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Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics

The quasiclassical approximation to Dirac equation. I

Kenji Yajima

Published Mar 31, 1982• 6 citations
Publisher

Abstract

The mathematical foundation to the quasi-classical approximation to the Dirac equation is discussed. After constructing the propagator $U^\kappa(t,s)$ and the quasi-classical propagator for the Dirac equation in an external electro-magnetic field, the usual quasi-classical approximation to the wave function $U^\kappa(t,s)(\exp (iS(x)/h)f)$ is obtained with error estimates in the Hilbert $L^2(R^3,C^4)$.

DOI
10.15083/00039586

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The mathematical foundation to the quasi-classical approximation to the Dirac equation is discussed. After constructing the propagator Uκ(t,s)U^\kappa(t,s)Uκ(t,s) and the quasi-classical propagator for the Dirac equation in an external electro-magnetic field, the usual quasi-classical approximation to the wave function Uκ(t,s)(exp⁡(iS(x)/h)f)U^\kappa(t,s)(\exp (iS(x)/h)f)Uκ(t,s)(exp(iS(x)/h)f) is obtained with error estimates in the Hilbert L2(R3,C4)L^2(R^3,C^4)L2(R3,C4).
The quasi-classical spin theory is a modified Hamilton-Jacobi theory for a classical relativistic dipole whose space-time trajectory is coupled to its spin motion. The equations of this quasi-classical theory are almost identical with the second-order (squared) Dirac equation, and, in fact, form a new WKB approximation to the second-order Dirac equation. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single valued, and that it satisfy an eigenvalue equation. We find for a particle in a uniform magnetic field, and in a Coulomb field, that this quasi-classical theory predicts the same energy and angular momentum eigenvalues as the Dirac theory. The quasi-classical theory is invariant under Lorentz transformation, spin rotations, and charge conjugation. The spinor wave functions are form invariant with respect to arbitrary canonical transformations.
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An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.
This chapter considers an approximate method of solving the Schrodinger equation in the case when the wavelength of the particle is small by comparison with the distance over which the potential changes appreciably. Since the limit of short wavelength corresponds to the limit of classical mechanics, the quasiclassical approximation allows to explore the relation between classical and quantum mechanics. The chapter discusses how to match the quasiclassical wave function f in the region accessible to classical motion with the wave function in the forbidden region. Since the action function is a minimum along the classical trajectory, the pencil of trajectories adjacent to the classical one gives the minimum phase factor. The chapter examines the problem of the escape of a particle from a potential barrier; this problem reduces to the problem of the spreading of a wave packet which describes a particle which at the initial time was inside the barrier.