The Relativistic Kinetic-Energy Correction

The gross structure of hydrogen, the $E_n=-Z^2\mathrm{Ry}/n^2$ ladder produced by solving the radial equation, rests on the non-relativistic kinetic energy $T=p^2/2m$. That expression is the first term of an expansion. The electron in the ground state moves with a characteristic speed $v\sim Z\alpha c$, where $\alpha =e^2/4\pi {ϵ}_0\hslash c\approx 1/137$ is the fine-structure constant, so $(v/c{)}^2\sim (Z\alpha {)}^2$ is a small but nonzero fraction. Keeping the next term in the relativistic kinetic energy shifts every level by a fractional amount of order $(Z\alpha {)}^2$, splitting lines that the gross theory leaves coincident. This lesson computes that shift for hydrogenic states; the spin-orbit and Darwin corrections, of the same order, follow in the next two lessons.

The size of the effect

The Bohr model fixes the electron's speed. In the $n$-th circular orbit the quantized angular momentum $m{v}_n{r}_n=n\hslash$ and the force balance $m{v}_n^2/r_n=Z{e}^2/4\pi {ϵ}_0{r}_n^2$ combine to

For hydrogen ($Z=1$) in the ground state, $v_1/c=\alpha \approx 7.3\times 10^{-3}$, so $(v/c{)}^2\approx 5.3\times 10^{-5}$. The relativistic correction to the kinetic energy is smaller than the kinetic energy itself by this factor, and the kinetic energy is comparable to the binding energy, so the level shifts land near $5\times 10^{-5}\times 13.6\text{eV}\approx 10^{-4}\text{eV}$. That number sets the scale of fine structure and explains why the effect is invisible in the coarse Balmer spectrum and prominent only under high resolution.

Expanding the relativistic kinetic energy

The exact relativistic energy of a free particle of rest mass $m$ and momentum magnitude $p$ is

The kinetic energy is this minus the rest energy $m{c}^2$. Factor out $m{c}^2$ and expand the square root in the small parameter $(p/mc{)}^2$:¹

Written out in powers of $p$,

The leading term is the familiar non-relativistic kinetic energy already in the hydrogen Hamiltonian. The next term is the lowest relativistic correction,

a perturbation to be added to the unperturbed hydrogen Hamiltonian $H_0=p^2/2m+V(r)$. The relative size of the two, $(p^2/2m)$ versus the correction, is $(p/mc{)}^2/4\sim (Z\alpha /n{)}^2$, confirming the estimate above.

First-order shift and the p⁴ trick

Because hydrogen's gross-structure levels are degenerate in $\ell$ and $m_{\ell }$, one might worry about degenerate perturbation theory. The saving fact is that $H_r^{\prime }$ commutes with $L^2$ and $L_z$: it is built from $p^4$, a scalar under rotations, and carries no spin or angular dependence beyond what $p^2=2m(E_0-V)$ already respects. The unperturbed eigenstates $|n\ell m_{\ell }⟩$ are therefore already the good states that diagonalize $H_r^{\prime }$ within each degenerate shell, and ordinary first-order theory applies:¹

Evaluating $⟨p^4⟩$ directly requires four derivatives of the wave function and is laborious. The efficient route uses the unperturbed Schrödinger equation to trade momentum for energy. Since $H_0|\psi ⟩=E_n|\psi ⟩$ and $H_0=p^2/2m+V$,

The operator $p^2$ is Hermitian, so it may act to the left on the bra and to the right on the ket:

This replaces the fourth-order differential operator with a simple function of $r$, at the cost of needing two radial expectation values. Expanding the square,

Evaluating on Coulomb states

For a hydrogenic atom the potential is $V(r)=-Z{e}^2/4\pi {ϵ}_{0}r$. Write $\beta \equiv Z{e}^2/4\pi {ϵ}_0$, so $V=-\beta /r$ and $⟨V⟩=-\beta ⟨1/r⟩$, $⟨V^2⟩={\beta }^2⟨1/r^2⟩$. The two radial expectation values are the standard hydrogenic results

with $a_0=4\pi {ϵ}_0{\hslash }^2/m{e}^2$ the Bohr radius. Two identities streamline the algebra. First, $\beta /a_0=Z{e}^2/4\pi {ϵ}_0{a}_0$, and comparing with $E_n=-\beta /2a_0{n}^2\cdot Z=-Z^2{e}^2/8\pi {ϵ}_0{a}_0{n}^2$ gives $\beta /a_0=-2n^2{E}_n/Z\cdot Z=-2n^2{E}_n$ once $Z$ is carried inside $\beta$. Concretely,

The first is the virial theorem in disguise: $⟨V⟩=2E_n$ and $⟨T⟩=-E_n$. Substituting both into the bracket,

The second form uses $E_n=-(Z\alpha {)}^{2}m{c}^2/2n^2$, which recasts $E_n^2/2m{c}^2$ as $(Z\alpha {)}^{4}m{c}^2/8n^4$. The correction is manifestly of order $(Z\alpha {)}^2$ times the gross energy $E_n$, the promised fine-structure scale.

Dependence on n and ℓ

The shift lowers every level, but not uniformly. At fixed $n$ the magnitude falls as $\ell$ grows, because $1/(\ell +\frac{1}{2})$ shrinks and high-$\ell$ states are held away from the nucleus by the centrifugal barrier, where the electron moves slower and the relativistic correction is weaker. The penetrating $\ell =0$ states, which sample the region of large $p$ near the origin, are pulled down the most.

A short table makes the pattern quantitative. Writing the shift as $E_r^{(1)}=-\frac{1}{2}(Z\alpha {)}^{4}m{c}^2{n}^{-4}f(n,\ell )$ with $f=n/(\ell +\frac{1}{2})-\frac{3}{4}$:

The bare $\ell$-dependence here is not the full story: the spin-orbit correction carries its own $\ell$-dependence of the same size, and when the two are added the combined shift reorganizes to depend only on $n$ and the total angular momentum $j$. The relativistic term taken alone does not respect that final simplicity, so its $\ell$-pattern is physical only in combination with the others.

Where it sits among the corrections

The relativistic kinetic term is one of three corrections of order $(Z\alpha {)}^2{E}_n$ that together make up the fine structure. It is worth fixing the hierarchy of scales before the pieces are assembled, because the same power counting recurs for hyperfine structure and the Lamb shift.

Because all three fine-structure terms scale as $(Z\alpha {)}^{4}m{c}^2$, none can be neglected relative to the others; the Darwin term acts only on $\ell =0$ states, the spin-orbit term only on $\ell \ge 1$, and the relativistic term on all of them. The remarkable outcome, derived once all three are in hand, is that their sum collapses to the single fine-structure formula depending on $n$ and $j$ alone, the same expression that the exact Dirac equation reproduces to this order. The relativistic kinetic correction is the piece that carries the velocity content of that agreement: it is what the $-p^4/8m^3{c}^2$ term of the classical energy becomes once the electron is quantized.