

Zhiguo Zhang

Published Jul 19, 2006• 3 citations
Publisher

Abstract

Exact solutions to the Dirac equations with a time-dependent mass and a static magnetic field or a time-dependent linear potential are given. Matrix elements of the coordinate, momentum and velocity operator are calculated. In the large quantum number limit, these matrix elements give the classical solution.
DOI
10.1088/0031-8949/74/2/012

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According to Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Applying HCP to Dirac equations in relativistic realm,for the relativistic free particle or the charged particle moving in a constant magnetic field,it is shown that quantum operators in Dirac theory reduces to the classical quantities in the classical limit and the classical limit of quantum expectation value is the time average of the classical quantity.
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The Dirac equation has been studied in which the Dirac matrices $\\hat{\\boldmath$\\alpha$}, \\hat\\beta$ have space factors, respectively fff and f1f_1f1​, dependent on the particle's space coordinates. The fff function deforms Heisenberg algebra for the coordinates and momenta operators, the function f1f_1f1​ being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schr\\"odinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomnb field with a linear dependence of the fff function on the distance rrr to the force centre and the inverse dependence on rrr for the f1f_1f1​ function has been found.
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.