Atomic and laser physics course of the University of Oxford (Physics Part B, B3)

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Source bundle pages 129–133. The cover establishes 2022 despite the attachment filename. Solutions use the paper's rounded numerical data.

In LS coupling, first add the individual orbital angular momenta and spins separately, then add their totals:The Coulomb interaction between electrons must separate different atomic terms more strongly than the spin-orbit interaction mixes them. Then $L,S,J,M_J$ are useful quantum numbers, with $J=|L-S|,\dots ,L+S$. This approximation is usually better for lighter chemical atoms and well-separated atomic terms; strong configuration interaction or strong spin-orbit coupling can spoil it. The term symbol is ${}^{2S+1}{L}_J$.

The general electric-dipole selection rules, for isolated chemical atoms with definite parity and total electronic angular momentum, are a change of parity, $\Delta J=0,\pm 1$ excluding $J=0\leftrightarrow 0$, and $\Delta M_J=0,\pm 1$, with the accessible magnetic sublevels determined by the light's polarization. In pure LS coupling, the additional selection rules are $\Delta S=0$ and $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$. In a one-electron description the electric dipole operator also requires $\Delta l=\pm 1$ for the active electron. These latter statements can fail as exact labels when atomic terms mix; the general parity and total-angular momentum restrictions still apply. Unresolved hyperfine structure is neglected here.

For strontium, the closed core contributes $L=S=0$. The Pauli exclusion principle makes the two equivalent $5s$ electrons a singlet state. Different radial orbitals permit both a singlet state and a triplet state. Thus the complete lists of atomic fine-structure levels areThe $5s5p$ electron configuration has odd parity; the other two electron configurations have even parity.

Measure each energy level by the wavenumber $\tilde{E}=(E-E_g)/(hc)$ in ${\mathrm{m}}^{-1}$. An upward electronic transition of vacuum wavelength $\lambda$ adds $1/\lambda$. Treat the quoted wavelengths as vacuum values to their limited accuracy; air corrections would be smaller than the precision justified here.

The strong ground-state absorption must be $5s^2{}^1{S}_0\to 5s5p{}^1{P}_1$ at $461\mathrm{nm}$. The weak ground-state electronic transition at $689\mathrm{nm}$ must reach $5s5p{}^3{P}_1$: the electric-dipole selection rule on $J$ excludes the other two triplet state energy levels, while weak spin-orbit coupling admixes singlet state character and relaxes the approximate spin selection rule.

Assume the usual ordering $E({}^3{P}_0)<E({}^3{P}_1)<E({}^3{P}_2)$, as expected from a positive effective spin-orbit coupling constant. The three nearby strong spectral lines then join these energy levels to a common $5s6s{}^3{S}_1$ energy level. Larger lower-state energy means a smaller gap and longer wavelength, so $679,688,707\mathrm{nm}$ correspond to $J=0,1,2$ respectively. The remaining $1.12\mu \mathrm{m}$ spectral line joins $5s5p{}^1{P}_1$ and $5s6s{}^1{S}_0$. Assume the latter is the upper energy level. These assignments and orderings agree with independently tabulated strontium energy levels.^[ref]

Starting with the two ground-state gaps, the construction isThe resulting energy levels, rounded to avoid suggesting spectroscopic precision, are

Both $461\mathrm{nm}$ and $1.12\mu \mathrm{m}$ exhibit the normal Zeeman effect. Each connects two singlet state energy levels, with Landé g-factor $g_J=1$ for the $J=1$ energy level and no first-order shift for $J=0$. Their frequency shifts are $0,\pm {\mu }_{B}B/h$. The weak $689\mathrm{nm}$ spectral line also has three components, but $g({}^3{P}_1)=3/2$ gives shifts $0,\pm 3{\mu }_{B}B/(2h)$; a three-component pattern alone does not establish the normal Zeeman effect. The $679\mathrm{nm}$ spectral line likewise has three components with $g({}^3{S}_1)=2$.

For $5s5d$, addition of angular momentum gives $L=2$ and $S=0,1$. The even-parity atomic terms and atomic fine-structure levels areThe electric-dipole selection rules exclude all connections to the even-parity $5s^2$ and $5s6s$ electron configurations. To the odd-parity $5s5p$ electron configuration, all seven allowed connections in pure LS coupling areHere a connection is listed independently of whether the $5s5d$ energy level is above or below its partner.

In a weak magnetic field, use the Landé g-factorFor the three ${}^3{D}_J$ energy levels, this gives $1/2,7/6,4/3$; for ${}^3{P}_1,{}^3{P}_2$ it gives $3/2$. The six-component spectral line is thereforeA component has frequency displacement $({\mu }_{B}B/h)(m_D/2-3m_P/2)$. The electric dipole transition with $m_D=m_P=0$ has zero matrix element, since the relevant $J=1\leftrightarrow 1$ rank-one Clebsch-Gordan coefficients vanish there. The remaining allowed pairs giveThus all six frequency positions $\pm 1/2,\pm 1,\pm 3/2$ are distinct. Perpendicular to the magnetic field, both the $\pi$ and $\sigma$ components are visible, with mutually perpendicular linear polarizations. Along the magnetic field the $\pi$ components disappear. This is the weak-field Zeeman effect, before the Paschen-Back effect changes the coupling scheme.

The closest components above differ by $\Delta {\nu }_{\min }={\mu }_{B}B/(2h)\simeq 7.0\mathrm{GHz}{\mathrm{T}}^{-1}B$. At $B=0.1\mathrm{T}$, their separation is about $0.70\mathrm{GHz}$. A visible spectral line near $500\mathrm{nm}$ would consequently require spectral resolving power of order $\nu /\Delta {\nu }_{\min }\simeq 9\times 10^5$. An ordinary low-resolution spectrometer would merge the components.

A suitable arrangement places an excited low-pressure strontium discharge or a transversely observed atomic beam between wide electromagnet pole pieces. Collect the emission perpendicular to the magnetic field, select the desired spectral line with a prefilter or monochromator, and send it through a high-finesse scanning Fabry–Pérot interferometer to a detector. A rotatable linear polarizer separates the two $\pi$ components from the four $\sigma$ components. Calibrate both the interferometer's free spectral range and the local magnetic field; take zero-field and several field-strength scans, and fit component positions against $B$.

The magnetic field should be uniform across the viewed volume, with its associated Zeeman splitting larger than the combined natural linewidth, Doppler broadening, pressure broadening and instrumental width, but much smaller than the separations of the relevant atomic fine-structure levels. A collimated transverse atomic beam, low collision rate and restricted collection angle reduce unwanted spectral line broadening. Avoid appreciable power broadening, resolve or model isotope and hyperfine structure, and check for self-absorption. A zero-field scan and the predicted linear scaling help distinguish actual Zeeman splitting components from neighbouring spectral lines.

Let the dimensionful angular momentum operators obey $\mathbf{F}=\mathbf{I}+\mathbf{J}$ and take the interaction Hamiltonian as $H^{\prime }=A\mathbf{I}\cdot \mathbf{J}$. Sincethe energy eigenvalues at fixed $I,J$ arewhere $F=|I-J|,\dots ,I+J$. Subtraction gives the Landé interval ruleMany atomic conventions use dimensionless angular momentum operators and call the energy-valued constant itself $A$; then the right-hand side is $AF$. Each energy level has $2F+1$ magnetic sublevels with the same energy in zero field.

For fine structure, the effective interaction is $a_{LS}\mathbf{L}\cdot \mathbf{S}$ with dimensionless angular momentum operators. It originates in the spin-orbit interaction, including the magnetic interaction of electron spin with orbital motion and its relativistic correction. Within an isolated LS coupling atomic term, the labels are the electron configuration, $L,S,J,M_J$, and the interval between adjacent $J$ energy levels is $a_{LS}J$. This is smaller than the electrostatic separation of atomic terms.

For magnetic-dipole hyperfine structure, the nuclear magnetic dipole moment interacts with the magnetic field generated by the electrons. Contributions include the Fermi contact interaction of penetrating $s$ electrons, orbital magnetic fields, and the dipolar field of electron spin. Projected onto one electronic $J$ energy level, their rotationally invariant magnetic-dipole part has the form $a_{IJ}\mathbf{I}\cdot \mathbf{J}$. The labels are $I,J,F,M_F$, together with the approximate electronic labels such as $L,S$. This splitting is generally smaller than fine structure, reflecting the much smaller nuclear magnetic dipole moment. An electric nuclear quadrupole moment can add a different interaction and violate the simple Landé interval rule.

At the larger electrostatic energy scale, an effective two-spin exchange interaction can also be written $a_s{\mathbf{s}}_1\cdot {\mathbf{s}}_2$ within fixed orbitals. Its origin is the Coulomb interaction combined with the Pauli exclusion principle, rather than a large direct magnetic interaction. It separates singlet state and triplet state atomic terms; before spin-orbit coupling, useful labels include the electron configuration, $L,M_L,S,M_S$. General electrostatic atomic term separations need not obey a single Landé interval rule as $L$ varies.

The $4f^7$ shell carries a large spin (its ground atomic term under Hund's rules has $L=0,S=7/2$), and the $6s$ electron penetrates the inner shells. Its nonzero probability density at the atomic nucleus supplies a potentially large Fermi contact interaction with the nuclear magnetic dipole moment. The $6s$ shell here is singly occupied, so the obligatory paired-$6s^2$ cancellation is absent. Both stable europium isotopes have nonzero nuclear spin, as the spectrum below also establishes.

The term symbol ${}^8{D}_{11/2}$ specifies $S=7/2$, $L=2$, $J=11/2$. Its large $J$, combined with appreciable nuclear spin, permits a wide range of total angular momentum $F$ and hence a substantial total hyperfine structure span. For $I=5/2$, $F=3,\dots ,8$ and the span is $30|a|$. The penetrating $6s$ contribution and the available range of $F$ explain the large hyperfine structure; the term symbol alone does not determine its constant, which also depends on electronic recoupling, radial wavefunctions and configuration interaction.

Write $a_{\nu }=a/h$, so the Landé interval rule for hyperfine structure intervals in ordinary frequency is ${\nu }_F-{\nu }_{F-1}=a_{\nu }F$. For the first four selected positions, the successive positive differences areTheir ratios are approximately $4:5:6$, sinceThe decreasing frequencies therefore correspond to increasing $F$, with $a_{\nu }\simeq -0.706\mathrm{GHz}$, if increasing printed position means increasing physical frequency. Continuing the intervals predicts additional positions near $10.60-7(0.706)\simeq 5.66$ and then $5.66-8(0.706)\simeq 0.01\mathrm{GHz}$. They identify peaks $i$ and $l$ unambiguously. The full assignment for ${}^{151}\mathrm{Eu}$ isWith $J=11/2$, these endpoints requireEquivalently, a complete set of six hyperfine levels gives $2I+1=6$ when $I<J$. This inference assumes that every upper hyperfine level contributes an observed position and the negligible lower hyperfine structure lets coincident transitions be grouped by their upper energy. A line count without these completeness assumptions would not alone fix nuclear spin.

The supplied positions obey the Landé interval rule only approximately. For example, ${\nu }_e-{\nu }_i=4.97\mathrm{GHz}$, whereas $7({\nu }_a-{\nu }_b)/4=4.935\mathrm{GHz}$. Even allowing each printed position a rounding error of $0.005\mathrm{GHz}$ does not fully reconcile these two relations. A least-squares dipole-only fit has a largest position residual of about $0.017\mathrm{GHz}$. Small additional hyperfine interactions, measurement uncertainty beyond the displayed precision, or an error in the tabulated data could account for this; it is inappropriate to claim an exact identity. The grouping and nuclear spin are nevertheless robust. Reversing the convention for the printed frequency axis reverses the inferred signs of both constants, without changing the following ratio.

The remaining six positions give the other isotope's sequenceThe adjacent differences divided by $4,5,6,7,8$ are $0.3125,0.3140,0.3133,0.3143,0.3138\mathrm{GHz}$, respectively. Thus $a_{\nu ,153}\simeq -0.314\mathrm{GHz}$ and the same endpoint argument gives $\boxed{I_{153}=5/2}$.

For the same electronic energy level, the magnetic-dipole hyperfine structure constant is proportional to ${\mu }_I/I$, to leading order in the small isotope-dependent change of the electronic wavefunction. Since the two nuclear spins are equal,Using the full spans averages over the approximate Landé interval rule: ${\sum }_{F=4}^{8}F=30$, soThe constants have the same sign, so the nuclear magnetic dipole moments have the same relative sign. Their absolute sign cannot be obtained from their ratio without the electronic field's sign. Finite nuclear-size corrections to the hyperfine interaction, often called the hyperfine anomaly, and electric-quadrupole corrections are neglected at this accuracy.

Choose the relative phases of the two energy eigenstates so that $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$ is real and nonnegative. Define the resonant Rabi angular frequency $\Omega =d/\hslash$, the Rabi detuning $\delta ={\omega }_0-\omega$, and $\Sigma ={\omega }_0+\omega$. Substitution into the Schrödinger equation, followed by projection onto the two eigenstates, gives the exact interaction picture equationsKeep both probability amplitudes dynamical: replacing $c_1$ by one would incorrectly impose weak excitation and miss complete population transfer.

For $|\delta |,\Omega \ll {\omega }_0$, apply the rotating-wave approximation to discard the rapidly oscillating terms at $\Sigma$. Differentiating the second remaining equation and using the first givesThe initial conditions are $c_2(0)=0$ and ${\dot{c}}_2(0)=-i\Omega /2$. With ${\Omega }_R=\sqrt{{\Omega }^2+{\delta }^2}$, their solution isConsequently,This result permits order-one excitation, including full population transfer on resonance, even though the drive is small compared with the optical angular frequency.

For an average from switch-on to time $\tau$, integrate the squared sine:On resonance this limiting average is $1/2$, although the instantaneous probability repeatedly reaches one. The zero-coupling, zero-Rabi detuning case is understood separately as $P_2=0$.

Near resonance alone does not make this an exact result for arbitrarily strong driving. The stated sinusoidal Hamiltonian also needs the rotating-wave approximation. A concrete counterexample is $\omega ={\omega }_0$ and $\Omega ={\omega }_0$: the exact equations give $P_2(t)={\Omega }^2{t}^2+O(t^4)$ at switch-on, while the claimed rotating-wave sinusoid gives ${\Omega }^2{t}^2/4+O(t^4)$. Thus “without assuming weak radiation” can mean without assuming small excitation, but cannot remove the drive-to-carrier scale separation. The infinite-time average above belongs to the rotating-wave model; omitted corrections can accumulate over sufficiently long times.

Rabi oscillations require a coherent drive with a stable phase over several population cycles and an isolated, appreciably coupled pair of quantum states. For large-amplitude oscillations, the Rabi detuning must be comparable to or smaller than the Rabi angular frequency. The population lifetime, optical coherence time, transit time and drive coherence time should all exceed $2\pi /{\Omega }_R$ by enough to resolve the cycles. An ensemble also needs sufficiently uniform Rabi frequencies and Rabi detunings that phase averaging does not wash the oscillations out. The usual sinusoidal formula additionally assumes the rotating-wave approximation.

The Einstein coefficients describe incoherent transition rates, rather than the evolving optical probability amplitudes. Their simplest use is appropriate for broadband, randomly phased radiation, with field correlation time short relative to population evolution, or for rapidly dephased chemical atoms whose optical coherences can be eliminated. The radiation energy density per unit frequency must be smooth over the relevant spectral line. Stimulated upward and downward transitions and spontaneous emission then enter rate equations, without predicting coherent population return. A short observation time can give a small excitation probability even for a coherent field; that condition alone does not justify an Einstein coefficients rate description. Coherent laser irradiation can also admit steady rate equations after optical coherence has relaxed, with the appropriate saturation response.

Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower energy level to be stable. Spontaneous emission gives survival probability $e^{-\Gamma t}$, hence a field or excited-state probability amplitude proportional to $e^{-\Gamma t/2}{e}^{-i{\omega }_{0}t}$ for $t\ge 0$. Its Fourier transform at angular frequency $\omega$ is proportional toThe squared modulus therefore has the Lorentzian line shapeHere $L$ is normalized over the full mathematical frequency axis. The full width at half maximum in ordinary frequency is $1/(2\pi T_1)$. If both energy levels decay, their population decay rates add in the natural linewidth; pure dephasing adds to the amplitude-decay rate instead.

Impact-regime pressure broadening provides another example. Independent phase-randomizing collisions with rate ${\gamma }_c$ give optical correlation proportional to $e^{-{\gamma }_c|t|}$, whose Fourier transform is a Lorentzian line shape of full width at half maximum $2{\gamma }_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a Lorentzian line shape.

Use first-order time-dependent perturbation theory, setting $c_1=1$ in the equation for $c_2$ and retaining the generally complex matrix element $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$. With $\delta ={\omega }_0-\omega$ and $\Sigma ={\omega }_0+\omega$, the probability amplitude isFor $t>0$, split the integral at the pulse maximum. For either $x=\delta$ or $x=\Sigma$,Thus the full first-order answer, without discarding the counter-rotating contribution, isAt long times it givesThis full expression is not exactly one squared Lorentzian line shape. To obtain the intended near-resonance spectral line profile, additionally assume $b,|\delta |\ll {\omega }_0$, so the pulse has a narrow positive-frequency lobe and $F(\Sigma ,t)$ is negligible. ThenThe weak-excitation condition near resonance is $|d|/(\hslash b)\ll 1$. The squared Lorentzian line shape occurs because the symmetric exponential pulse has a Lorentzian line shape amplitude Fourier transform, which is then squared to obtain probability.

At half the maximum, $[1+(\delta /b{)}^2{]}^2=2$, soThe full width at half maximum refers to varying the drive frequency through the positive-frequency resonance. For a concrete failure of the unqualified squared-Lorentzian line shape claim, set $b=\omega ={\omega }_0$ while keeping $|d|/(\hslash b)\ll 1$. The full first-order resonant probability is $36/25$ times the one-lobe prediction. Weak radiation alone therefore does not justify dropping the second Fourier transform lobe.

Define $r=g_2/g_1$ and let $\sigma ={\sigma }_{21}(\omega )$ be the stimulated-emission cross section, averaged over upper magnetic sublevels. The Einstein coefficients relation $g_1{B}_{12}=g_2{B}_{21}$ gives the absorption cross section ${\sigma }_{12}=r\sigma$. Assume equally populated magnetic sublevels within each energy level, a steady pump, and a monochromatic beam. The photon flux is $I/(\hslash \omega )$; define $W=\sigma I/(\hslash \omega )$ and the effective population inversion $D=N_2-r{N}_1$. Then the rate equations areThe $WD$ term includes both stimulated emission and absorption. In steady state, adding the two rate equations gives $N_1=R_2{\tau }_1$. The first equation then yieldsThe directed beam gains energy at rate $\hslash \omega WD$ per volume, giving $dI/dz=\sigma DI$ and the optical gain coefficient $\alpha =\sigma D$. At line centre, write ${\sigma }_0={\sigma }_{21}({\omega }_{21})$. ThereforeHere ${\alpha }_0$ is the small-signal optical gain coefficient and $I_s$ is the saturation intensity, at which the local optical gain is half its small-signal value. Positive optical gain requires $r{A}_{21}{\tau }_1<1$: the lower energy level must empty sufficiently quickly.

The cancellation of ${\tau }_1$ from $I_s$ is specific to this model's steady fixed pump and the fact that every upper spontaneous emission feeds the lower energy level. Adding an extra lower-state accumulation factor to $I_s$ would contradict these rate equations. Alternative conventions using the absorption cross section must replace ${\sigma }_0$ by ${\sigma }_{12}/r$ throughout. Homogeneous spectral broadening permits the whole selected ensemble to share this local saturation response.

In a uniform steady amplifier, with no extra propagation losses, the local optical gain law givesIntegrate between the two faces of length $L$ to obtainFor $I\ll I_s$, this reduces to exponential amplification. In deep saturation, the intensity increment approaches ${\alpha }_0{I}_{s}L$: finite pumping limits the extracted power.

Let $x=I_0/I_s$ and $G={\alpha }_{0}L$. Applying the integrated amplifier law to each section givesEliminating $G$ givesAs a consistency check, both amplifiers have $G\simeq 26.1175$. The second section receives a larger intensity and adds $80I_0$, compared with $69I_0$ in the first, while its multiplicative optical gain is much smaller. This is the expected approach to constant extracted power in saturation.

For a closed absorbing two-level system, write $N=N_1+N_2$, let $A=1/\tau$, and use ${\sigma }_a={\sigma }_{12}$ for the absorption cross section. This differs from the stimulated-emission cross section convention in part (a). With $r=g_2/g_1$, the rate equations areAt steady state, $N_2=W_{a}N/[A+(1+1/r)W_a]$ for $W_a={\sigma }_{a}I/(\hslash \omega )$. Hence the positive absorption coefficient isFor the given sodium fine-structure transition, the statistical weights are $g_1=2$, $g_2=4$, so $r=2$ and $1+1/r=3/2$. The small-signal transmission gives the optical depthTo account for the finite decrease of intensity through the sample, integrate $dI/dz=-\kappa I$ rather than treating the stronger beam as exactly uniform. If $T=I_{\mathrm{out}}/I_{\mathrm{in}}$, this givesTaking $270\mathrm{W}{\mathrm{m}}^{-2}$ as the incident intensity and $T=0.967$,Using the photon energy $hc/\lambda$ then givesIgnoring the small variation of the stronger intensity would instead give about $1.11\times 10^{-13}{\mathrm{m}}^2$; the integrated value uses the stated transmissions more consistently. The length is not needed to eliminate $N$, but gives the check $N\simeq 9.3\times 10^{14}{\mathrm{m}}^{-3}$.

The resonant absorption cross section is of optical-wavelength-squared size, much larger than a typical atomic geometric collision area. Pressure broadening cross sections involve phase-changing interatomic encounters and need not equal hard-sphere collision areas; their numerical value depends on collision partners and speed. No precise ratio follows from the supplied data. A useful distinction is that resonant absorption involves coherent response on the optical scale, so its cross section need not resemble an chemical atom's geometric area.

For homogeneous spectral broadening, all participating chemical atoms must share the same resonance and single-atom response width. In a dilute sample dominated by natural linewidth, the weak-beam requirement isUsing the sodium atomic mass, equality occurs near $T=0.017\mathrm{K}$, so $T\ll 17\mathrm{mK}$ is a sufficient thermal condition. Spatial Zeeman energy shifts, light shifts and transit effects should also be small, the laser should address the same selected component, and the interaction should last long enough for steady populations. Uniform impact pressure broadening can also be homogeneous, but would change the width and effective cross section; it is not included in the lifetime-only numerical model.

Real sodium also has hyperfine structure and optical pumping among magnetic sublevels. The inferred ${\sigma }_a$ is the effective cross section of the stated $g_J=2J+1$ two-level system model, assuming statistical sublevel populations and no loss into unaddressed states. A measurement on a resolved, polarized closed hyperfine structure component requires its actual transition dipole moment and selection rules; a universal polarization-independent sodium cross section cannot be inferred from this simplified model. The dependence of resonant cross sections and saturation intensities on sublevel preparation is tabulated explicitly for the real sodium D lines.^[ref]

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The source bundle filename differs from the examination year; the cover identifies Trinity Term 2021. The solutions below retain the printed conventions where consistent and explain the corrections needed elsewhere.

The printed magnetic dipole moment uses dimensionless angular momentum operators: their physical counterparts are $\hslash \mathbf{L}$ and $\hslash \mathbf{S}$. Choose the magnetic field along $z$. Then the Zeeman energy is $H_Z={\mu }_{B}B(L_z+g_s{S}_z)$, whose characteristic size is ${\mu }_{B}B$. In LS coupling, treating this as a time-independent perturbation theory requires it to be much smaller than the relevant fine structure separation, rather than merely smaller than the Coulomb interaction energy.

For a concrete hydrogen atom estimate take the $2p_{1/2},2p_{3/2}$ pair. The hydrogen fine-structure correction iswhere ${\alpha }_{\mathrm{fs}}$ is the fine-structure constant, distinguished from the later potential coefficient. ThusConsequently a suitable weak-magnetic field condition for this pair isThere is no single threshold for every hydrogen energy level: the relevant fine structure separations decrease with $n$. The $1s$ energy level has no same-$n$ spin-orbit interaction partner, so it cannot itself define this comparison. This calculation neglects hyperfine structure; resolving that smaller scale requires an additional weak-magnetic field condition.

Within a fixed $L,S,J$ angular momentum multiplet, the component of $\mathbf{L}$ along $\mathbf{J}$ follows from $2\mathbf{L}\cdot \mathbf{J}={\mathbf{J}}^2+{\mathbf{L}}^2-{\mathbf{S}}^2$. The corresponding result for $\mathbf{S}$ follows by interchanging $L$ and $S$. For the quantum state $|LSJ{M}_J⟩$,These are dimensionless expectation values in the convention of part (a). Substituting into the first-order time-independent perturbation theory formula $\Delta E=⟨H_Z⟩$ givesFor $g_s=2$, the Landé g-factor becomesThe magnetic sublevels must first be diagonalized within a degenerate multiplet; choosing $M_J$ along the magnetic field already does this. For $J=0$ the displayed fraction is undefined, but all vector expectation values vanish and there is no first-order Zeeman splitting. A higher-order shift may remain.

Write $k=e^2/(4\pi {ϵ}_0)$ and retain the nuclear Coulomb potential. In spherical coordinates, the total potential energy iswhen $\beta =-\alpha /2$; $P_2(u)=(3u^2-1)/2$ is a Legendre polynomial. Treat the added term using quadratic anisotropic perturbation of hydrogen and degenerate perturbation theory in the $n=2$ hydrogen energy eigenbasis. It preserves parity and $m$, so it cannot mix $2s$ with $2p$, or distinct $2p$ magnetic sublevels.

For a hydrogen atom, the hydrogen radial second moment givesThe angular expectation values of ${\cos }^2\theta$ are $1/3$ for an $s$ orbital, $3/5$ for $p_0$, and $1/5$ for $p_{\pm 1}$, obtained by integrating the squared spherical harmonics. Therefore the five lowest spatial quantum states, counting statistical weight, have first-order energy levelsEach spatial quantum state has two electron spin states when spin-orbit coupling and the magnetic field are omitted. Thus these are five spatial states but only three distinct first-order energies; five distinct energies would be an incorrect reading. The accidental $2s,2p_{\pm 1}$ equality can be lifted at higher order. Quantitatively the added shifts must also be small compared with neighboring principal-shell gaps; $\alpha \ll k/a_0^3$ is an asymptotic condition, not a guarantee for arbitrary large $n$.