Spin–Orbit Coupling

The electrostatic Dirac spin–orbit interaction couples spin to the cross product of the potential-energy gradient and momentum. Its coefficient contains the Thomas factor: transforming the magnetic moment into an instantaneous rest frame without accounting for that frame’s rotation gives twice the minimal-Dirac result. The controlled coefficient comes from the Foldy–Wouthuysen Expansion; this page explains its interpretation and applies it to a central potential.

Required background. The Foldy–Wouthuysen Expansion supplies the Hamiltonian coefficient. Spin–Orbit Angular-Momentum Algebra owns the coupled-basis diagonalization. Helpful background. Wigner Rotations fixes the transport convention for Thomas precession.

Electrostatic spin–orbit operator

Consider a positive-energy, massive, minimally coupled Dirac particle in a static electric potential energy $V(\mathbf r)=q\Phi(\mathbf r)$, with $\mathbf A=0$. Use the low-momentum and smooth-field assumptions of the FW expansion. The spin-dependent term through order $1/c^2$ is

$$H_{\rm SO} =\frac{\hbar}{4m^2c^2} \boldsymbol\sigma\cdot(\nabla V\times\mathbf p) =\frac{1}{2m^2c^2} \mathbf S\cdot(\nabla V\times\mathbf p).$$

Here $\mathbf S=\hbar\boldsymbol\sigma/2$ is the two-component spin in the FW representation. The derivative acts on the potential energy, including the sign of the charge. For an electron in an attractive Coulomb field, $V$ is negative but its outward radial derivative is positive.

For a smooth scalar $V$, $\nabla\times\nabla V=0$ makes this operator Hermitian on the common domain. The ordered form before specializing to electrostatics is given on the FW owner page; a spatially varying coefficient cannot in general be moved through a momentum operator without checking its commutator.

This is a relativistic correction within an effective low-energy Hamiltonian. It does not require a classical orbit or a rotating charged sphere. It also does not include an anomalous magnetic moment, recoil, radiation, or a general time-dependent electromagnetic background.

Why the Thomas factor is one half

A semiclassical consistency check explains the coefficient without replacing the FW derivation. Let the particle have laboratory velocity $\mathbf v$ and let the laboratory magnetic field vanish. To leading order in $v/c$, its instantaneous rest frame sees

$$\mathbf B_{\rm rest} =-\frac{\mathbf v\times\mathbf E}{c^2} =\frac{\mathbf E\times\mathbf v}{c^2}.$$

With minimal $g=2$, the magnetic precession angular velocity is

$$\boldsymbol\omega_{\rm mag} =-\frac{q}{m}\mathbf B_{\rm rest} =\frac{\nabla V\times\mathbf v}{mc^2}.$$

The sign follows from $d\mathbf S/dt=\boldsymbol\omega_{\rm mag}\times\mathbf S$, as in the Pauli Equation. This contribution alone would correspond to a spin–orbit coefficient twice the Dirac value.

Successive instantaneous rest frames rotate. Using the rotationless-boost comparison specified on Wigner Rotations, the low-speed Thomas angular velocity is

$$\boldsymbol\omega_T =\frac{\mathbf a\times\mathbf v}{2c^2}.$$

At leading order in this electrostatic problem, $m\mathbf a=q\mathbf E=-\nabla V$, so

$$\boldsymbol\omega_T =-\frac{\nabla V\times\mathbf v}{2mc^2}.$$

The total precession is thus

$$\boldsymbol\omega_{\rm SO} =\boldsymbol\omega_{\rm mag}+\boldsymbol\omega_T =\frac{\nabla V\times\mathbf p}{2m^2c^2},$$

where $\mathbf p=m\mathbf v$ suffices at this order. A spin Hamiltonian $H=\boldsymbol\omega_{\rm SO}\cdot\mathbf S$ generates precisely this precession and agrees with the FW result. Thomas (1926) identified the kinematic correction behind the factor-of-two discrepancy.

The Thomas term must not be added a second time to a Hamiltonian already obtained from the Dirac FW expansion. It explains the coefficient already present there. The argument also uses a particular rest-frame spin comparison; it is not an exact laboratory evolution equation for the untransformed Dirac matrix $\hbar\boldsymbol\Sigma/2$.

Central potentials and resolved doublets

For $V=V(r)$,

$$\nabla V\times\mathbf p =\frac{V'(r)}{r}\mathbf L, \qquad H_{\rm SO}=\xi(r)\mathbf L\cdot\mathbf S,$$

with

$$\xi(r)=\frac{V'(r)}{2m^2c^2r}.$$

Rotational symmetry preserves total angular momentum. For a spin-half particle, the eigenvalues from the angular-momentum owner are

Total angular momentum	Eigenvalue of $\mathbf L\cdot\mathbf S$
$j=\ell+\tfrac12$	$\hbar^2\ell/2$
$j=\ell-\tfrac12$, $\ell\geq1$	$-\hbar^2(\ell+1)/2$

For first-order perturbation theory in a resolved radial state, define $\bar\xi_{n\ell}=\langle n\ell|\xi(r)|n\ell\rangle$. The two spin–orbit shifts are

$$\Delta E_+ =\frac{\hbar^2\ell}{2}\bar\xi_{n\ell}, \qquad \Delta E_- =-\frac{\hbar^2(\ell+1)}{2}\bar\xi_{n\ell}.$$

Their separation is

$$\Delta E_+-\Delta E_- =\frac{\hbar^2(2\ell+1)}{2}\bar\xi_{n\ell}.$$

If $V'(r)>0$ throughout the sampled region, the $j=\ell+\tfrac12$ member has the higher spin–orbit energy. For a general effective potential, the sign is determined by the radial expectation rather than by the name of the interaction. When additional degeneracies allow mixing, the perturbation must be diagonalized in that space.

For $V=-\kappa_{\rm C}/r$, with $\kappa_{\rm C}>0$, the coefficient is $\xi(r)=\kappa_{\rm C}/(2m^2c^2r^3)$. For $\ell=0$, the original operator annihilates the orbital state because $\mathbf L=0$. One must not calculate a divergent S-state $\langle r^{-3}\rangle$ and then multiply by zero. The Darwin term supplies a different, nonzero S-state correction.

Symmetry and the limits of the coefficient

With no imposed magnetic field, the electrostatic operator is time-reversal even: both $\mathbf p$ and $\mathbf S$ reverse. For a parity-symmetric potential, $\nabla V$ and $\mathbf p$ are polar vectors, and their cross product and spin are axial vectors, so their scalar product is parity even. A central potential also preserves the $m_j$ degeneracy.

The free-space Dirac coefficient is only one ingredient in atomic fine structure. The relativistic kinetic and Darwin terms occur at the same order, while nuclear motion and QED effects enter the precision spectrum under their own counting. Atomic Fine Structure owns that spectroscopic hierarchy.

Likewise, projecting microscopic interactions onto crystal bands can produce effective spin–orbit coefficients that depend strongly on the material. Spin–Orbit Coupling in Solids owns those band Hamiltonians. Their coefficients cannot be inferred by substituting a band effective mass into the free-space expression.

Exercises

A smooth harmonic potential. For $V=m\omega^2r^2/2$, find $\xi$ and the separation of the $\ell\geq1$ doublet in first-order perturbation theory.

Solution

Here $V'(r)/r=m\omega^2$, so

$$\xi=\frac{\omega^2}{2mc^2},\qquad \Delta E_+-\Delta E_- =\frac{\hbar^2\omega^2}{4mc^2}(2\ell+1).$$

This is a local low-energy expansion in the sampled part of the potential. It does not assert a globally confining, stable Dirac spectrum for an arbitrarily large time-component vector potential.

A P-state doublet. For $\ell=1$ and $\bar\xi>0$, list the two shifts and their degeneracies. Check their degeneracy-weighted mean.

Solution

The $j=3/2$ level has shift $+\hbar^2\bar\xi/2$ and degeneracy four; the $j=1/2$ level has shift $-\hbar^2\bar\xi$ and degeneracy two. The weighted sum is zero: $4(\hbar^2\bar\xi/2)+2(-\hbar^2\bar\xi)=0$. The spin–orbit interaction splits this multiplet without changing its first-order centroid.

Keeping the frame correction. In the same semiclassical electrostatic argument, replace the minimal magnetic moment by a specified $g$. Find the coefficient multiplying $(\nabla V\times\mathbf p)\cdot\mathbf S$.

Solution

The magnetic precession becomes $g\,\nabla V\times\mathbf v/(2mc^2)$. The kinematic Thomas term is unchanged. Their sum gives the coefficient $(g-1)/(2m^2c^2)$. Thus a covariantly matched anomaly changes spin–orbit response as well as the Zeeman term. This check is limited to the stated low-speed electrostatic setting; a full effective Hamiltonian requires the corresponding Pauli-operator reduction.

References

• Foldy, Leslie L., and Siegfried 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.
• Littlejohn, Robert G. “The Foldy-Wouthuysen Transformation.” Physics 221B, Notes 49, University of California, Berkeley, academic year 2021–22 (2021). Lecture notes.
• Thomas, Llewellyn H. “The Motion of the Spinning Electron.” Nature 117, 514 (1926). doi:10.1038/117514a0.