Codex @codex  0

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.

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]

The Lamb shift is a radiative displacement of atomic energy levels beyond the values from the Dirac equation in a Coulomb potential. In hydrogen, the $2S_{1/2}$ and $2P_{1/2}$ levels are degenerate for a point Coulomb potential in the Dirac equation, yet the $2S_{1/2}$ level lies higher by roughly a gigahertz. This is distinct from the fine structure splitting between different $J$ levels and from nuclear-spin hyperfine structure.

Electron self-energy in the quantized electromagnetic field is the dominant contribution to the familiar hydrogen splitting; vacuum polarization contributes too. Renormalization separates the physical electron mass from its divergent free-electron self-energy. A useful leading-log scale for a hydrogenic $nS$ level isnot an exact numerical formula. The enhanced sensitivity of S states follows from their nonzero wavefunction at the nucleus; leading short-distance corrections act like contact perturbations. A precision prediction also includes finite nuclear size and recoil, and must consistently separate those effects from the radiative contribution.^[ref]

The original Lamb-Retherford experiment used a beam of hydrogen atoms, electron-impact excitation into metastable $2S$, and microwave-induced transfer into short-lived $2P$ states. Those atoms decayed before reaching the detector. Metastable atoms surviving to a metal target ejected electrons, so a drop in the measured current signalled a resonance. Varying the microwave excitation and magnetic field established the displacement from the Dirac equation expectation. Microwave quenching converted a small energy difference into a measurable loss of a long-lived population, avoiding optical Doppler widths that obscure the splitting.^[ref]

For hydrogen, the leading characteristic scales areOrder-one spin and nuclear magnetic factors, and state-dependent factors, are suppressed here. Gross atomic structure is Coulomb binding. Fine structure includes relativistic kinetic and spin-orbit effects, with Thomas precession affecting the spin-orbit coefficient. Hyperfine structure couples electronic motion and spin to the nuclear magnetic moment, whose inverse nuclear mass makes it much weaker. The contact spin interaction can remain in an s state where orbital spin-orbit coupling vanishes.

For positronium, the reduced mass is $m_e/2$. Its gross levels and separation scale areThus the mean ground-state electron–Positron separation is twice the infinite-proton-mass hydrogen value: both means have the same factor $3/2$ multiplying their respective Bohr radii. Retaining proton recoil makes the ratio $2m_p/(m_p+m_e)$, very slightly below two. Separation is not the distance of either positronium constituent from its centre of mass.

Interpret the spin operators in units of $\hslash$, as required by the printed Zeeman term, so $A$ has units of energy. Two spin halves give four eigenstates. At zero field, total spin $F=0,1$ and projection $m_F$ label the singlet and triplet. At high field relative to the hyperfine scale, the individual projections $m_{s,e},m_{s,p}=\pm 1/2$ become approximately good quantum numbers. Total projection remains exact at every field, but $F^2$ does not for the two mixed states.

Choose the quantization axis along $\mathbf{B}$. The aligned-spin states $|\uparrow \uparrow ⟩$ and $|\downarrow \downarrow ⟩$ both have $E=A/4$ because their Zeeman shifts cancel. In the other product-state sector,For physical positronium $A>0$. At $B=0$ there is a triplet at $A/4$ and a singlet at $-3A/4$. At large positive field the mixed states approach $|\uparrow \downarrow ⟩$ and $|\downarrow \uparrow ⟩$, with energies $-A/4+2{\mu }_{B}B$ and $-A/4-2{\mu }_{B}B$ respectively, while the aligned pair remains degenerate at $A/4$.

The dimensionless mixing parameter is $x=4{\mu }_{B}B/A$. A crossover estimate is $x\sim 1$, rather than a sharp transition. The scale $A\sim {\alpha }^4{m}_e{c}^2$ already gives a field of a few tesla. Using the physical hyperfine interval $A/h\simeq 203.39\mathrm{GHz}$ givesThe numerical hyperfine interval is external physical input, not a number supplied by the symbolic Hamiltonian.^[ref] The high-field limit here still presumes that orbital distortion is negligible compared with Coulomb binding.

In the ideal point-nucleus Dirac-Coulomb problem, hydrogen levels depend on $n$ and $j$, so $2S_{1/2}$ and $2P_{1/2}$ are degenerate despite their different orbital angular momenta. The Lamb shift removes this degeneracy: the $2S_{1/2}$ level lies about $h\times 1.06\mathrm{GHz}$ above $2P_{1/2}$, an energy of roughly $4.4\mu \mathrm{eV}$. This is distinct from the much larger $2P_{3/2}-2P_{1/2}$ fine-structure splitting.

quantum electrodynamics changes the electron's interaction with the Coulomb field. electron self-energy and vacuum polarization give state-dependent radiative corrections; a complete prediction also keeps specified recoil and nuclear-size contributions. An $S$ wave samples the nucleus differently from a $P$ wave, so equal $n,j$ no longer ensures equal energy. A leading-log estimate for an $nS$ radiative contribution scales as $\alpha (Z\alpha {)}^4{m}_e{c}^2\ln [(Z\alpha {)}^{-2}]/n^3$, with a numerical prefactor and further state-dependent terms. This is a scale estimate, not a replacement for renormalized bound-state QED. The measurable shift is a difference relative to a specified reference Hamiltonian, rather than an unregulated absolute vacuum energy.^[ref]

The original microwave experiment used a tungsten oven to dissociate hydrogen, an electron beam to populate metastable $2S$ atoms, and electron ejection from a metal target to measure surviving metastable atoms. Microwaves transferred population to rapidly decaying $2P$, reducing detector current. At a fixed microwave frequency, interruption of the radiation and a scan of static magnetic field revealed Zeeman-tuned resonances. Comparing resonance locations with the Dirac-level prediction showed the $S$ level shifted upward. Field calibration and homogeneity, excessive microwave intensity, unresolved hyperfine structure and stray-electric-field quenching limit the measurement; the signal is not direct absorption by a large equilibrium excited population.^[ref]

A good atomic qubit needs two reproducible, orthogonal internal states whose populations and relative quantum phase survive for many gate times. Both should be long lived, with weak spontaneous emission and little decoherence from fluctuating fields or atomic motion. Their transition must nevertheless permit fast, selective, coherent control with little leakage to other states. Reliable initialization and state-selective quantum measurement are also required; optical pumping and fluorescence readout can supply these functions.

For rubidium-87, use the two ground state hyperfine levels in the $5{}^2{S}_{1/2}$ electronic configuration. The electronic angular momentum is $J=1/2$, and addition of angular momentum to the given nuclear spin $I=3/2$ gives $F=|I-J|,\dots ,I+J=1,2$. A suitable Rubidium-87 clock qubit isThese are coupled electron-nucleus angular momentum eigenstates, rather than two bare nuclear-spin projections. Their hyperfine structure supplies a resolvable splitting ${\nu }_{01}\simeq 6.835\mathrm{GHz}$. Both belong to the electronic ground state, avoiding a short optical excited-state lifetime. Since the weak-field Zeeman shift is proportional to $g_F{m}_{F}B$, both have zero linear shift at $B=0$: their rubidium-87 clock transition suppresses first-order magnetic-field dephasing near zero field. A quadratic Zeeman shift of a clock transition and differential optical shifts remain. A small stable bias field can define the quantization axis and help resolve unwanted transitions, at the cost of some residual magnetic sensitivity.^[ref]

Drive the magnetic dipole transition with a resonant microwave field of suitable polarization, here $\Delta m_F=0$, or use two phase-coherent optical fields for a far-detuned stimulated Raman transition with their frequency difference resonant with ${\nu }_{01}$. The optical excited state is then only virtually populated, reducing spontaneous emission. In the rotating frame and rotating-wave approximation, define the drive phase $\varphi$ by the effective HamiltonianHere $X,Y,Z$ are the Pauli matrices on the chosen qubit, and $\Omega$ is the Rabi angular frequency. Pulse duration controls the rotation gate angle, while phase selects the axis in the equatorial plane of the Bloch sphere. A $\pi$ pulse exchanges the populations; a $\pi /2$ pulse makes an equal quantum superposition from either basis state. Controlled Rabi detuning, a differential energy shift, or compositions of these pulses give $z$ rotations. For example, $R_z(\alpha )R_y(\beta )R_z(\gamma )$ realizes any one-qubit unitary operator up to a global phase. Thus the atomic encoding admits arbitrary single-qubit quantum gates.

In a small trapped-ion quantum computer, use two long-lived internal states of each ion, for example a pair of hyperfine ground states or a narrow optical transition, as $|0⟩,|1⟩$. Electric trapping fields confine a short ion string, laser cooling reduces its motion, and optical pumping initializes the internal states. State-dependent fluorescence supplies readout: one qubit level participates in a bright cycling transition and the other is dark.

A resonant microwave, Raman or optical drive performs a single-qubit Rabi rotation. The drive phase fixes a rotation axis in the Bloch equatorial plane and its pulse area fixes the angle; detuning or phase-frame control supplies Z rotations. Individual spatial or frequency addressing selects the intended ion.

For a two-qubit gate, couple internal states to shared quantized vibrational modes. Resolved motional sidebands let one ion's spin influence the collective oscillator and then the second ion's spin. For example, bichromatic red- and blue-sideband drives produce a spin-dependent force; a closed motional phase-space trajectory returns the oscillator to its initial state while accumulating an entangling interaction proportional to $X_i{X}_j$. Together with single-qubit rotations this implements a CNOT. The short chain has well-separated normal-mode frequencies, and sideband cooling and the Lamb-Dicke regime suppress unwanted motion and make coherent bus control practical.

The five DiVincenzo criteria are: a scalable collection of well-characterized qubits; preparation in a known fiducial quantum state; relevant coherence times long compared with quantum gate times; a universal quantum gate set; and qubit-specific quantum measurement. The criterion is useful control of the whole computation, including errors, rather than merely observing two resolved levels.^[ref]

For trapped ions, the implementation is:

For trapped neutral atoms, the corresponding implementation is:

Thus neither platform is forbidden from meeting a criterion in principle. Weak neutral-ground-state interactions make entangling control harder, but claiming that trapped neutral atoms cannot implement a universal quantum gate set would be false. Gate-model quantum algorithms using Rydberg blockade had already been demonstrated in a programmable neutral-atom processor by 2022.^[ref]

A spectral line has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish homogeneous spectral broadening of each emitter from inhomogeneous spectral broadening across the ensemble.

Finite lifetime and dephasing. For an upper-state population lifetime $\tau$ and a stable lower state, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Fourier spectrum is Lorentzian, with full width at half maximumMore generally, an optical coherence time $T_2$ gives $\Delta {\omega }_{\mathrm{hom}}=2/T_2$. Both level lifetimes and pure dephasing can contribute to $1/T_2$. Distinguishing population lifetime from amplitude coherence time prevents the usual factor-of-two error.

Thermal motion. To first order in velocity, $\delta \nu ={\nu }_0{v}_{\parallel }/c$. A Maxwell-Boltzmann velocity distribution therefore produces a Gaussian profile, withThis is inhomogeneous doppler broadening: different velocity classes have different resonance frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.

Interactions with the surroundings. Collisions can interrupt the dipole wave phase and perturb the energy levels, producing pressure broadening and shifts. In the impact approximation, approximately exponential wave phase relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and resonance interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.^[ref]

Finite observation time limits coherence; transit-time broadening also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes power broadening: for an ideal lifetime-limited two-level transition, $\Delta \omega =\Gamma \sqrt{1+s}$ with population decay rate $\Gamma$ and saturation parameter $s$. Unresolved fine structure, hyperfine structure, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.

The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a Voigt profile; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.

For the nonrelativistic Coulomb Hamiltonian with reduced electron-proton mass $\mu$, separation in Spherical coordinate system gives ${\psi }_{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi )$. The quantum numbers obey $n\ge 1$, $0\le \ell \le n-1$ and $-\ell \le m\le \ell$. These are joint eigenstates of energy, $L^2$ and $L_z$, with the latter two eigenvalues $\ell (\ell +1){\hslash }^2$ and $m\hslash$.

The Coulomb bound-state energies and characteristic length areFor example, ${\psi }_{100}=(\pi a_{\mu }^3{)}^{-1/2}{e}^{-r/a_{\mu }}$ is spherically symmetric and has no radial node. In general the radial part contains an exponential times a Generalized Laguerre polynomial, with $n-\ell -1$ radial nodes. The spherical harmonic specifies angular structure and azimuthal phase; real linear combinations of degenerate $m$ states give familiar directional orbitals.

The leading Coulomb energy depends on $n$ alone, giving $n^2$ spatial states for each principal shell, or $2n^2$ including electron spin. Fine structure, Lamb shift and Hyperfine structure lift parts of this ideal degeneracy. An atomic orbital is a probability amplitude, not a classical planetary path.^[ref]

Operators, quantization and rotations. Angular momentum is the generator of spatial rotations. In quantum mechanics, its components are Hermitian operators on their appropriate domains and obeyThese angular momentum commutation relations prevent all three components from having simultaneous sharp values. For example, the Robertson uncertainty principle gives $\Delta J_x\Delta J_y\ge \hslash |⟨J_z⟩|/2$. One conventionally diagonalizes ${\hat{J}}^2$ and a selected component ${\hat{J}}_z$:The allowed labels are $j=0,\frac{1}{2},1,\frac{3}{2},\dots$ and $m=-j,-j+1,\dots ,j$. An angular-momentum multiplet has $2j+1$ states. The magnitude $\hslash \sqrt{j(j+1)}$ differs from the largest possible component $\hslash j$; the remaining components fluctuate even in a state with maximal $m$.

The angular momentum lowering operator and its raising counterpart, ${\hat{J}}_{\pm }={\hat{J}}_x\pm i{\hat{J}}_y$, satisfyThe ladder terminates at $m=\pm j$. A rotation through angle $\theta$ about unit vector $\hat{\mathbf{n}}$ acts through the unitary operator $\exp [-i\theta \hat{\mathbf{n}}\cdot \hat{\mathbf{J}}/\hslash ]$. Thus quantized components and the continuous rotation of a state are compatible.

Orbital angular momentum. The orbital contribution is associated with motion in space:For a spinless spatial wavefunction, the simultaneous angular eigenfunctions are the spherical harmonics $Y_{\ell m}(\theta ,\varphi )$, with $\ell =0,1,2,\dots$ and $m=-\ell ,\dots ,\ell$. Single-valued ordinary scalar spatial wavefunctions give integer orbital labels. In a central potential energy, $[\hat{H},{\hat{L}}^2]=[\hat{H},{\hat{L}}_z]=0$, so stationary states may be labelled by these angular quantum numbers. The angular kinetic contribution appears in the radial equation as $\ell (\ell +1){\hslash }^2/(2m{r}^2)$, the centrifugal potential. Noncentral interactions need not conserve these orbital labels.

Intrinsic spin and addition. Spin angular momentum is an intrinsic internal degree of freedom with the same commutation algebra. It is not the angular momentum of a small classical charged sphere literally rotating about its centre. For an electron, $s=1/2$ and $m_s=\pm 1/2$, so ${\hat{S}}_i=(\hslash /2){\sigma }_i$ in terms of the Pauli matrices. Its squared spin is $3{\hslash }^2/4$. Other particles can have integer or half-integer spin, including spin zero.

For one particle with orbital and spin degrees of freedom, $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$; the two component sets commute with each other. Addition of angular momentum permits $j=|\ell -s|,|\ell -s|+1,\dots ,\ell +s$. The coupled states $|j,m_j⟩$ are linear combinations of the uncoupled $|\ell ,m_{\ell }⟩|s,m_s⟩$ with $m_j=m_{\ell }+m_s$, determined by Clebsch-Gordan coefficients. A rotationally invariant spin-orbit interaction can conserve $J^2,J_z$ while mixing the separate component labels.

Angular momentum also carries a magnetic moment. For an electron's orbital motion and intrinsic spin,The minus signs reflect negative electric charge. With the Bohr magneton ${\mu }_B=e\hslash /(2m_e)$, a spin projection $m_s$ has ${\mu }_z=-g_s{\mu }_B{m}_s$. A magnetic field can therefore distinguish angular-momentum states through their interaction energy $-\mathbf{\mu }\cdot \mathbf{B}$.

Stern–Gerlach measurement. In the Stern-Gerlach experiment, atoms emerge from a heated source, pass through collimating apertures, traverse a strongly inhomogeneous magnetic field, and strike a position-sensitive screen. The beam is neutral, so its separation is interpreted through the atomic magnetic moment rather than the ordinary charged-particle Lorentz force. Near a chosen quantization direction, the interaction givesA uniform field can exert a torque but would not separate the beam by this translational force; the gradient is essential. Different allowed magnetic-moment components become correlated with different trajectories. A classical ensemble with continuously varying orientations would yield a continuous spread, whereas discrete quantum components yield separated channels.

In the modern electronic description of ground-state silver, filled shells cancel their electronic angular momenta and the unpaired $5s$ electron has $L=0$, $S=1/2$, $J=1/2$. The effective electronic measurement therefore has two components, approximately ${\mu }_z=\pm {\mu }_B$, rather than an orbital $\ell =1/2$, which is not allowed. For positive field gradient the electronic $m_J=+1/2$ component has negative ${\mu }_z$ and deflects downward in this coordinate convention. Nuclear spin and hyperfine structure are neglected in this simplified two-channel account. Historically the original splitting preceded the electron-spin interpretation, so the modern explanation should not be confused with the original investigators' interpretation.

More generally a prepared spin state $a|+z⟩+b|-z⟩$ gives the two outcomes with probabilities $|a{|}^2,|b{|}^2$. Selecting one channel prepares a state for further measurements. An ideal second analyser along the same axis then gives a definite result, whereas one along a perpendicular axis gives two equally likely outcomes for a spin-one-half eigenstate. This connects the apparatus directly to incompatible spin components and the Born rule. Angular momentum is quantized through its operator algebra; the Stern–Gerlach gradient turns that discrete internal information into spatially distinguishable outcomes.

In the weak-field Zeeman effect, spin-orbit coupling keeps $L,S,J$ good and a level shifts by ${\mu }_B{g}_{J}B{m}_J$. At fields large compared with fine-structure coupling but still within the linear electronic Zeeman approximation, $L$ and $S$ decouple: the Paschen-Back effect gives shifts ${\mu }_{B}B(m_L+2m_S)$. Electric-dipole transitions obey $\Delta m_L=0,\pm 1$, $\Delta m_S=0$ in that limit, recovering an orbital triplet pattern. Intermediate fields mix the coupled states.

For the weak-field sodium D-line Zeeman pattern, the given Landé g-factor formula givesAllowed photon shifts are $\delta E={\mu }_{B}B(g_u{m}_u-2m_l)$ with $m_u-m_l=0,\pm 1$. ThusThese are the four- and six-component patterns. Both $\pi$ and $\sigma$ components are represented, so an observation strictly along the field, which removes $\pi$ emission, would not show this complete pattern. Hyperfine structure is neglected.

To determine the magnetic field from a Zeeman diagram, the unsplit separation provides its energy scale:The smallest adjacent split spacing $s_E=2{\mu }_B{B}_1/3$ is approximately $0.070$ of that separation in the sketch. ThereforeThis is a graphical estimate; the drawing does not warrant high precision. Its splitting energy is small compared with $\Delta E_D$, consistent with the weak-field approximation.

For Doppler resolution of a Zeeman pattern, thermal velocities give Gaussian Doppler broadening with widthAt $350\mathrm{K}$, the photon-energy FWHM near $589\mathrm{nm}$ is $5.88\times 10^{-6}\mathrm{eV}$. Equating the smallest adjacent Zeeman spacing to this width gives $B\sim 0.15\mathrm{T}$ for loss of individually resolved components. There is no sharp Gaussian resolution threshold; using whole-pattern width instead gives a smaller field of order $0.03$–$0.04\mathrm{T}$ for complete disappearance into a single envelope. Instrumental and collisional broadening would also affect the estimate.

In the regime of weak-field Zeeman splitting, $J,m_J$ remain good labels and $\Delta E={\mu }_B{g}_J{m}_{J}B$. For the 393.4 nm line, $J_u=3/2$ in $4p{}^2{P}_{3/2}$ and $J_l=1/2$ in $4s{}^2{S}_{1/2}$. The Landé g-factors are $g_u=4/3$ and $g_l\simeq 2$. Apply $\Delta m_J=0,\pm 1$ to the four upper and two lower sublevels. The allowed pairs $(m_u,m_l)$ areTheir frequency displacements in units of ${\mu }_{B}B/h$ are respectively $1,-1/3,-5/3,5/3,1/3,-1$, so none coincide. Looking perpendicular to the field permits both the $\pi$ radiation from $q=0$ transitions and the $\sigma$ radiation from $q=\pm 1$ transitions. ConsequentlyA linear polarizer parallel to the field isolates the two $\pi$ components; a transverse polarizer selects the four $\sigma$ components. Without polarization selection both sets are present. This count assumes a field small compared with fine structure coupling and neglects isotope-dependent hyperfine structure; the strong-field Paschen-Back effect or a resolved nonzero-spin isotope would require a different model. The spectral spacings are unnecessary for the count, but the distinct shifts verify that counting allowed transitions does not inadvertently count degenerate lines twice.

For two valence electrons, the central-field shell structure should be the largest scale, residual electrostatic interactions should establish a reasonably definite configuration and atomic term, and spin-orbit coupling should be weaker than the separations between the relevant terms. Then $L$ and $S$ remain approximately good quantum numbers. Within a well-isolated term the leading fine structure interaction can be represented as $A\mathbf{L}\cdot \mathbf{S}$. Its coefficient must be effectively constant for all $J$ levels in that term. Mixing of neighboring terms, spin-spin interactions, higher-order corrections or strong applied fields can spoil the simple interval rule; hyperfine structure must be negligible or separately resolved.

A suitable approximate example is the magnesium $3s3p{}^3{P}_J$ term, with $L=1$, $S=1$ and $J=0,1,2$. The interval-rule prediction isAn arbitrary common term energy may be added. For $A>0$ the levels ascend in the order ${}^3{P}_0,{}^3{P}_1,{}^3{P}_2$, with gap ratio one to two. Each level still contains $2J+1$ degenerate magnetic sublevels in zero external field. The superscript three labels the spin multiplicity $2S+1$, while the different subscripts identify fine structure components of that same term. This is an approximate example of LS coupling, not a claim that every real two-electron atom has exact one-to-two fine splitting.

For a hyperfine multiplet governed by the magnetic-dipole hyperfine interaction, adjacent frequency gaps obey ${\nu }_F=|A|F/h$. The five measured gaps for the first isotope are approximately in the ratios $1:2:3:4:5$. Therefore the six levels have $F=0,1,2,3,4,5$: the first measured gap corresponds to higher $F=1$. In particular $F_{\min }=|I-J|=0$ and $F_{\max }=I+J=5$, givingThis inference uses both the number of intervals and their successive ratios. Merely seeing six levels would specify only $\min (I,J)=5/2$ and would not by itself establish equality.

Solving a linear least-squares problem for a fit through the origin, taking the rounded frequencies as equally weighted, givesThus the scale of the hyperfine structure is about $4.80\times 10^{-26}$ J, or $3.00\times 10^{-7}$ eV. The displayed frequencies have unequal rounding precision, so additional fit digits are not physically warranted. The gap pattern supports a leading dipole interval rule, with small residuals that do not establish higher-multipole constants without uncertainties.

The total width between $F=0$ and $F=5$ is $15|A_{55}|$, about $1.086$ GHz in frequency units. The sign of $A_{55}$, the absolute nuclear g-factor and the electronic field coefficient cannot be separated from these gap magnitudes alone. Measuring the level ordering or using another known magnetic property would supply additional information. The figure in b shows the positive-$A$ ordering as an illustration.

For the electronic ${}^2{S}_{1/2}\to {}^2{P}_{1/2}$ transition, neglect hyperfine structure and use weak-field LS coupling. The Landé g-factors are $g_g=2$ and $g_e=2/3$, so a transition from $m_g$ to $m_e$ shifts by $({\mu }_{B}B/h)(g_e{m}_e-g_g{m}_g)$.

With propagation perpendicular to the magnetic field, linear polarization parallel to the field drives the $\pi$ electric-dipole selection rule $\Delta m=0$. The two shifts are $\pm 2{\mu }_{B}B/(3h)$. Perpendicular linear polarization is an equal-amplitude combination of the two field-axis circular components, driving ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$ and shifts $\pm 4{\mu }_{B}B/(3h)$. Each geometry therefore selects a doublet, with twice the separation in the perpendicular-polarization case. Equal line heights within a pair require equal lower-state populations and an unsaturating probe; line positions do not require that extra population assumption.