Probability Current in Hydrogen with Spin-Orbit Interaction

Seogjoo Jang

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.

Federico Bottegoni, H.-J. Drouhin, G. Fishman +1 more

We propose a procedure, based on momentum power series expansion Hamiltonian of nth general order, to obtain a coherent expression of the probability current operator that is valid also when Spin-Orbit Interaction (SOI) terms are included. We prove that we recover the standard definition when the free electron-like term is included in the Hamiltonian, but when taking into account higher order Spin-Orbit Interaction (SOI) terms, a more general definition of the probability current operator is mandatory, due to its different symmetrization, compared to the Hermitian velocity operator expression.

V. V. Dixit, L. P. S. Singh

Arjun Sharma

The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.

Victor M. Galitski, Ian B. Spielman