Spin and relativity, inseparably joined

In chapter 5 the experiments held our heads down and made us accept two things: the electron has spin $\hbar/2$; and its magnetic moment carries a strange factor $g\approx2$ — for the same angular momentum, spin produces twice the magnetic moment that orbital motion does (section 5.4). In chapter 6 we then stuffed three “relativistic patches” into the hydrogen atom by hand to piece together the fine structure (section 6.6). Each time the numbers came out right; each time we could not say why.

This section settles all the accounts at once: every one of those “hand additions” is an automatic consequence of the Dirac equation. There is only one method — put the equation in an electromagnetic field and expand order by order in powers of $v/c$.

The zeroth-order surprise: g = 2

The electromagnetic field is plugged in exactly as in classical mechanics and Schrödinger theory (minimal coupling): $\hat{\vec p}\to\hat{\vec\pi}=\hat{\vec p}-q\vec A$, plus the scalar-potential energy $q\varphi$:

$$\ii\hbar\frac{\partial\psi}{\partial t} =\bigl[c\,\vec\alpha\cdot\hat{\vec\pi}+\beta mc^2+q\varphi\bigr]\psi\tag{12.5.1}$$

Note carefully: no magnetic moment has been put in — the coupling says not one word about spin.

From the Dirac equation to the Pauli equation: the birth of g=2advanced~12 minexpand

Step 1: split into upper and lower halves. Following the block structure of section 12.3, write the four-component spinor as two two-component spinors $\psi=\begin{pmatrix}\chi\\ \eta\end{pmatrix}$, and peel off the dominant rest-energy phase: $\psi\to\ee^{-\ii mc^2t/\hbar}\psi$. Substitute into the equation, use the block forms of $\alpha_k,\beta$, and obtain the coupled pair

$$\ii\hbar\,\partial_t\chi=c\,\vec\sigma\cdot\hat{\vec\pi}\,\eta+q\varphi\,\chi\tag{12.5.2}$$$$\ii\hbar\,\partial_t\eta=c\,\vec\sigma\cdot\hat{\vec\pi}\,\chi+q\varphi\,\eta-2mc^2\eta\tag{12.5.3}$$

Step 2: identify the large and small components. In the second equation, $-2mc^2\eta$ dwarfs everything else: at low speeds both $\ii\hbar\partial_t\eta$ and $q\varphi\eta$ (an energy of order $\sim mv^2$ times $\eta$) are far smaller than it, so

$$\eta\approx\frac{\vec\sigma\cdot\hat{\vec\pi}}{2mc}\,\chi \;\sim\;\frac{v}{2c}\,\chi\tag{12.5.4}$$

$\eta$ is smaller than $\chi$ by a factor of $v/c$ — call $\chi$ the large component (the Pauli two-component wavefunction of the non-relativistic electron) and $\eta$ the small component.

Step 3: substitute back. Insert $\eta$ into the first equation:

$$\ii\hbar\,\partial_t\chi =\left[\frac{(\vec\sigma\cdot\hat{\vec\pi})^2}{2m}+q\varphi\right]\chi\tag{12.5.5}$$

Step 4: the single most important piece of algebra in the whole story. Use the Pauli-matrix identity $(\vec\sigma\cdot\vec a)(\vec\sigma\cdot\vec b)=\vec a\cdot\vec b+\ii\vec\sigma\cdot(\vec a\times\vec b)$:

$$(\vec\sigma\cdot\hat{\vec\pi})^2=\hat\pi^2+\ii\,\vec\sigma\cdot(\hat{\vec\pi}\times\hat{\vec\pi})\tag{12.5.6}$$

An ordinary vector crossed with itself is zero — but the components of $\hat{\vec\pi}$ do not commute with one another ($\hat{\vec p}$ does not commute with $\vec A(\vec x)$)! Compute component by component:

$$(\hat{\vec\pi}\times\hat{\vec\pi})_z=\hat\pi_x\hat\pi_y-\hat\pi_y\hat\pi_x =-q\bigl([\hat p_x,A_y]+[A_x,\hat p_y]\bigr) =\ii\hbar q\Bigl(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\Bigr) =\ii\hbar q\,B_z\tag{12.5.7}$$

Assembling the three components: $\hat{\vec\pi}\times\hat{\vec\pi}=\ii\hbar q\vec B$. Substituting back:

$$(\vec\sigma\cdot\hat{\vec\pi})^2=\hat\pi^2-q\hbar\,\vec\sigma\cdot\vec B\tag{12.5.8}$$

Step 5: read off the Pauli equation and the g factor.

$$\ii\hbar\,\partial_t\chi =\left[\frac{\hat\pi^2}{2m}+q\varphi -\frac{q\hbar}{2m}\,\vec\sigma\cdot\vec B\right]\chi\tag{12.5.9}$$

The last term is a magnetic-moment energy $-\vec\mu\cdot\vec B$, where (using $\hat{\vec S}=\frac\hbar2\vec\sigma$)

$$\vec\mu=\frac{q\hbar}{2m}\vec\sigma=\underbrace{2}_{g}\cdot\frac{q}{2m}\hat{\vec S}\tag{12.5.10}$$

Compare the orbital moment $\vec\mu_L=\frac{q}{2m}\hat{\vec L}$ (coefficient $\frac{q}{2m}$, no factor of 2): the g factor of the spin magnetic moment comes out exactly 2. At no step was a magnetic moment put into the theory — it grew by itself out of the algebra of $(\vec\sigma\cdot\hat{\vec\pi})^2$, and its value matches the Stern-Gerlach and anomalous-Zeeman measurements. Before 1928, $g=2$ was a purely empirical number; after 1928, it is a homework derivation.

First order: the fine-structure trio arrives complete

Push the approximation one order further (keep the $1/c^2$ corrections in the equation for $\eta$; the systematic version of this expansion is called the Foldy-Wouthuysen transformation — tedious algebra, no new ideas), and for the hydrogen atom ($q\varphi=V(r)$ the Coulomb potential) the effective Hamiltonian gains three extra terms:

$$\hat H_{1/c^2} =\underbrace{-\frac{\hat p^4}{8m^3c^2}}_{\text{relativistic kinetic correction}} +\underbrace{\frac{1}{2m^2c^2}\frac1r\frac{\dd V}{\dd r}\,\hat{\vec S}\cdot\hat{\vec L}}_{\text{spin-orbit coupling}} +\underbrace{\frac{\hbar^2}{8m^2c^2}\nabla^2V}_{\text{Darwin term}}\tag{12.5.11}$$

Check the books term by term — every one is an old acquaintance from section 6.6:

∎Key formulas

• Pauli equation (derived automatically)

  $$\ii\hbar\,\partial_t\chi=\Bigl[\frac{(\hat{\vec p}-q\vec A)^2}{2m}+q\varphi-\frac{q\hbar}{2m}\vec\sigma\cdot\vec B\Bigr]\chi$$

  Non-relativistic limit of the large component; the magnetic-moment term needs no hand insertion

• The key algebra

  $$(\vec\sigma\cdot\hat{\vec\pi})^2=\hat\pi^2-q\hbar\,\vec\sigma\cdot\vec B$$

  From π×π=iħqB (non-commuting components); g=2 comes from here

• Fine structure (arriving automatically)

  $$\hat H_{1/c^2}=-\frac{\hat p^4}{8m^3c^2}+\frac{1}{2m^2c^2}\frac1r\frac{\dd V}{\dd r}\hat{\vec S}\cdot\hat{\vec L}+\frac{\hbar^2}{8m^2c^2}\nabla^2V$$

  Thomas 1/2 born correct; the Darwin term touches only s states

• Hydrogen levels

  $$E_{nj}=-\frac{13.6\ \text{eV}}{n^2}\Bigl[1+\frac{\alpha^2}{n^2}\Bigl(\frac{n}{j+1/2}-\frac34\Bigr)\Bigr]$$

  Depends only on n and j, identical to the section 6.6 formula

?Self-check4 questions

1. 1.

   Which piece of algebra in the derivation is the direct source of g=2?

   • (σ·π)² = π² − qħσ·B, where π×π = iħqB is non-zero because p does not commute with A(x)
   • Adding a magnetic-moment term −μ·B to the Hamiltonian with its experimental value
   • Modelling the electron as a charged spinning ball and computing its rotational magnetic moment
   • First-order perturbation theory on the Coulomb potential

   Submit
2. 2.

   Compared with the hand-built approach of section 6.6, the advantages of obtaining fine structure by expanding the Dirac equation are: (select all that apply)

   Select all that apply

   • All three corrections come from a single expansion, no longer each fending for itself
   • The spin-orbit coefficient automatically carries Thomas’s 1/2, with no extra correction needed
   • The Darwin term gains a definite origin, and automatically acts only on s states (∇²V is concentrated at the origin)
   • The expansion shows that fine structure does not actually exist

   Submit
3. 3.

   Experiment measures g=2.00232 rather than exactly 2. This deviation shows that:

   • The Dirac single-particle equation has reached its limit of validity: the electron’s interaction with quantum fluctuations of the electromagnetic field dresses the magnetic moment, requiring quantum field theory (first-order correction α/2π)
   • The Dirac equation has been falsified and must be torn down and rebuilt
   • It is a systematic error caused by insufficient experimental precision
   • The electron is actually a spin-1 particle

   Submit
4. 4.

   Among the n=2 levels of hydrogen, the fine-structure splitting between 2p₃/₂ and 2p₁/₂ is ΔE=13.6 eV×α²/16. Taking α=1/137.0, find ΔE (in units of 10⁻⁵ eV).

   ×10⁻⁵ eV20% relative tolerance

   Submit

What comes next

Count the IOUs this chapter has signed: KG’s negative probability is redeemed by the field’s charge interpretation (12.2); the Dirac sea’s infinite background is smoothed away by the field’s recasting of the vacuum (12.4); the little tail of $g-2$ is computed from virtual-photon fluctuations (this section); and the proof of the spin-statistics theorem lives entirely inside field theory. Every thread points the same way: promote “the wavefunction of one particle” to “a quantum field vibrating at every point of spacetime”, with particles merely the field’s excitations. The door to field theory now stands ajar. In the next chapter we no longer derive line by line; instead, carrying this complete quantum-mechanical education, we go and look at the frontier scenery through the crack — from topological phases to black-hole information, where every single problem speaks the language you have already learned.