Pauli Hamiltonian

This is a compact Hamiltonian card. The low-energy equation, magnetic-factor convention, Zeeman and Landau structure, and validity regime are owned by the Pauli Equation; the $g=2$ reduction is derived on Dirac to Pauli.

Hamiltonian

For charge $q$ and

$$\boldsymbol\pi = \hat{\mathbf p}-q\mathbf A,$$

the Pauli Hamiltonian with $g=2$ is

$$H = \frac{\boldsymbol\pi^2}{2m} + q\Phi - \frac{q\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B.$$

Equivalently,

$$H = \frac{1}{2m} \left( \boldsymbol\sigma\cdot\boldsymbol\pi \right)^2 + q\Phi.$$

Operator setting

The free Pauli operator acts on $\mathcal H_P=L^2(\mathbb R^3,\mathbb C^2)$ with dense domain $H^2(\mathbb R^3,\mathbb C^2)$ and codomain $\mathcal H_P$. For prescribed backgrounds, the self-adjoint realization is determined by the regularity and growth of $\mathbf A$, $\Phi$, and $\mathbf B$; a magnetic Sobolev domain is the natural replacement when the covariant derivatives are unbounded. The formula alone does not determine that domain.

On a bounded region, Dirichlet, magnetic Neumann, or another stated self-adjoint boundary condition is part of the Hamiltonian and affects its spectrum. Gauge-related potentials must be compared with the corresponding unitary phase transformation and consistently transformed boundary data.

The common positive-energy phase $e^{-imc^2t/\hbar}$ has been removed, so the displayed Pauli energy zero is shifted downward by $mc^2$ relative to the positive-energy Dirac branch. If the backgrounds depend on time, a unitary propagator additionally requires suitable time regularity and compatible domains for $H(t)$.

Assumptions

• Nonrelativistic two-component spinor in the operator setting stated above.
• External electromagnetic potentials are prescribed.
• The displayed magnetic coupling uses $g=2$ and omits anomalous magnetic moment terms.
• The sign of $q$ is explicit.

Reliability checks

• The identity $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-q\hbar\boldsymbol\sigma\cdot\mathbf B$ reproduces the displayed Zeeman term.
• Setting $\Phi=\mathbf A=\mathbf B=0$ gives the spin-degenerate free Hamiltonian $\hat{\mathbf p}^{2}/(2m)$.
• Every term has dimensions of energy, and the signed-$q$ electron term has the correct Zeeman sign.
• Gauge-related potentials give unitarily equivalent dynamics when the spinor and boundary data are transformed consistently.

Common Mistakes

• Losing the sign of the Zeeman term for electrons.
• Forgetting that $\pi_i$ components fail to commute in a magnetic field.
• Treating the Pauli Hamiltonian as fully relativistic.
• Omitting spinor structure and using a scalar wavefunction.

Canonical owner and released companions

• Pauli Equation
• Dirac to Pauli
• Pauli Equation formula card
• Minimal Coupling in Wave Mechanics

References

• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons”, Zeitschrift für Physik 43, 601-623, 1927.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.