Spin Force and Torque in Non-relativistic Dirac Oscillator on a Sphere

TL;DR

This paper derives the spin force and torque operators in a non-relativistic Dirac oscillator confined to a sphere, revealing their effects on angular momentum and linking them to phenomena like Zitterbewegung and spin Hall effect.

Contribution

It introduces explicit calculations of spin force and torque in a non-relativistic Dirac oscillator on a sphere, highlighting anomalous parts and their physical implications.

Findings

Spin force operator vanishes when confined to a sphere.

Spin torque equals the rate of change of kinetic orbital angular momentum.

Anomalous spin parts induce oscillatory behavior similar to Zitterbewegung.

Abstract

The spin force operator on a non-relativistic Dirac oscillator ( in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin-orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle . When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggest that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the \textit{Zitterbewegung}. It is suggested that the underlying physics that gives rise to the spin force and the \textit{Zitterbewegung} is one and the same in NRDO and in systems that manifest spin Hall effect.