A unit vector for characterizing the spin polarization of free electron

TL;DR

This paper introduces a new unit vector-based method to characterize free electron spin polarization, leveraging the azimuthal angle of an auxiliary vector to provide a more complete description of spin states.

Contribution

It proposes a novel approach using a unit vector to characterize electron spin polarization, incorporating the azimuthal angle of an auxiliary vector for enhanced description.

Findings

The unit vector $ extbf{I}$ effectively characterizes spin polarization.

Rotation of $ extbf{I}$ corresponds to twice the spin rotation.

Application to free electrons yields a new representation of spin states.

Abstract

New degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector $\mathbf I$, with respect to the quantization direction to characterize the spin polarization. The rotation of $\mathbf I$ through an angle about the quantization axis leads to a rotation of the spin polarization vector through twice the angle about the same axis. Discussions are also made in Heisenberg picture as well. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector $\mathbf I$ functions as an independent index to characterize the spin polarization.