Consider a neutral spin one-half Dirac particle with mass...

Consider a neutral spin one-half Dirac particle with mass and with an intrinsic magnetic moment, and assume the Hamiltonian $$ H=c \boldsymbol{\alpha} \cdot \frac{\hbar}{i} \nabla+\beta m c^2+\lambda B \beta \Sigma_z $$ in the presence of a uniform constant magnetic field along the $z$ axis. Determine the important constants of the motion, and derive the energy eigenvalues. Show that orbital and spin motions are coupled in the relativistic theory but decoupled in a nonrelativistic limit. The coefficient $\lambda$ is a constant, proportional to the gyromagnetic ratio.

Consider a neutral spin one-half Dirac particle with mass and with an intrinsic magnetic moment, and assume the Hamiltonian
$$
H=c \boldsymbol{\alpha} \cdot \frac{\hbar}{i} \nabla+\beta m c^2+\lambda B \beta \Sigma_z
$$
in the presence of a uniform constant magnetic field along the $z$ axis. Determine the important constants of the motion, and derive the energy eigenvalues. Show that orbital and spin motions are coupled in the relativistic theory but decoupled in a nonrelativistic limit. The coefficient $\lambda$ is a constant, proportional to the gyromagnetic ratio.

Key Concepts

Dirac Hamiltonian

The Dirac Hamiltonian is the fundamental operator describing relativistic spin-1/2 particles. It incorporates terms for kinetic energy, rest mass energy, and intrinsic magnetic moments interacting with external fields. The use of Dirac matrices (? and ?) allows the equation to capture both particle and antiparticle solutions as well as relativistic spin dynamics.

Constants of Motion

Constants of motion are quantities whose corresponding operators commute with the Hamiltonian, meaning they are conserved in time. In the context of a Dirac particle in a uniform magnetic field, conserved quantities often include momentum components (such as pz along the field direction) and total angular momentum (modified by spin contributions), which arise from the symmetries of the Hamiltonian.

Spin-Orbit Coupling

Spin-orbit coupling refers to the interaction between the intrinsic spin of a particle and its orbital motion. In the relativistic framework provided by the Dirac equation, this coupling is naturally embedded in the structure of the Hamiltonian, resulting in energy eigenvalues that reflect a combined effect of both orbital and spin degrees of freedom. This phenomenon does not simply separate into independent parts as seen in nonrelativistic treatments.

Nonrelativistic Limit

The nonrelativistic limit of a relativistic theory like the Dirac equation is achieved when the particle's velocity is much less than the speed of light. In this limit, the coupling between orbital and spin is diminished, and the dynamics reduce to those described by the Pauli equation, where orbital motion and spin interactions (such as the Zeeman effect) effectively decouple, leading to simpler energy eigenvalue expressions.

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