Relativistic Schrödinger equation including spin-orbit interaction

N. J. Ionesco‐Pallas

William Hodge, S. V. Migirditch, W. C. Kerr

This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time-dependent Schrödinger equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrödinger equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus.

N. I. Ionescu-Pallas

M. M. Moya, J. S. Helman

We consider the problem of an electron interacting with a surface potential barrier which includes the spin-orbit interaction. An exact solution of the Schrödinger equation is given for the case of a step potential. For smooth barrier profiles, the time- dependent Schrödinger equation is numerically solved, and the time evolution of the spin-up and spin-down components of a wave packet is displayed in a series of pictures.

Frank V. Kowalski

Yasushi Takahashi, C. Ropchan

The consequences of reducing the Schrödinger equation to two first order equations are examined in detail. Explicit calculations of the symmetrization of the stress tensors are given. It is pointed out that the spin-orbit coupling emerges in this formalism without stepping into the relativistic regime.