--- title: The Bohr-Sommerfeld Old Quantum Theory module: Early Atomic Models and the Old Quantum Theory moduleNumber: 1 lessonNumber: 4 order: 104 summary: > Bohr fixed the hydrogen levels with a single quantum number by quantizing angular momentum. Sommerfeld replaced that ad hoc rule with a general prescription: quantize the action of each separable coordinate. The rule produces elliptical orbits, a second (azimuthal) quantum number, space quantization, and — once the relativistic mass variation is included — a fine-structure splitting that matches experiment to order alpha squared. topics: [Early Atomic Models and the Old Quantum Theory] sources: - book: Bransden & Joachain ref: "Ch. 2 — The Old Quantum Theory; §2.5 The Bohr-Sommerfeld Rules, §2.6 Fine Structure" - book: Foot ref: "Ch. 1 — Early Atomic Physics; Appendix on the old quantum theory" - book: Demtröder ref: "Ch. 3 — Development of Quantum Physics; §3.4 The Bohr-Sommerfeld Model" draft: false --- Bohr's condition $L = n\hbar$ quantizes one degree of freedom, the angle swept by a circular orbit. A general bound motion has more degrees of freedom, and a Coulomb orbit is generically an ellipse, not a circle. In 1916 Sommerfeld supplied the missing generalization: quantize the **action** of every separable coordinate independently. The prescription reproduces the Bohr energies, introduces a second quantum number that labels the shape of the orbit, quantizes the orientation of the orbital plane in space, and — with the relativistic correction to the electron mass — splits each Bohr level into a fine-structure multiplet. This is the high-water mark of the old quantum theory, the last atomic model built from classical orbits before wave mechanics replaced them. ## The action-integral quantization rule For a periodic coordinate $q$ with conjugate momentum $p$, the **action variable** is the phase-space area enclosed over one period, $$ J = \oint p\,\d q . $$ Sommerfeld and Wilson postulated that each such action is an integer multiple of Planck's constant.[^bj-rule] > **Postulate (Wilson-Sommerfeld quantization).** For a separable system whose > motion is periodic in each coordinate $q_i$, the allowed states are those for > which every action integral is quantized, > $$ > \oint p_i\,\d q_i = n_i h, \qquad n_i = 0, 1, 2, \dots, > $$ > the integral running over one full period of $q_i$. The rule is dimensionally forced: $p\,\d q$ has units of action, the same as $h$, and no other combination of the orbital constants is dimensionless when divided by $h$. Its deeper justification is Ehrenfest's adiabatic principle, taken up at the end of this lesson: the action integrals coincide with the mechanical quantities that stay constant when the system is deformed slowly, so they are the only quantities that can carry an unchanging integer label. $$ % caption: The phase-space trajectory of a one-dimensional bound motion is a % closed loop; the enclosed area equals the action J, and quantization admits % only the loops whose area is an integer multiple of h. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-3.4,0) -- (3.6,0) node[right, font=\scriptsize] {$q$}; \draw[->, black] (0,-2.3) -- (0,2.5) node[above, font=\scriptsize] {$p$}; \draw[acc, very thick, fill=acc!10] (0,0) ellipse [x radius=2.7, y radius=1.7]; \draw[black, thick] (0,0) ellipse [x radius=1.8, y radius=1.13]; \draw[black, thick] (0,0) ellipse [x radius=0.9, y radius=0.57]; \node[acc, font=\scriptsize] at (1.35,1.2) {area $= nh$}; \end{tikzpicture} $$ > **Worked example (The harmonic oscillator).** A one-dimensional oscillator of > mass $m$ and angular frequency $\omega$ has energy > $E = p^2/2m + \tfrac12 m\omega^2 q^2$. At fixed $E$ the phase-space curve is the > ellipse > $$ > \frac{p^2}{2mE} + \frac{q^2}{2E/m\omega^2} = 1, > $$ > with semi-axes $\sqrt{2mE}$ and $\sqrt{2E/m\omega^2}$. Its area is > $$ > \oint p\,\d q = \pi\sqrt{2mE}\,\sqrt{\frac{2E}{m\omega^2}} = \frac{2\pi E}{\omega}. > $$ > Setting this equal to $nh = 2\pi n\hbar$ gives $E_n = n\hbar\omega$. The result > is off by the zero-point energy $\tfrac12\hbar\omega$; the old rule cannot > supply the half-integer, a defect repaired only by the > [WKB correction](/atomic-physics/early-models-and-old-quantum-theory/old-quantum-theory-limits-wkb). ## The circular orbit recovered For the hydrogen atom, use plane polar coordinates $(r, \varphi)$ in the orbital plane. The Lagrangian $L = \tfrac12 m(\dot r^2 + r^2\dot\varphi^2) + \kappa/r$, with the Coulomb strength $\kappa \equiv Ze^2/4\pi\epsilon_0$, gives the conjugate momenta $$ p_\varphi = m r^2\dot\varphi, \qquad p_r = m\dot r . $$ Because $\varphi$ is cyclic, $p_\varphi$ is a constant of the motion — it is the orbital angular momentum $L$. Its action integral over one revolution is immediate: $$ \oint p_\varphi\,\d\varphi = p_\varphi \int_0^{2\pi}\d\varphi = 2\pi L . $$ Quantizing with the azimuthal integer $k$ gives $2\pi L = k h$, that is $L = k\hbar$. For a circular orbit $p_r \equiv 0$, so the radial action vanishes and $k$ alone labels the state. Identifying $k$ with Bohr's $n$ reproduces $L = n\hbar$ and every result of the [Bohr model](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen). The content of Sommerfeld's extension is what happens when $p_r$ does not vanish. ## Elliptical orbits and the radial quantum number An orbit with $p_r \neq 0$ is an ellipse with the nucleus at one focus. The radial momentum follows from energy conservation. Writing the energy with the centrifugal term, $$ E = \frac{p_r^2}{2m} + \frac{L^2}{2mr^2} - \frac{\kappa}{r} \quad\Longrightarrow\quad p_r = \sqrt{2mE + \frac{2m\kappa}{r} - \frac{L^2}{r^2}} . $$ The motion oscillates between the perihelion $r_{\min}$ and aphelion $r_{\max}$, the two roots of the radicand, where $p_r = 0$. The radial action is the integral over one in-and-out excursion,[^bj-radial] $$ J_r = \oint p_r\,\d r = 2\int_{r_{\min}}^{r_{\max}} \sqrt{2mE + \frac{2m\kappa}{r} - \frac{L^2}{r^2}}\,\d r . $$ This is a standard integral of the form $\displaystyle\oint\sqrt{A + \frac{2B}{r} - \frac{C}{r^2}}\,\d r$ with $A = 2mE < 0$, $B = m\kappa$, and $C = L^2$. For a bound orbit ($A < 0$) its value is $$ J_r = 2\pi\left(\frac{B}{\sqrt{-A}} - \sqrt{C}\right) = 2\pi\left(\kappa\sqrt{\frac{m}{2\lvert E\rvert}} - L\right). $$ > **Proof.** Write the radicand over a common denominator, > $2mE + 2m\kappa/r - L^2/r^2 = (A r^2 + 2B r - C)/r^2$, with the quadratic > $A r^2 + 2B r - C = A(r - r_{\min})(r - r_{\max})$. Since $A < 0$, the quadratic > is positive between its roots, the two turning points. The integrand becomes > $\sqrt{\lvert A\rvert(r - r_{\min})(r_{\max} - r)}\,/\,r$, so > $$ > J_r = 2\sqrt{\lvert A\rvert}\int_{r_{\min}}^{r_{\max}} > \frac{\sqrt{(r - r_{\min})(r_{\max} - r)}}{r}\,\d r > = 2\sqrt{\lvert A\rvert}\,\pi\!\left(\frac{r_{\min} + r_{\max}}{2} > - \sqrt{r_{\min} r_{\max}}\right), > $$ > using the standard definite integral. The root sum and product read off the > quadratic are $r_{\min} + r_{\max} = -2B/A = \kappa/\lvert E\rvert$ and > $r_{\min} r_{\max} = -C/A = L^2/2m\lvert E\rvert$. Substituting, > $$ > J_r = 2\pi\sqrt{2m\lvert E\rvert} > \left(\frac{\kappa}{2\lvert E\rvert} - \frac{L}{\sqrt{2m\lvert E\rvert}}\right) > = 2\pi\left(\kappa\sqrt{\frac{m}{2\lvert E\rvert}} - L\right), > $$ > the quoted result. Quantizing $J_r = n_r h = 2\pi n_r\hbar$ and inserting $L = k\hbar$ gives $$ n_r\hbar = \kappa\sqrt{\frac{m}{2\lvert E\rvert}} - k\hbar \quad\Longrightarrow\quad (n_r + k)\,\hbar = \kappa\sqrt{\frac{m}{2\lvert E\rvert}} . $$ Define the **principal quantum number** $n \equiv n_r + k$. Squaring isolates the energy, > **Definition (Bohr-Sommerfeld energy levels).** With $n = n_r + k$, the bound > energies of a hydrogen-like atom are > $$ > E_n = -\frac{m\kappa^2}{2\hbar^2}\frac{1}{n^2} > = -\frac{Z^2 e^4 m}{2(4\pi\epsilon_0)^2\hbar^2}\frac{1}{n^2} > = -\frac{Z^2}{n^2}\,(13.6\,\text{eV}), > $$ > depending only on $n$ and not separately on $n_r$ or $k$. The energy is exactly Bohr's, but the state is now labelled by two integers. The radial quantum number $n_r = 0, 1, 2, \dots$ counts the radial oscillations, and the azimuthal quantum number $k = 1, 2, \dots, n$ fixes the angular momentum $L = k\hbar$. The value $k = 0$ is excluded: it describes a degenerate straight line through the nucleus, a collision orbit with no angular momentum, which the old theory rejects. For each $n$ there are $n$ allowed values of $k$, all degenerate in energy. ## The shape of the orbit The eccentricity of the ellipse is set by the ratio $k/n$. The angular momentum of a Kepler ellipse of semi-major axis $a$ and eccentricity $\varepsilon$ is $L = \sqrt{m\kappa a(1 - \varepsilon^2)}$, while $a = \kappa/2\lvert E\rvert$ fixes the semi-major axis from the energy alone. Both the circular orbit ($k = n$) and the ellipse of the same $n$ share this $a$, so $$ \frac{b}{a} = \sqrt{1 - \varepsilon^2} = \frac{L}{L_{\text{circ}}} = \frac{k}{n}, $$ where $b$ is the semi-minor axis and $L_{\text{circ}} = n\hbar$ is the angular momentum of the circle of the same energy. The orbit is roundest when $k = n$ and most elongated when $k = 1$; the perihelion of the $k = 1$ ellipse dips close to the nucleus, which is exactly where the relativistic correction below will bite hardest. $$ % caption: The three Sommerfeld orbits with n = 3 share the same semi-major axis % and energy but differ in azimuthal quantum number k; the ratio b/a equals k/n, % so k = 3 is the Bohr circle and k = 1 is the most eccentric ellipse, all with % the nucleus at a common focus. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \fill[black] (0,0) circle (2.6pt); \node[font=\scriptsize, below left] at (0,-0.08) {nucleus}; % k=3 circle, radius 2.4, centered at origin (focus = center) \draw[acc, thick] (0,0) circle (2.4); \node[acc, font=\scriptsize] at (-1.9,1.75) {$k=3$}; % k=2 ellipse: a=2.4, b=1.6, c=1.79, center at (1.79,0) \draw[black, thick, dashed] (1.79,0) ellipse [x radius=2.4, y radius=1.6]; \node[black, font=\scriptsize] at (1.79,2.1) {$k=2$}; % k=1 ellipse: a=2.4, b=0.8, c=2.26, center at (2.26,0) \draw[black, thick, densely dotted] (2.26,0) ellipse [x radius=2.4, y radius=0.8]; \node[black, font=\scriptsize, anchor=west] at (4.75,0.4) {$k=1$}; \end{tikzpicture} $$ ## Space quantization In three dimensions the orbital plane can tilt, adding a third degree of freedom. Working in spherical coordinates $(r, \theta, \varphi)$ with the polar axis fixed by an external field, the azimuthal angle $\varphi$ about that axis is cyclic, and its conjugate momentum is the projection $L_z$ of the angular momentum on the axis. Its action integral gives a third condition,[^bj-space] $$ \oint p_\varphi\,\d\varphi = 2\pi L_z = m_\ell\, h \quad\Longrightarrow\quad L_z = m_\ell\,\hbar, $$ with the **magnetic quantum number** $m_\ell$ an integer. The polar action then constrains the total angular momentum magnitude to $L = k\hbar$ as before, with the requirement $\lvert m_\ell\rvert \le k$. The orbital plane cannot point in an arbitrary direction: only the $2k + 1$ orientations with integer $L_z/\hbar$ are allowed. This **space quantization** is the old-theory ancestor of the directional quantization confirmed by the Stern-Gerlach experiment, treated with [electron spin](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession). Counting the states of a given $n$ reproduces a result the full quantum theory recovers exactly. For each azimuthal number $k$ there are $2k - 1$ allowed orientations (the values $m_\ell = -(k-1), \dots, k-1$), and $k$ runs from $1$ to $n$, so the total is $$ \sum_{k=1}^{n} (2k - 1) = n^2. $$ Identifying the Sommerfeld azimuthal number with the modern orbital number through $k = \ell + 1$ turns this into the familiar hydrogenic count: $\ell$ runs from $0$ to $n-1$, each with $2\ell + 1$ values of $m_\ell$, again summing to $n^2$. The old theory already carries the degeneracy structure of the [quantum hydrogen atom](/atomic-physics/quantum-hydrogen-atom/schrodinger-3d-hydrogen), though it assigns the states to orbits rather than to wavefunctions. $$ % caption: Space quantization for k = 2: the angular-momentum vector of fixed % length k hbar may take only the orientations whose projection on the field axis % is an integer multiple of hbar, giving the five allowed values m = 2, 1, 0, -1, % -2 marked on the vertical axis. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,-3.0) -- (0,3.3) node[above, font=\scriptsize] {axis}; % five projection lines at heights +2,+1,0,-1,-2 (scale 1.1) \foreach \h in {2.2,1.1,0,-1.1,-2.2} \draw[black, dashed] (-0.15,\h) -- (2.9,\h); % vector length L = k hbar = sqrt(5)*1.1 ~ 2.46 ; draw arrows to each height \foreach \h in {2.2,1.1,0,-1.1,-2.2} { \pgfmathsetmacro{\xx}{sqrt(2.46*2.46 - \h*\h)} \draw[->, black, thick] (0,0) -- (\xx,\h); } \node[black, font=\scriptsize, anchor=west] at (0.65,2.55) {$m=2$}; \node[black, font=\scriptsize, anchor=west] at (2.3,1.35) {$m=1$}; \node[black, font=\scriptsize, anchor=west] at (2.6,0.22) {$m=0$}; \end{tikzpicture} $$ ## The relativistic fine-structure correction The Bohr-Sommerfeld levels are exactly degenerate in $k$: every ellipse of a given $n$ has the same energy. Experiment shows a small splitting. Sommerfeld found its origin by keeping the relativistic dependence of the electron mass on speed. The elliptical orbit runs fastest at perihelion, where the mass is largest, so the orbit no longer closes: the perihelion precesses and the ellipse traces a slowly rotating rosette.[^bj-fine] $$ % caption: With the relativistic mass variation the elliptical orbit fails to % close; the perihelion advances a little each revolution and the path fills a % rosette between an inner and an outer radius, lifting the k-degeneracy of the % Bohr level. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \fill[black] (0,0) circle (2.4pt); \node[font=\scriptsize, below] at (0,-0.12) {nucleus}; \draw[acc, thick, samples=320, domain=0:1440, variable=\t] plot ({ 1.5/(1+0.5*cos(0.82*\t)) * cos(\t) }, { 1.5/(1+0.5*cos(0.82*\t)) * sin(\t) }); \node[acc, font=\scriptsize, anchor=west] at (2.05,2.15) {precessing orbit}; \end{tikzpicture} $$ Evaluating the action integrals with the relativistic momentum, Sommerfeld obtained a closed form for the energy that keeps $Z\alpha$ to all orders, where $\alpha = e^2/4\pi\epsilon_0\hbar c \approx 1/137$ is the fine-structure constant: > **Definition (Sommerfeld fine-structure formula).** With radial quantum number > $n_r$ and azimuthal quantum number $k$, the relativistic energy is > $$ > E_{n_r,k} = mc^2\left[ > \left(1 + \frac{(Z\alpha)^2}{\bigl(n_r + \sqrt{k^2 - (Z\alpha)^2}\bigr)^2}\right)^{-1/2} > - 1\right]. > $$ Expanding in powers of $(Z\alpha)^2$ and writing $n = n_r + k$ gives the form used to compare with spectra, $$ E_{n,k} \approx -\frac{Z^2\alpha^2 mc^2}{2n^2} \left[1 + \frac{(Z\alpha)^2}{n^2}\left(\frac{n}{k} - \frac{3}{4}\right)\right]. $$ The leading bracketed term is the Bohr energy; the correction is smaller by a factor $(Z\alpha)^2 \sim 10^{-5}$ for hydrogen. It depends on $k$, so it lifts the degeneracy: within a level $n$, the small-$k$ (eccentric) orbits are shifted down more than the large-$k$ (round) ones, because the eccentric orbit spends part of its period deep in the potential where the electron moves relativistically. $$ % caption: The relativistic correction splits a single Bohr level of principal % quantum number n = 3 into three sublevels labelled by k; the most eccentric % orbit (k = 1) is bound most tightly, and the spacing scales as alpha squared % times the gross energy. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % unsplit level on the left \draw[black, thick] (0,0) -- (2.4,0); \node[black, font=\scriptsize, anchor=east] at (0,0) {$n=3$}; \node[black, font=\scriptsize, anchor=south] at (1.2,0.05) {non-relativistic}; % split levels on the right, exaggerated \draw[acc, thick] (4.6,0.55) -- (7.0,0.55); \node[acc, font=\scriptsize, anchor=west] at (7.05,0.55) {$k=3$}; \draw[acc, thick] (4.6,0.05) -- (7.0,0.05); \node[acc, font=\scriptsize, anchor=west] at (7.05,0.05) {$k=2$}; \draw[acc, thick] (4.6,-0.75) -- (7.0,-0.75); \node[acc, font=\scriptsize, anchor=west] at (7.05,-0.75) {$k=1$}; % connectors \draw[black, dashed] (2.4,0) -- (4.6,0.55); \draw[black, dashed] (2.4,0) -- (4.6,0.05); \draw[black, dashed] (2.4,0) -- (4.6,-0.75); \end{tikzpicture} $$ > **Worked example (Fine-structure splitting of $n = 2$).** For hydrogen > ($Z = 1$) the $n = 2$ level admits $k = 1$ and $k = 2$. The relativistic shift > relative to the gross energy $\lvert E_2\rvert = 3.40\,\text{eV}$ is > $$ > \Delta E_k = -\lvert E_2\rvert\,\frac{\alpha^2}{n^2} > \left(\frac{n}{k} - \frac34\right), \qquad \frac{\alpha^2}{n^2} > = \frac{5.33\times10^{-5}}{4} = 1.33\times10^{-5}. > $$ > The bracket is $\tfrac14$ for $k = 2$ and $\tfrac54$ for $k = 1$, giving > $\Delta E_{k=2} = -1.1\times10^{-5}\,\text{eV}$ and > $\Delta E_{k=1} = -5.7\times10^{-5}\,\text{eV}$. The eccentric orbit is bound > more tightly, and the two components are separated by > $$ > \Delta E_{k=1} - \Delta E_{k=2} > = -\lvert E_2\rvert\,\frac{\alpha^2}{n^2}\left(\frac54 - \frac14\right) > = -4.5\times10^{-5}\,\text{eV}, > $$ > or about $0.37\,\text{cm}^{-1}$ in wavenumber, the observed order of the > hydrogen fine structure. The numerical agreement was a triumph. The success is partly a coincidence: the Sommerfeld formula agrees with the exact [Dirac result](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen) once the azimuthal number is reinterpreted as $k \to j + \tfrac12$, with $j$ the total angular momentum. Sommerfeld got the right levels from the wrong physics: the splitting is dominated by electron spin, absent from his purely orbital calculation, and two errors compensate to give the correct answer. ## Adiabatic invariance The action integrals were quantized without justification beyond dimensional consistency. Ehrenfest supplied the principle that singles them out. A quantity is an **adiabatic invariant** if it is unchanged when the parameters of the Hamiltonian are varied slowly compared with the orbital period. For a one-dimensional periodic system the action $J = \oint p\,\d q$ is such an invariant to all orders in the rate of change.[^bj-adiabatic] > **Theorem (Adiabatic invariance of the action).** If the Hamiltonian > $H(q, p; \lambda)$ depends on a parameter $\lambda$ that changes slowly, then the > action $J = \oint p\,\d q$ of a bound periodic orbit is conserved: > $\d J/\d t \to 0$ as the rate $\dot\lambda \to 0$. The relevance is this. A quantum number must be a discrete label that cannot change under a smooth, slow deformation of the system, since a slow deformation excites no transition. Only an adiabatic invariant qualifies. The action integrals are the adiabatic invariants of a separable bound motion, so they are the correct quantities to set equal to $n h$. For the harmonic oscillator, slowly changing $\omega$ leaves $E/\omega = J/2\pi$ fixed, the classical shadow of the quantum statement that the integer $n$ does not change when a trap is squeezed gently. This principle carries into the wave theory as the invariance of the quantum number under adiabatic changes of a Hamiltonian. The Bohr-Sommerfeld theory is the furthest the orbit picture reaches. It handles one-electron atoms, fixes the level count, and produces a fine structure of the right magnitude, but it fails the moment a second electron makes the motion non-separable, and it cannot supply the half-integers that a correct theory requires. Those limits, and the semiclassical rule that survives them, are the subject of the [next lesson](/atomic-physics/early-models-and-old-quantum-theory/old-quantum-theory-limits-wkb). [^bj-rule]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §2.5 — the Wilson-Sommerfeld action-quantization rule and its application to periodic systems. [^bj-radial]: **Bransden & Joachain**, §2.5 — the radial and azimuthal action integrals for the Coulomb problem and the emergence of $n = n_r + k$; the closed form of the radial integral is worked in the appendix. See also **Demtröder**, _Atoms, Molecules and Photons_, 2nd ed., §3.4. [^bj-space]: **Bransden & Joachain**, §2.5 — space quantization and the magnetic quantum number from the third action integral; **Foot**, _Atomic Physics_, Ch. 1. [^bj-fine]: **Bransden & Joachain**, §2.6 — Sommerfeld's relativistic treatment, the precessing orbit, and the fine-structure formula; the reinterpretation $k \to j + \tfrac12$ matching the Dirac spectrum. The fine-structure constant value $\alpha^{-1} = 137.035999$ is the CODATA recommended value. [^bj-adiabatic]: **Bransden & Joachain**, §2.5 — Ehrenfest's adiabatic principle and the role of adiabatic invariants in selecting the quantized action variables.