Fine Structure of Hydrogen

Derivation

Fine Structure of Hydrogen

D-223 Home PU-301 Threads energy · matter · symmetry · fields Depends on Hydrogen Atom Bound-State Spectrum, Degenerate Perturbation Theory, Addition of Angular Momenta, Spin-1/2 and the Pauli Matrices

Statement

Treating the leading relativistic corrections to the non-relativistic hydrogen Hamiltonian as a perturbation, we derive the three order-\( \alpha^4 \) shifts — the relativistic-kinetic correction \( H_{\text{rel}}=-\hat{p}^4/(8m^3c^2) \), the spin-orbit coupling \( H_{\text{SO}}=\dfrac{1}{2m^2c^2}\dfrac{1}{r}\dfrac{dV}{dr}\,\vec{S}\cdot\vec{L} \), and the Darwin term \( H_{\text{D}}=\dfrac{\pi\hbar^2}{2m^2c^2}\dfrac{e^2}{4\pi\varepsilon_0}\,\delta^3(\vec{r}) \) — and show they collapse to a single closed form depending only on \( n \) and the total angular momentum \( j \), \[ E^{\text{fs}}_{nj}=\frac{(E_n)^2}{2mc^2}\left(3-\frac{4n}{\,j+\tfrac12\,}\right), \] then add the weak-field Zeeman splitting \( E_Z=\mu_B g_J B\,m_j \) with the Landé factor \( g_J \).

Why it matters

The Bohr spectrum \( E_n=-13.6\,\text{eV}/n^2 \) leaves every level of fixed \( n \) exactly degenerate in \( \ell \) and \( j \). Fine structure is the first place that degeneracy breaks: it is the observable fingerprint of special relativity and of electron spin acting inside a real atom, and it sets the scale — a few parts in \( 10^{5} \) of the binding energy — at which spectroscopy stops agreeing with the simple Coulomb picture.

The same machinery underlies the sodium doublet, the design of atomic clocks, and the interpretation of the anomalous Zeeman effect that first forced spin onto physics. Getting the \( j \)-dependence and the Landé factor right is the difference between a toy spectrum and one that matches a spectrometer to five significant figures.

Assumptions

The electron is bound in a pure Coulomb potential with \( V(r)=-\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r} \); if screening, finite nuclear size, or QED vacuum polarization are included, \( \langle 1/r^3\rangle \) and \( \psi(0) \) shift and the Lamb shift (not captured here) lifts the residual \( \ell \)-degeneracy of equal \( j \).

The corrections are small compared with the Bohr spacing so first-order perturbation theory suffices; if \( \alpha \) were \( O(1) \) the expansion of \( \sqrt{p^2c^2+m^2c^4} \) would not converge and only the full Dirac equation would do.

The unperturbed states may be re-chosen within each degenerate \( n \)-shell so that \( H_{\text{SO}} \) is diagonal; because \( \vec{S}\cdot\vec{L} \) does not commute with \( L_z \) and \( S_z \) separately, using the uncoupled basis \( |m_\ell m_s\rangle \) would give a non-diagonal perturbation and wrong first-order energies — degenerate perturbation theory forces the coupled basis \( |j\,m_j\rangle \).

The proton is an infinitely heavy spinless point charge so that hyperfine structure (proton spin) and reduced-mass and recoil effects are dropped; restoring proton spin adds a further \( \sim 10^{3} \) times smaller splitting on top of every fine-structure level.

Derivation

\[ T=\sqrt{\hat{p}^2c^2+m^2c^4}-mc^2=\frac{\hat{p}^2}{2m}-\frac{\hat{p}^4}{8m^3c^2}+\cdots \]

Expand the relativistic kinetic energy in powers of \( (\hat{p}/mc)^2 \); the first term is the Bohr kinetic energy, the second is the leading correction \( H_{\text{rel}} \). A

\[ E^1_{\text{rel}}=-\frac{1}{8m^3c^2}\langle n\ell m|\hat{p}^4|n\ell m\rangle=-\frac{1}{2mc^2}\big\langle (E_n-V)^2\big\rangle \]

On an eigenstate \( \hat{p}^2\psi=2m(E_n-V)\psi \); since \( H_{\text{rel}} \) commutes with \( L^2 \) and \( L_z \) it is already diagonal in \( |n\ell m\rangle \), so first-order PT applies directly. B

\[ E^1_{\text{rel}}=-\frac{1}{2mc^2}\Big[E_n^2-2E_n\,\kappa\langle 1/r\rangle+\kappa^2\langle 1/r^2\rangle\Big]=-\frac{(E_n)^2}{2mc^2}\left[\frac{4n}{\ell+\tfrac12}-3\right] \]

Insert \( V=-\kappa/r \) with \( \kappa=e^2/4\pi\varepsilon_0 \) and the Coulomb expectation values \( \langle 1/r\rangle=1/(n^2a) \), \( \langle 1/r^2\rangle=1/[n^3(\ell+\tfrac12)a^2] \); use \( E_n=-\kappa/(2n^2 a) \) to eliminate \( a \). C

\[ H_{\text{SO}}=\frac{1}{2m^2c^2}\,\frac{1}{r}\frac{dV}{dr}\,\vec{S}\cdot\vec{L}=\frac{1}{2m^2c^2}\,\frac{\kappa}{r^3}\,\vec{S}\cdot\vec{L} \]

In the electron rest frame the nuclear Coulomb field appears partly magnetic, \( \vec{B}'=-\vec{v}\times\vec{E}/c^2 \), coupling to \( \vec{\mu}_s=-(e/m)\vec{S} \); the factor \( \tfrac12 \) is the Thomas precession correction for the non-inertial frame. A

\[ \vec{S}\cdot\vec{L}=\tfrac12\left(\hat{J}^2-\hat{L}^2-\hat{S}^2\right)=\frac{\hbar^2}{2}\Big[j(j+1)-\ell(\ell+1)-\tfrac34\Big] \]

Diagonalize the angular part by adding \( \vec{L} \) and \( \vec{S} \) (Clebsch-Gordan): in the coupled basis \( |n\,\ell\,j\,m_j\rangle \) the operator \( \vec{S}\cdot\vec{L} \) is diagonal, which is exactly the basis degenerate PT selects. B

\[ E^1_{\text{SO}}=\frac{\kappa}{2m^2c^2}\,\frac{\hbar^2}{2}\big[j(j+1)-\ell(\ell+1)-\tfrac34\big]\,\langle 1/r^3\rangle=\frac{(E_n)^2}{mc^2}\,\frac{n\big[j(j+1)-\ell(\ell+1)-\tfrac34\big]}{\ell\,(\ell+\tfrac12)(\ell+1)} \]

Insert \( \langle 1/r^3\rangle=1/[n^3\ell(\ell+\tfrac12)(\ell+1)a^3] \) (valid for \( \ell\neq 0 \)) and re-express \( a,\kappa \) through \( E_n \). C

\[ H_{\text{D}}=\frac{\hbar^2}{8m^2c^2}\nabla^2 V=\frac{\pi\hbar^2\kappa}{2m^2c^2}\,\delta^3(\vec{r}),\qquad E^1_{\text{D}}=\frac{\pi\hbar^2\kappa}{2m^2c^2}\,|\psi_{n00}(0)|^2=\frac{(E_n)^2}{2mc^2}\,4n \]

The Dirac reduction smears the electron over a Compton wavelength (Zitterbewegung), giving a contact term \( \propto\nabla^2 V=4\pi\kappa\,\delta^3(\vec{r}) \); only \( \ell=0 \) states have \( \psi(0)\neq 0 \), with \( |\psi_{n00}(0)|^2=1/(\pi n^3 a^3) \). C

\[ E^{\text{fs}}_{nj}=E^1_{\text{rel}}+E^1_{\text{SO}}+E^1_{\text{D}}=\frac{(E_n)^2}{2mc^2}\left(3-\frac{4n}{\,j+\tfrac12\,}\right) \]

For \( \ell\neq0 \) add steps 3 and 6 (Darwin absent); for \( \ell=0 \) add steps 3 and 7 (spin-orbit absent, \( j=\tfrac12 \)). Both cases give the identical \( j \)-only result — the algebraic miracle of hydrogen fine structure. B

\[ H_Z=-\vec{\mu}\cdot\vec{B}=\frac{e}{2m}\big(\vec{L}+2\vec{S}\big)\cdot\vec{B}=\frac{\mu_B}{\hbar}\big(\hat{L}_z+2\hat{S}_z\big)B,\quad \mu_B=\frac{e\hbar}{2m} \]

Add a uniform field \( \vec{B}=B\hat{z} \); the electron magnetic moment is \( \vec{\mu}=-\dfrac{e}{2m}(\vec{L}+g_s\vec{S}) \) with \( g_s\simeq2 \). In the weak-field regime treat \( H_Z \) as a perturbation on the fine-structure eigenstates \( |n\ell j m_j\rangle \). A

\[ \langle \vec{L}+2\vec{S}\rangle=g_J\,\langle\vec{J}\rangle,\quad g_J=1+\frac{j(j+1)-\ell(\ell+1)+\tfrac34}{2j(j+1)}\ \Rightarrow\ E^1_Z=\mu_B\,g_J\,B\,m_j \]

Within a fixed-\( j \) multiplet the projection (Wigner-Eckart) theorem replaces \( \vec{L}+2\vec{S} \) by its component along \( \vec{J} \); the coefficient is the Landé factor with \( s=\tfrac12 \). Each \( j \)-level fans into \( 2j+1 \) equally spaced sublevels. C

Result

\[ E_{nj}=-\frac{mc^2\alpha^2}{2n^2}\left[\,1+\frac{\alpha^2}{n^2}\left(\frac{n}{\,j+\tfrac12\,}-\frac34\right)\right],\qquad E^{\text{weak}}_{Z}=\mu_B\,g_J\,B\,m_j \]

Reading. The Bohr energy acquires a fractional correction of order \( \alpha^2\approx 5.3\times10^{-5} \) that depends on \( n \) and \( j \) but not on \( \ell \): states of equal \( n \) and \( j \) (e.g. \( 2S_{1/2} \) and \( 2P_{1/2} \)) remain degenerate at this order. Larger \( j \) sits higher. Switching on \( B \) splits each \( j \)-level into \( 2j+1 \) uniformly spaced lines whose spacing \( \mu_B g_J B \) carries the Landé factor, so different terms split at visibly different rates — the anomalous Zeeman effect.

Units check. \( (E_n)^2/mc^2 \) is (energy)²/(energy) = energy. \( \mu_B B \) is \( (\text{J T}^{-1})(\text{T})=\text{J} \); \( g_J \) and \( m_j \) are dimensionless. \( \alpha \) is dimensionless, so \( E_{nj} \) has the units of \( mc^2 \), i.e. energy. Consistent throughout.

Limiting cases

\( \alpha\to0 \): the bracket \( \to 1 \) and the Coulomb spectrum \( E_n=-mc^2\alpha^2/2n^2 \) is recovered with all fine structure switched off.
\( j=n-\tfrac12 \) (maximal, circular orbits): the correction is smallest in magnitude, \( \propto (\tfrac{n}{n}-\tfrac34)=\tfrac14 \); these near-classical states are least perturbed.
Weak field \( \mu_B B\ll E^{\text{fs}} \): \( j,m_j \) are good quantum numbers and \( E_Z=\mu_B g_J B m_j \) (anomalous Zeeman).
Strong field \( \mu_B B\gg E^{\text{fs}} \) (Paschen-Back): \( \vec{L} \) and \( \vec{S} \) decouple, \( m_\ell,m_s \) become good, and \( E_Z\to\mu_B B(m_\ell+2m_s) \) with fine structure as the small correction.
Singlet-like \( s=0 \) (hypothetical): \( g_J\to1 \) and the normal Zeeman triplet returns; the anomaly is entirely a spin effect.

Breaks when

High \( Z \) or strong binding (\( Z\alpha\sim1 \)): the perturbative \( \sqrt{p^2c^2+m^2c^4} \) expansion diverges; the corrections are no longer small and one must solve the full Dirac equation, whose exact levels depend on \( j \) through \( \sqrt{(j+\tfrac12)^2-(Z\alpha)^2} \) rather than the truncated formula.
The \( 2S_{1/2}\text{–}2P_{1/2} \) degeneracy is probed: fine structure predicts these exactly equal, but the measured Lamb shift (\( \sim1058 \) MHz) from QED radiative corrections breaks it — an effect entirely outside this derivation.
Intermediate field \( \mu_B B\sim E^{\text{fs}} \): neither \( j \) nor \( m_\ell,m_s \) are good quantum numbers; \( H_{\text{fs}} \) and \( H_Z \) must be diagonalized together (Breit-Rabi), and the simple \( \mu_B g_J B m_j \) formula fails.
Hyperfine resolution: at splittings below \( \sim10^{-6} \) eV the proton-spin coupling dominates and the electron-only \( j \) levels are no longer the eigenstates.

Failure modes

Forgetting the Thomas \( \tfrac12 \): the naive rest-frame boost gives a spin-orbit term twice too large; the factor \( \tfrac12 \) from Thomas precession is essential and doubling it destroys agreement with experiment.
Using the uncoupled basis for spin-orbit: evaluating \( \langle m_\ell m_s|\vec{S}\cdot\vec{L}|m_\ell m_s\rangle \) with non-degenerate PT gives zero off-diagonal handling and wrong energies; \( \vec{S}\cdot\vec{L} \) requires the coupled \( |j m_j\rangle \) basis.
Dividing by \( \ell \) at \( \ell=0 \): plugging \( \ell=0 \) into the spin-orbit formula gives \( 0/0 \); the \( \ell=0 \) shift comes entirely from the Darwin term, not from taking a limit of \( E^1_{\text{SO}} \).
Assigning \( g_J=2 \) to everything: \( g_s\simeq2 \) is the spin \( g \)-factor, not the Landé factor; the observable splitting uses \( g_J \), which varies with the term (e.g. \( \tfrac23 \) for \( P_{1/2} \), \( \tfrac43 \) for \( P_{3/2} \)).
Treating fine structure and Zeeman on equal footing at all fields: applying \( \mu_B g_J B m_j \) in a strong field, where \( j \) is not good, gives spurious level orderings; the regime must be checked first.
Sign slips in \( E_n^2 \): since \( E_n<0 \) but \( E_n^2>0 \), students often carry an extra minus and invert the \( j \)-ordering; larger \( j \) must come out higher.

Discussion

The deep surprise is that three physically distinct effects — a relativistic mass increase, a spin-orbit magnetic torque, and a contact smearing of the wavefunction — conspire to produce a result that depends only on \( j \). This is not coincidence: all three are the order-\( (v/c)^2 \) pieces of a single object, the Dirac equation for an electron in a Coulomb field. Expanding Dirac's exact energy \( E_{nj}=mc^2[1+(Z\alpha)^2/(n-j-\tfrac12+\sqrt{(j+\tfrac12)^2-(Z\alpha)^2})^2]^{-1/2} \) in powers of \( Z\alpha \) reproduces our perturbative formula term by term, with \( \ell \) nowhere in sight because the Dirac Hamiltonian's conserved quantum numbers are \( n \) and \( j \), not \( \ell \).

The \( \ell \)-independence is exact only at this order and only for the pure Coulomb problem. It is the reason \( 2S_{1/2} \) and \( 2P_{1/2} \) coincide in the fine-structure picture — a degeneracy so clean that its eventual breaking, the Lamb shift, became the experimental doorway to quantum electrodynamics. Fine structure thus sits at a hinge in the history of physics: it is the last thing a "semiclassical spin plus relativity" story explains, and its failures point directly at field-theoretic corrections.

The Zeeman half of the result encodes how angular momentum is shared. The Landé factor \( g_J \) is nothing but the projection of the magnetic moment \( \vec{L}+2\vec{S} \) onto the total \( \vec{J} \): because spin contributes twice its share to the moment but the same share to \( \vec{J} \), the ratio is not simply 1, and the resulting non-uniform splitting of multiplets was the very anomaly that Uhlenbeck and Goudsmit resolved by inventing spin. Measuring \( g_J \) is a direct read-out of the coupling scheme.

At the most rigorous level, the Darwin term resists a purely classical picture: it arises because in the Foldy-Wouthuysen reduction of the Dirac equation the position operator that couples to \( V \) is the mean position averaged over the rapid \( 2mc^2/\hbar \) Zitterbewegung oscillation, so the electron effectively samples \( V \) over a region of size the Compton wavelength \( \hbar/mc \). The Taylor expansion of that averaging is precisely \( \tfrac{\hbar^2}{8m^2c^2}\nabla^2 V \), and since \( \nabla^2(1/r)=-4\pi\delta^3(\vec{r}) \) the correction lives entirely at the origin — which is why only penetrating \( s \)-states feel it. Common misconceptions: (i) that spin-orbit alone explains all hydrogen fine structure — it does not touch \( \ell=0 \); (ii) that the splitting scales as \( \alpha \) — it scales as \( \alpha^4 mc^2 \), i.e. \( \alpha^2 \) relative to \( E_n \); (iii) that fine structure removes all \( n=2 \) degeneracy — the \( j=\tfrac12 \) pair survives until QED intervenes.

Worked examples

Fine-structure splitting of the \( n=2 \) shell: \( 2P_{3/2} \) versus \( 2P_{1/2}=2S_{1/2} \).

Use \( E^{\text{fs}}_{nj}=\dfrac{(E_n)^2}{2mc^2}\left(3-\dfrac{4n}{j+\tfrac12}\right) \) with \( n=2 \), \( mc^2=5.110\times10^5\,\text{eV} \). A

\[ E_2=-\frac{13.6\ \text{eV}}{4}=-3.40\ \text{eV},\qquad \frac{(E_2)^2}{2mc^2}=\frac{(3.40)^2}{2(5.110\times10^5)}=1.131\times10^{-5}\ \text{eV} \]

Symbols before numbers; the common prefactor is computed once. A

\[ j=\tfrac32:\ 3-\frac{8}{2}=-1\ \Rightarrow\ E=-1.131\times10^{-5}\,\text{eV};\qquad j=\tfrac12:\ 3-\frac{8}{1}=-5\ \Rightarrow\ E=-5.66\times10^{-5}\,\text{eV} \]

Evaluate the bracket for each \( j \). A

\[ \Delta E_{3/2-1/2}=4.53\times10^{-5}\ \text{eV}\ \approx\ 10.9\ \text{GHz}\ \approx\ 0.365\ \text{cm}^{-1} \]

Reading. The \( 2P_{3/2} \) level lies \( 45\,\mu\text{eV} \) above \( 2P_{1/2} \); this is the textbook hydrogen fine-structure doublet, in agreement with spectroscopy (the residual \( 2S_{1/2}\text{–}2P_{1/2} \) Lamb shift of \( \sim4\,\mu\text{eV} \) is a separate QED effect). Units check. eV throughout; conversion \( 1\,\text{eV}=2.418\times10^{14}\,\text{Hz} \) gives the frequency.

Weak-field Zeeman splitting of \( 2P_{1/2} \) in \( B=0.10\ \text{T} \).

First confirm the regime, then use \( E_Z=\mu_B g_J B m_j \). Here \( \ell=1,\ s=\tfrac12,\ j=\tfrac12 \). A

\[ \mu_B B=(5.788\times10^{-5}\,\text{eV/T})(0.10\,\text{T})=5.79\times10^{-6}\,\text{eV}\ \ll\ E^{\text{fs}}\sim6\times10^{-5}\,\text{eV} \]

Weak-field condition satisfied, so \( j,m_j \) are good quantum numbers. A

\[ g_J=1+\frac{\tfrac12\cdot\tfrac32-1\cdot2+\tfrac34}{2\cdot\tfrac12\cdot\tfrac32}=1+\frac{0.75-2+0.75}{1.5}=1-\frac13=\frac23 \]

Landé factor for the \( ^2P_{1/2} \) term. B

\[ m_j=\pm\tfrac12:\quad E_Z=\frac23(5.788\times10^{-5})(0.10)\left(\pm\tfrac12\right)=\pm1.93\times10^{-6}\,\text{eV} \]

Insert numbers. A

\[ \Delta E_{\text{Zeeman}}=E(m_j{=}{+}\tfrac12)-E(m_j{=}{-}\tfrac12)=3.86\times10^{-6}\ \text{eV}\ \approx\ 933\ \text{MHz} \]

Reading. The single \( 2P_{1/2} \) level splits into two sublevels separated by \( \sim0.9 \) GHz at \( 0.1\,\text{T} \); note this is smaller than the \( g_J=\tfrac43 \) splitting a \( P_{3/2} \) level would show at the same field, which is the observable signature of the anomalous Zeeman effect. Units check. \( (\text{eV T}^{-1})(\text{T})=\text{eV} \); \( g_J,m_j \) dimensionless.

Problems

Compute the relativistic-kinetic shift \( E^1_{\text{rel}} \) of the hydrogen ground state \( 1S_{1/2} \) in eV.
Solution
For \( n=1,\ell=0 \): \( E^1_{\text{rel}}=-\dfrac{(E_1)^2}{2mc^2}\left[\dfrac{4n}{\ell+\tfrac12}-3\right]=-\dfrac{(13.6)^2}{2(5.110\times10^5)}\left[\dfrac{4}{0.5}-3\right] \). The prefactor is \( \dfrac{185.0}{1.022\times10^6}=1.810\times10^{-4}\,\text{eV} \); the bracket is \( 8-3=5 \). Thus \( E^1_{\text{rel}}=-1.810\times10^{-4}\times5=-9.05\times10^{-4}\,\text{eV} \).
Show that for \( \ell=0,\ j=\tfrac12 \) the relativistic and Darwin terms together reproduce the general fine-structure formula (spin-orbit being absent).
Solution
For \( \ell=0 \) the spin-orbit shift vanishes, so \( E^{\text{fs}}=E^1_{\text{rel}}+E^1_{\text{D}} \). From step 3 with \( \ell=0 \), \( \ell+\tfrac12=\tfrac12 \), so \( E^1_{\text{rel}}=-\dfrac{(E_n)^2}{2mc^2}\left[\dfrac{4n}{1/2}-3\right]=-\dfrac{(E_n)^2}{2mc^2}(8n-3) \). From step 7, \( E^1_{\text{D}}=\dfrac{(E_n)^2}{2mc^2}(4n) \). Their sum is \( \dfrac{(E_n)^2}{2mc^2}\big[-(8n-3)+4n\big]=\dfrac{(E_n)^2}{2mc^2}(3-4n) \). The general formula at \( j=\tfrac12 \) gives \( \dfrac{(E_n)^2}{2mc^2}\left(3-\dfrac{4n}{1}\right)=\dfrac{(E_n)^2}{2mc^2}(3-4n) \). The two agree, so the closed \( j \)-only form holds for \( s \)-states, with the Darwin contact term supplying exactly what spin-orbit cannot.
Evaluate the Landé factor \( g_J \) for the \( 3D_{5/2} \) term and give the number of Zeeman sublevels.
Solution
\( D \) means \( \ell=2 \); \( j=\tfrac52,\ s=\tfrac12 \). \( g_J=1+\dfrac{\tfrac52\cdot\tfrac72-2\cdot3+\tfrac34}{2\cdot\tfrac52\cdot\tfrac72}=1+\dfrac{8.75-6+0.75}{17.5}=1+\dfrac{3.5}{17.5}=1+0.2=1.2=\dfrac{6}{5} \). Sublevels: \( 2j+1=6 \), with \( m_j=-\tfrac52,\dots,+\tfrac52 \).
At what field \( B \) does the weak-field Zeeman splitting of \( 2P_{3/2} \) (\( g_J=\tfrac43 \)) between adjacent \( m_j \) equal the \( 2P_{3/2}\text{–}2P_{1/2} \) fine-structure gap of \( 4.53\times10^{-5}\,\text{eV} \)? Comment on the regime.
Solution
Adjacent-\( m_j \) spacing \( =\mu_B g_J B=(5.788\times10^{-5})(\tfrac43)B=7.72\times10^{-5}B \) eV (B in tesla). Set equal to \( 4.53\times10^{-5} \): \( B=4.53\times10^{-5}/7.72\times10^{-5}=0.587\,\text{T} \). At \( B\approx0.6\,\text{T} \) the Zeeman and fine-structure energies are comparable, so this is the intermediate (Breit-Rabi) regime — neither the weak-field \( \mu_B g_J B m_j \) nor the Paschen-Back formula is accurate, and \( H_{\text{fs}}+H_Z \) must be diagonalized jointly.
Using \( E_{nj}=-\dfrac{mc^2\alpha^2}{2n^2}\left[1+\dfrac{\alpha^2}{n^2}\left(\dfrac{n}{j+1/2}-\dfrac34\right)\right] \), find the energy difference between \( 3D_{5/2} \) and \( 3P_{3/2} \) in hydrogen.
Solution
Both have \( n=3 \), so the Bohr and the \( -\tfrac34 \) pieces cancel; only the \( j \)-term differs. For \( 3D_{5/2} \), \( j=\tfrac52 \Rightarrow \dfrac{n}{j+1/2}=\dfrac{3}{3}=1 \). For \( 3P_{3/2} \), \( j=\tfrac32\Rightarrow\dfrac{3}{2}=1.5 \). Difference of the bracketed \( j \)-term: \( \Delta=-\dfrac{mc^2\alpha^2}{2n^2}\cdot\dfrac{\alpha^2}{n^2}(1-1.5)=-\dfrac{mc^2\alpha^4}{2n^4}(-0.5) \). With \( mc^2=5.110\times10^5\,\text{eV} \), \( \alpha^4=2.836\times10^{-9} \), \( n^4=81 \): \( \dfrac{mc^2\alpha^4}{2n^4}=\dfrac{5.110\times10^5\times2.836\times10^{-9}}{162}=8.94\times10^{-6}\,\text{eV} \). Times \( 0.5 \) gives \( E(3D_{5/2})-E(3P_{3/2})=+4.47\times10^{-6}\,\text{eV} \): the \( j=\tfrac52 \) state lies about \( 4.5\,\mu\text{eV} \) above \( j=\tfrac32 \), consistent with larger \( j \) sitting higher.