Pauli Equation

This is a compact lookup card. The magnetic-factor convention, gauge covariance, Zeeman and Landau structure, and regime of validity are owned by the Pauli Equation; the leading reduction is derived on Dirac to Pauli.

Formula

For a spin-$1/2$ particle of charge $q$ in electromagnetic potentials $\Phi$ and $\mathbf A$, define

$$\boldsymbol\pi = -i\hbar\nabla-q\mathbf A.$$

The Pauli Hamiltonian with $g=2$ is

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

Using

$$\mathbf B=\nabla\times\mathbf A,$$

this becomes

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

The time-dependent equation is

$$i\hbar\frac{\partial\psi}{\partial t} = H\psi,$$

where $\psi$ is a two-component spinor.

Assumptions

• The motion is nonrelativistic, but spin and electromagnetic coupling are retained.
• The displayed magnetic term corresponds to the Dirac value $g=2$ and omits anomalous magnetic moment corrections.
• $\boldsymbol\pi=-i\hbar\nabla-q\mathbf A$ uses the charge sign $q$ explicitly.
• The electromagnetic fields are prescribed external backgrounds.

Validity

The Pauli equation is the leading nonrelativistic spin-$1/2$ equation with minimal electromagnetic coupling. It is the natural bridge between Pauli spinors and the Dirac equation’s nonrelativistic limit. Higher-order relativistic corrections include spin–orbit coupling, Darwin terms, and magnetic-moment corrections beyond $g=2$; their crystal-field and Bloch-band projections require additional effective-theory assumptions.

Common Mistakes

• Losing the sign of $q$ in the magnetic term.
• Treating $\boldsymbol\sigma\cdot\mathbf B$ as a scalar potential rather than a matrix acting on spin.
• Forgetting that $\boldsymbol\pi$ components do not commute in a magnetic field.
• Using the Pauli equation where pair creation, strong relativistic fields, or fully relativistic covariance matter.

Quick Check

For an electron with $q=-e$, what sign does the Pauli magnetic term have?

Solution

Insert $q=-e$ into

$$- \frac{q\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B.$$

The term becomes

$$+ \frac{e\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B.$$

The sign convention is tied to the electron’s negative charge and to the Hamiltonian convention used here.

Canonical owner and released companions

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

References

• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons”, Zeitschrift für Physik 43, 601-623, 1927.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.