Spin in Magnetic Field Hamiltonian

Hamiltonian

The magnetic-dipole Hamiltonian is

$$H = -\boldsymbol\mu\cdot\mathbf B.$$

For spin with magnetic moment

$$\boldsymbol\mu = g\frac{q}{2m}\mathbf S,$$

the Hamiltonian is

$$H = -g\frac{q}{2m} \mathbf S\cdot\mathbf B.$$

For spin $1/2$,

$$H = -g\frac{q\hbar}{4m} \boldsymbol\sigma\cdot\mathbf B.$$

Assumptions

• Only magnetic-dipole spin coupling is displayed.
• The charge sign $q$ is explicit.
• The $g$ factor and spin convention are specified.
• Orbital coupling to vector potential is omitted unless stated.

Common Mistakes

• Losing the sign for electrons by replacing $q$ with $e$ without saying $e>0$.
• Confusing spin Zeeman coupling with orbital magnetic coupling.
• Assuming $g=2$ for every particle or effective spin.
• Forgetting that a spatially varying magnetic field can couple spin and motion.

Canonical Links

• Spin in Magnetic Fields
• Magnetic Moments and g-Factors
• Larmor Precession
• Spin-1/2 as a Canonical System
• Spin Operator
• Zeeman Effect
• Spin and Pauli Matrix Conventions

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.