Pauli Equation

The Pauli equation is the leading low-energy wave equation for a spin-$1/2$ particle in a prescribed electromagnetic field. This page first treats the charged Dirac-particle class, for which the magnetic moment is parameterized by signed charge $q$, mass $m$, positive magnetic-factor magnitude $g$, and $\mathbf S=\hbar\boldsymbol\sigma/2$,

$$i\hbar\partial_t\varphi = H_P\varphi,$$

with

$$\boxed{ H_P = \frac{\boldsymbol\pi^2}{2m} +q\Phi -g\frac{q}{2m}\mathbf S\cdot\mathbf B }, \qquad \boldsymbol\pi=-i\hbar\nabla-q\mathbf A.$$

Equivalently, the spin term is

$$-\frac{gq\hbar}{4m} \boldsymbol\sigma\cdot\mathbf B.$$

Minimal Dirac theory gives $g=2$, exactly reproducing the Hamiltonian derived on Dirac to Pauli. Measured particles can have anomalous corrections, so $g=2$ is a tree-level Dirac result rather than an exact statement about the electron.

Required background. Dirac to Pauli, Minimal Coupling, and Minimal Coupling in Wave Mechanics supply the derivation, signed-charge gauge convention, and nonrelativistic operator structure. Pauli-matrix and spin-commutator algebra are assumed.

Magnetic moment and sign convention

Define the magnetic-moment operator by

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

Then

$$H_{\mathrm{spin}}=-\boldsymbol\mu\cdot\mathbf B.$$

Within this charged Dirac-particle convention, this page takes $g>0$ as a magnitude and leaves the sign in $q$. It is not a universal convention for all spin-$1/2$ systems: a neutral particle or a composite whose moment need not have the sign of its net charge requires an independently signed gyromagnetic ratio $\gamma$ or magnetic moment $\boldsymbol\mu=\gamma\mathbf S$. For an electron,

$$q=-e, \qquad \boldsymbol\mu_e = -g\frac{e}{2m_e}\mathbf S,$$

so its magnetic moment is antiparallel to its spin.

Another common convention tabulates a signed electron $g$ factor while using the positive elementary charge in the moment formula. The 2022 CODATA adjustment, published in 2025, lists the electron $g$ factor as negative; in this page’s convention its magnitude is

$$g_e=2.00231930436092(36).$$

Do not combine that signed tabulation with $q=-e$ a second time. Either use a positive $g$ and signed $q$, as here, or translate the entire alternative definition.

The anomaly is defined in this positive-magnitude convention by

$$a=\frac{g-2}{2}.$$

For a unit-charge lepton in pure QED, the leading correction is Schwinger’s result

$$a=\frac{\alpha}{2\pi}+O(\alpha^2).$$

For a point particle of charge $Qe$, the one-loop QED coefficient scales as $Q^2\alpha/(2\pi)$ in the corresponding charge-normalized anomaly. Higher QED loops and, depending on the particle, hadronic, weak, or composite-structure effects determine the precision value. They belong to QED and precision- physics treatments, not to the minimal Pauli equation.

Gauge-covariant two-component dynamics

Under

$$\mathbf A'=\mathbf A+\nabla\chi, \qquad \Phi'=\Phi-\partial_t\chi, \qquad \varphi'=e^{iq\chi/\hbar}\varphi,$$

the kinetic momentum acts covariantly:

$$\boldsymbol\pi'\varphi' = e^{iq\chi/\hbar}\boldsymbol\pi\varphi.$$

The magnetic field is unchanged, so the spin term transforms with the same phase. Including the time derivative and scalar potential gives

$$\left(i\hbar\partial_t-q\Phi'\right)\varphi' = e^{iq\chi/\hbar} \left(i\hbar\partial_t-q\Phi\right)\varphi.$$

Therefore the Pauli equation is gauge covariant. The two-component spin does not modify the gauge phase; it adds a gauge-invariant coupling to $\mathbf B$.

For $g=2$, the Hamiltonian has the compact form

$$H_P = \frac{1}{2m} (\boldsymbol\sigma\cdot\boldsymbol\pi)^2 +q\Phi,$$

because

$$(\boldsymbol\sigma\cdot\boldsymbol\pi)^2 = \boldsymbol\pi^2 -q\hbar\boldsymbol\sigma\cdot\mathbf B.$$

This factorization is special to the minimal $g=2$ coefficient.

Uniform-field Zeeman splitting

Ignore orbital motion temporarily and take $\mathbf B=B\hat{\mathbf z}$ with $B>0$. For $\sigma_z|s\rangle=s|s\rangle$, $s=\pm1$,

$$E_s = -\frac{gq\hbar B}{4m}s.$$

The signed level difference is

$$E_{+}-E_{-} = -\frac{gq\hbar B}{2m},$$

and its magnitude is

$$\Delta E = \frac{g|q|\hbar B}{2m}.$$

For a positive charge, spin parallel to $\mathbf B$ is lower. For an electron, $q=-e$, spin antiparallel to $\mathbf B$ is lower. Stating only “spin up is lower” without defining the charge and quantization axis is ambiguous.

In a nonuniform field, the magnetic energy produces the Stern–Gerlach force

$$\mathbf F \simeq \nabla(\boldsymbol\mu\cdot\mathbf B),$$

when the spin follows a resolved local projection and field gradients can be treated semiclassically. The force depends on the moment, not on a classical picture of the particle literally rotating.

Spin precession

For a uniform, time-independent field, the Heisenberg equation and $[S_i,S_j]=i\hbar\epsilon_{ijk}S_k$ give

$$\frac{d\mathbf S}{dt} = \frac{gq}{2m}\mathbf S\times\mathbf B.$$

Equivalently,

$$\frac{d\mathbf S}{dt} = \boldsymbol\Omega_L\times\mathbf S, \qquad \boldsymbol\Omega_L = -\frac{gq}{2m}\mathbf B.$$

The sign fixes the sense of precession; the angular-frequency magnitude is $g|q|B/(2m)$. Resonance experiments measure this splitting or precession and thereby determine the magnetic moment.

Landau and spin energies together

For a uniform magnetic field and free motion along $z$, the orbital spectrum is organized by the cyclotron frequency

$$\omega_c=\frac{|q|B}{m}.$$

Adding the Pauli term gives

$$E_{n,s}(p_z) = \frac{p_z^2}{2m} +\hbar\omega_c\left(n+\frac12\right) -\frac{gq\hbar B}{4m}s,$$

where $n=0,1,2,\ldots$ and $s=\pm1$. For $g=2$, one spin orientation cancels the orbital zero-point contribution. If $q>0$, the $s=+1$ branch begins at $n\hbar\omega_c$; for $q<0$, the cancelling orientation is reversed.

This pairing is the nonrelativistic trace of the minimally coupled Dirac structure. Boundary conditions and geometry still determine degeneracies and the allowed longitudinal momenta. The full orbital derivation belongs to Landau Levels.

Relativistic Landau Levels constructs the exact Dirac spinors and energy branches, and distinguishes their internal multiplicity from the orbital degeneracy.

Anomalous moment as an effective interaction

When $g=2(1+a)$, separate the Hamiltonian into the minimal Dirac prediction and an anomaly:

$$H_{\mathrm{spin}} = -\frac{q\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B -a\frac{q\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B.$$

The second term is the low-energy magnetic effect of a nonminimal covariant operator proportional to $\bar\psi\sigma^{\mu\nu}F_{\mu\nu}\psi$. Its coefficient is particle and scale dependent. For a charged composite particle, a conventionally defined $g$ can differ substantially from $2$ even before radiative corrections are discussed; its sign convention must be stated rather than inferred from the positive-$g$ convention above. For a neutral particle, write the Pauli term directly as $-\boldsymbol\mu\cdot\mathbf B$ with an independently signed moment, because $gq\mathbf S/(2m)$ would vanish identically.

The anomaly does not license arbitrary modification of the kinetic term or gauge phase. It supplements the gauge-invariant field-strength coupling while minimal coupling continues to govern $\boldsymbol\pi$ and $q\Phi$.

Magnetic Moment matches the covariant Pauli coefficient explicitly, including its electric partner and the static form-factor interpretation.

Regime of validity

The Pauli equation is a two-component, fixed-particle, nonrelativistic theory in a prescribed classical electromagnetic background. It assumes:

• characteristic momenta $|\boldsymbol\pi|\ll mc$;
• gauge-covariant residual energies and relevant potential-energy differences small compared with the particle–antiparticle gap $2mc^2$;
• fields weak and slowly varying enough that pair creation and strong positive/negative energy mixing are negligible;
• a specified $g$ factor appropriate to the particle and precision level.

It is not Lorentz covariant as a standalone two-component wave equation. It does not describe antiparticles, pair creation, dynamical photons, radiation reaction, or the full set of relativistic $1/c^2$ corrections. Spin–orbit, Darwin, and relativistic kinetic terms require a systematic reduction such as Foldy–Wouthuysen theory; radiative anomalies require QED or an empirically matched effective theory.

The static Foldy–Wouthuysen expansion supplies the ordered corrections. Their interpretations are developed on Spin–Orbit Coupling and Darwin Term.

Common pitfalls

Using $\boldsymbol\sigma$ as the physical spin. The Pauli matrices are dimensionless; $\mathbf S=\hbar\boldsymbol\sigma/2$ carries angular momentum.

Double counting the electron sign. A positive $g$ with $q=-e$ already fixes the moment. Do not also insert a signed negative $g$ from another convention.

Calling $g=2$ exact. It is the minimal tree-level Dirac value. Precision particles have anomalous contributions, and composite particles encode internal structure.

Appending higher-order terms without assumptions. Relativistic corrections depend on field time dependence, gradients, ordering, and the chosen block-diagonalization scheme.

Exercises

1. Zeeman energies

Find the two spin energies for a particle of charge $q$ in $\mathbf B=B\hat{\mathbf z}$. Identify the lower state for $q>0$ and for an electron.

Solution

For $\sigma_z=s$, the energies are

$$E_s=-\frac{gq\hbar B}{4m}s.$$

If $q>0$, $s=+1$ is lower. For an electron $q=-e$, $s=-1$ is lower. The splitting magnitude is $g|q|\hbar B/(2m)$.

2. Larmor precession

Use the spin commutator to derive the precession equation and its signed angular-velocity vector.

Solution

With $H=-gq\mathbf S\cdot\mathbf B/(2m)$,

$$\frac{dS_i}{dt} = \frac{i}{\hbar}[H,S_i] = \frac{gq}{2m} (\mathbf S\times\mathbf B)_i.$$

Since $\mathbf S\times\mathbf B=-\mathbf B\times\mathbf S$, this is $d\mathbf S/dt=\boldsymbol\Omega_L\times\mathbf S$ with $\boldsymbol\Omega_L=-gq\mathbf B/(2m)$.

3. Landau–Zeeman cancellation

Set $g=2$ in $E_{n,s}$ and show which spin branch cancels the orbital $\hbar\omega_c/2$ term for each sign of $q$.

Solution

Write $q=\operatorname{sgn}(q)|q|$. The spin term becomes

$$-\frac{q\hbar B}{2m}s = -\frac12\operatorname{sgn}(q)s\hbar\omega_c.$$

It cancels the $+\hbar\omega_c/2$ orbital term when $s=\operatorname{sgn}(q)$. The opposite branch adds another half quantum.

4. First-order anomalous shift

Let $g=2(1+a)$ with $|a|\ll1$. Find the anomaly’s first-order shift of a $\sigma_z=s$ state in a uniform field.

Solution

The anomalous Hamiltonian is

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

Therefore

$$\delta E_s = -a\frac{q\hbar B}{2m}s.$$

The shift changes sign with both charge and spin projection.

References

• L. L. Foldy and S. A. Wouthuysen, “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit,” Physical Review 78, 29–36, 1950, doi:10.1103/PhysRev.78.29.
• P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA Recommended Values of the Fundamental Physical Constants: 2022,” Journal of Physical and Chemical Reference Data 54, 033105, 2025, doi:10.1063/5.0279860.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601–623, 1927, doi:10.1007/BF01397326.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, doi:10.1017/9781108587280.
• J. Schwinger, “On Quantum-Electrodynamics and the Magnetic Moment of the Electron,” Physical Review 73, 416–417, 1948, doi:10.1103/PhysRev.73.416.