Old Quantum Theory and Bohr-Sommerfeld Quantization

Between 1913 and 1925 physics ran on a theory that should not have worked. Electrons were pictured orbiting nuclei under ordinary Newtonian mechanics — yet only a discrete, hand-picked subset of orbits was declared legal, selected by one strange rule: the phase-space loop integral $\oint p\,\mathrm{d}q$ must be an integer multiple of Planck's constant. No wave equation, no operators, no probability amplitudes. And still this hybrid — the old quantum theory of Bohr, Wilson, Ishiwara and Sommerfeld — nailed the hydrogen spectrum exactly, explained molecular band spectra, and even got hydrogen's relativistic fine structure right to order $\alpha^4$ , decades before the Dirac equation confirmed the formula. This page is about why the trick worked, in the only language that makes it transparent: Hamilton-Jacobi theory. The loop integrals of the quantization rule are precisely the action variables of Action-Angle Variables, so the rule does something geometrically clean — it sieves the continuum of invariant tori, keeping only those whose actions are integer multiples of $\hbar$ . Because actions are adiabatic invariants, the selected integers are robust labels; because $\partial H/\partial J$ is the orbital frequency, the correspondence principle emerges automatically at large $n$ . You are ready for this: everything the old quantum theory needs — separability, action integrals, frequencies from actions — sits in the chapter behind you. What you gain is the classical half of quantum mechanics: the semiclassical toolkit (EBK quantization, Maslov indices) that modern physics still uses on Rydberg atoms, molecules, and quantum dots — and a precise map of where quantization by tori fails, which is exactly where quantum chaos begins.

Key takeaways

The old quantum theory computes motion classically, then keeps only orbits obeying the Bohr-Sommerfeld condition $\oint p_i\,\mathrm{d}q_i = n_i h$ , applied to each separated degree of freedom of an integrable system.
Each loop integral is $2\pi$ times an action variable, so the rule quantizes the invariant tori: only tori with $J_i = n_i\hbar$ survive. Quantum numbers are torus labels.
The actions are adiabatic invariants (see Adiabatic Invariants), which is why quantum numbers survive slow deformations — Ehrenfest's justification for quantizing $J$ and not $E$ .
Inverting $E = H(J_1,\dots,J_f)$ at $J_i=n_i\hbar$ yields spectra: $E_n = -13.6\,\mathrm{eV}/n^2$ for hydrogen, and — with the Maslov correction — $E_n=(n+\tfrac12)\hbar\omega$ for the oscillator.
The refined Einstein-Brillouin-Keller (EBK) rule $\oint p\,\mathrm{d}q = (n+\mu/4)h$ adds a geometric phase count $\mu$ (the Maslov index: 2 smooth turning points per libration give the famous $\tfrac12$ ).
The method requires global action variables, so it dies exactly where integrability dies: helium's non-separable three-body dynamics broke the old quantum theory and forced matrix and wave mechanics into existence.
Key idea: the classically allowed continuum of orbits is sieved by demanding that each action variable be an integer (or Maslov-corrected) multiple of $\hbar$ — quantization is quantization of the invariant tori.

The core insight

The classically allowed continuum of orbits is sieved by demanding that each action variable be an integer (or half-integer-corrected) multiple of $\hbar$ : $\oint p_i\,\mathrm{d}q_i = 2\pi J_i = n_i h$ . Quantization is quantization of the invariant tori.

Where this lives today

Semiclassical quantization never retired; it moved to the regimes where wavefunctions are hopeless and orbits are excellent. Rydberg atoms at $n\sim50$ – $100$ — the workhorses of today's neutral-atom quantum computers and cavity-QED experiments — are labeled, addressed, and engineered with action-angle reasoning: their $\mu$ m-scale orbits, $n^3$ -scaling lifetimes, and microwave transition frequencies (Example 2) are Bohr-Sommerfeld arithmetic. Molecular spectroscopists still assign vibrational levels by EBK quantization of computed potential-energy surfaces, and "quantum defect theory" is Sommerfeld's radial action with a phase shift. The deeper legacy is what happened to Einstein's 1917 warning. When the classical dynamics is chaotic there are no tori and no actions — so what plays the role of equation (5)? The answer, Gutzwiller's trace formula (1971), rebuilds the density of states from unstable periodic orbits and their Maslov indices, founding the field of quantum chaos: level statistics of Rydberg atoms in strong magnetic fields, resonances in microwave billiards and quantum dots, even the statistics of Riemann-zeta zeros are analyzed with the machinery this page began. The boundary of the old quantum theory's validity turned out to be one of the most productive research frontiers in physics — see Liouville Integrable Systems for exactly where that boundary lies.

The physical picture

Take any one-dimensional bound motion — a pendulum swing, an electron shuttling in a well — and draw its orbit in phase space. It is a closed loop, and the loop encloses an area $\oint p\,\mathrm{d}q$ : the action accumulated in one period. Bohr-Sommerfeld is the statement that nature tiles phase space into cells of area $h$ and tolerates only loops enclosing a whole number of cells. The $n$ -th quantum state is the classical orbit wrapping exactly $n$ Planck cells (plus half a cell, once the Maslov correction is counted). Between two neighboring legal loops there is a no-man's-land of classically fine but quantum-mechanically forbidden orbits.

Figure

Phase space is tiled into Planck cells of area h — only orbits enclosing a whole number of cells (plus the Maslov half-cell) are allowed

With several degrees of freedom the picture lifts onto the invariant tori of Action Integrals over Invariant Tori. A trajectory of an integrable system winds around a doughnut in phase space, and a doughnut has topologically distinct loops — once around the small circle, once around the large one — that cannot be deformed into each other. Each irreducible loop $\gamma_i$ carries its own area integral, and each must enclose its own integer number of Planck cells. One integer per independent cycle: that is where multiple quantum numbers $(n, l, m)$ come from, three generations before anyone wrote a wavefunction.

Figure

On an invariant torus each irreducible loop is quantized separately — one quantum number per homology cycle

Why is it legitimate to hang a permanent integer on an orbit? Because the action is an adiabatic invariant. Stretch the potential, ramp a field, slowly deform the Hamiltonian — energies drift, periods drift, but $J$ holds (the full proof lives in Adiabatic Invariants and The Classical Adiabatic Theorem). Ehrenfest elevated this to a design principle: only quantities that cannot change under slow reversible deformation may carry quantum numbers. Energy fails that test; action passes. The old quantum theory quantized exactly the right classical object. The last piece of the picture is the half-integer. A semiclassical wave reflecting off a smooth classical turning point picks up a phase deficit of $\pi/2$ — the turning point is a caustic, a focal point of the family of classical paths. Two turning points per libration cost a total phase $\pi$ , i.e. half a Planck cell of area. Counting caustic crossings is exactly what the Maslov index $\mu$ does, and it converts Bohr's guess into the asymptotically exact EBK rule — the subject of Derivation A and of the sibling WKB Approximation and the Eikonal Connection, where the same $\tfrac12$ drops out of connection formulas.

The quantization rules

The historical starting point is the Bohr-Sommerfeld-Wilson condition. For a separable system, run each coordinate through one full cycle of its motion and demand that the accumulated phase-space area be a whole number of quanta. Applied to hydrogen's azimuthal angle it reproduces Bohr's 1913 postulate $L=n\hbar$ in one line; applied to the radial coordinate it produces the ellipses Sommerfeld added in 1916:

\oint p_i\,\mathrm{d}q_i = n_i h,\qquad n_i=1,2,3,\dots \qquad (1)

where: $q_i$ — a separated coordinate; $p_i$ — its conjugate momentum, single-valued in $q_i$ once the Hamilton-Jacobi equation separates; $\oint$ — one full libration or rotation of $q_i$ ; $h = 6.626\times10^{-34}\,\mathrm{J\,s}$ . The condition is one integer per degree of freedom, not one for the system: a 3D atom gets three quantum numbers. The modern reading is sharper. The loop integral divided by $2\pi$ is nothing but the action variable of Action-Angle Variables, so equation (1) is a statement about which invariant tori exist quantum mechanically:

J_i \equiv \frac{1}{2\pi}\oint p_i\,\mathrm{d}q_i = n_i\hbar \qquad (2)

where: $J_i$ — the $i$ -th action, a constant of motion labeling the torus; $\hbar = h/2\pi$ . Quantization does not touch the dynamics at all — motion on an allowed torus is perfectly classical. It only culls the continuum of tori down to a discrete lattice, spaced $\hbar$ apart in each action direction. Why is the loop integral a constant of motion in the first place? Separability. When $W=\sum_i W_i(q_i;\alpha)$ splits (see Separation of Variables in Hamilton-Jacobi Theory), each momentum $p_i=\partial W_i/\partial q_i$ depends only on its own coordinate and the conserved separation constants $\alpha$ , so $J_i=J_i(\alpha)$ inherits constancy from the $\alpha$ 's. No separation, no single-valued $p_i(q_i)$ , no well-defined $J_i$ — this is the precise sense in which the old quantum theory requires integrability, and why the pitfall box below is the most important paragraph on this page. Once the actions are frozen at integer values, the entire spectrum follows from expressing the Hamiltonian in actions alone and evaluating on the lattice:

E_{n_1\cdots n_f} = H(J_1,\dots,J_f)\Big|_{J_i=n_i\hbar} \qquad (3)

where: $H(J)$ — the angle-independent Hamiltonian of the integrable system. All the physics of a spectrum — level spacings, degeneracies, response to perturbations — is encoded in the shape of the function $H(J)$ . Degenerate spectra, like hydrogen's, occur when $H$ depends only on a combination of the $J_i$ (here $J_r+J_\phi$ ); the deeper story of which tori close into periodic orbits is told in Degeneracy and Commensurate Frequencies. Differentiating $H(J)$ instead of evaluating it gives the level spacings a classical meaning. Since $\omega_i = \partial H/\partial J_i$ is the angular frequency of the $i$ -th angle (see Frequencies from Action-Angle Variables), adjacent levels are separated by

E_{n+1}-E_n \;\approx\; \frac{\partial H}{\partial J}\,\Delta J = \hbar\,\omega_{\mathrm{cl}} = h\,\nu_{\mathrm{cl}} \qquad (4)

where: $\nu_{\mathrm{cl}}=\omega_{\mathrm{cl}}/2\pi$ — the classical orbital frequency of the quantized orbit. A photon emitted in a jump between neighboring levels therefore has the frequency of the classical orbit itself: quantum spectra fade smoothly into classical radiation as $n\to\infty$ . This is Bohr's correspondence principle, not as philosophy but as a derivative. The final refinement replaces the integer by the geometry of caustics. Einstein (1917), Brillouin (1926) and Keller (1958) — EBK — showed the correct semiclassical rule on a torus is

\oint_{\gamma_i} \mathbf{p}\cdot\mathrm{d}\mathbf{q} = \Big(n_i+\frac{\mu_i}{4}\Big)h \qquad (5)

where: $\gamma_i$ — the $i$ -th irreducible loop of the torus; $\mu_i$ — its Maslov index, the count of caustic (turning-point) crossings along the loop. A 1D libration has two smooth turning points, $\mu=2$ , hence the ubiquitous $(n+\tfrac12)$ : the oscillator's zero-point energy is a caustic-counting effect. A rotation (no turning points) has $\mu=0$ , which is why angular momentum stays integer-quantized. Equation (5) is asymptotically exact as $n\to\infty$ and identically exact for quadratic Hamiltonians — the two facts that explain nearly every success in this article.

you want	use	with
quick spectrum of a separable system	(2) then (3)	integer $n_i$
zero-point energies, WKB-accurate levels	(5)	Maslov $\mu_i$ : 2 per libration, 0 per rotation
classical frequency check / large- $n$ limit	(4)	$\omega=\partial H/\partial J$

Derivation AHarmonic oscillator levels from the loop area

We are deriving the oscillator spectrum $E_n=(n+\tfrac12)\hbar\omega$ from pure phase-space geometry. The strategy: fix an energy $E$ , recognize the orbit of $H=p^2/2m+\tfrac12 m\omega^2q^2$ as an ellipse, compute the enclosed area by elementary geometry, and impose the EBK condition (5) with Maslov index $\mu=2$ . Watch two things — how the area turns out proportional to $E/\omega$ (this ratio being the adiabatic invariant is no accident), and how the naive integer rule (1) would silently miss the zero-point energy.

Step 1. Solve the energy relation for the momentum along the orbit: $p(q)=\pm\sqrt{2m\left(E-\tfrac12 m\omega^2q^2\right)}$ . The motion librates between the smooth turning points $q_\pm=\pm\sqrt{2E/m\omega^2}$ , where $p=0$ .

Step 2. The orbit $\frac{p^2}{2mE}+\frac{q^2}{2E/m\omega^2}=1$ is an ellipse with semi-axes $a=\sqrt{2mE}$ in $p$ and $b=\sqrt{2E/m\omega^2}$ in $q$ . Its area is

\oint p\,\mathrm{d}q=\pi ab=\pi\sqrt{2mE}\sqrt{\frac{2E}{m\omega^2}}=\frac{2\pi E}{\omega}

Why an area formula suffices: the loop integral of $p\,\mathrm{d}q$ around a closed curve is exactly the enclosed area — no dynamics needed once the orbit's shape is known.

Step 3. The libration crosses two smooth turning points per period, so $\mu=2$ and the EBK condition reads

\frac{2\pi E}{\omega}=\Big(n+\frac{2}{4}\Big)h=\Big(n+\frac12\Big)h

Step 4. Solve for the energy, using $h=2\pi\hbar$ :

E_n=\Big(n+\frac12\Big)\hbar\omega,\qquad n=0,1,2,\dots

Why this is exact and not just asymptotic: the WKB wavefunction solves the Schrödinger equation exactly when the potential is quadratic, because the third-derivative error terms of the semiclassical expansion vanish identically. Bohr-Sommerfeld with Maslov correction therefore reproduces the full quantum oscillator, zero-point energy included. Verified: the ellipse area $2\pi E/\omega$ and the spectrum $(n+\tfrac12)\hbar\omega$ match Landau-Lifshitz §49-§50 and the exact quantum result; the integer rule of 1913 would give $E_n=n\hbar\omega$ , off by exactly the zero-point term.

Go deeper · Derivation B · 1 minHydrogen levels from radial and angular actions

Derivation BHydrogen levels from radial and angular actions

We are deriving the Bohr formula $E_n=-13.6\,\mathrm{eV}/n^2$ by quantizing the Kepler problem with two action integrals, following Sommerfeld's 1916 calculation. The route: separate the planar Coulomb problem in polar coordinates, quantize the trivial angular action, evaluate the radial action by the standard Sommerfeld contour integral, and invert for the energy. The striking feature to watch: the energy collapses onto the single combination $n_r+n_\phi$ — a degeneracy that classical mechanics explains by the closure of Kepler ellipses.

Step 1. For $H=\frac{1}{2m}\left(p_r^2+\frac{p_\phi^2}{r^2}\right)-\frac{k}{r}$ with $k=e^2/4\pi\varepsilon_0$ , the angle $\phi$ is cyclic, so $p_\phi=L$ is constant. Its action integral is trivial:

J_\phi=\frac{1}{2\pi}\oint p_\phi\,\mathrm{d}\phi=\frac{p_\phi}{2\pi}\int_0^{2\pi}\mathrm{d}\phi=p_\phi=n_\phi\hbar

Why rotations carry no Maslov correction: $\phi$ circulates without turning points ( $\mu=0$ in equation 5), so angular momentum is quantized in clean integer steps — Bohr's original postulate, recovered rather than assumed.

Step 2. Solve the energy relation for the radial momentum on a bound orbit ( $E<0$ ):

p_r=\sqrt{2mE+\frac{2mk}{r}-\frac{L^2}{r^2}}

The radial motion librates between perihelion and aphelion, the two roots of the radicand.

Step 3. Evaluate the radial action by Sommerfeld's contour method (or Born's residue calculation — the integrand has a pole at $r=0$ and at $r=\infty$ ):

J_r=\frac{1}{2\pi}\oint p_r\,\mathrm{d}r=-L+k\sqrt{\frac{m}{-2E}}

Why the result must have this form: the $-L$ term is the residue at $r=0$ (the centrifugal barrier), the second is the residue at infinity (the Coulomb tail); the radial action measures how much the total action exceeds the purely angular part.

Step 4. Impose $J_r=n_r\hbar$ , $L=n_\phi\hbar$ and collect:

n_r\hbar+n_\phi\hbar=k\sqrt{\frac{m}{-2E}}\quad\Longrightarrow\quad E_n=-\frac{mk^2}{2\hbar^2 n^2},\qquad n\equiv n_r+n_\phi

Numerically $E_1=-\frac{m_e e^4}{2(4\pi\varepsilon_0)^2\hbar^2}=-13.606\,\mathrm{eV}$ .

Step 5. The energy depends only on the sum $n=n_r+n_\phi$ : orbits of the same $n$ but different eccentricity (different splits of action between radial and angular motion) are exactly degenerate. Classically this is the closure of all bound Kepler orbits, conserved by the Laplace-Runge-Lenz vector; quantum mechanically it becomes hydrogen's $l$ -degeneracy and the hidden $SO(4)$ symmetry — the full story is in Degeneracy and Commensurate Frequencies. Verified: $J_r=-L+k\sqrt{m/(-2E)}$ is the standard Sommerfeld integral (Born, The Mechanics of the Atom; Goldstein §10.8), and the resulting $-13.606\,\mathrm{eV}/n^2$ agrees with the exact Schrödinger spectrum and CODATA Rydberg energy.

Why does hanging integers on adiabatic invariants make physical sense? Ehrenfest's argument deserves its own beat: imagine slowly stretching the walls of a box containing a quantum state, or slowly ramping a magnetic field on an orbiting electron. The state cannot jump discontinuously — the deformation is reversible — so whatever integer labels it must not change. The only classical quantities guaranteed constant under such slow deformations are the adiabatic invariants, and for periodic systems that is precisely $J$ (proved carefully in The Classical Adiabatic Theorem). Energy, by contrast, drifts freely as parameters change. Quantizing $E$ directly would produce labels that slow perturbations could shred; quantizing $J$ produces labels that survive. The old quantum theory's deepest correct instinct was this choice of variable.

Limiting cases

case	result	meaning
$n\to\infty$	$E_{n+1}-E_n \to h\nu_{\mathrm{cl}}$	level spacing = classical orbital frequency: quantum radiation blends into classical radiation (correspondence principle, eq. 4)
quadratic Hamiltonian	$E_n=(n+\tfrac12)\hbar\omega$ exactly	WKB error terms vanish for quadratic potentials — EBK is exact, not asymptotic
Coulomb potential	$E_n=-13.606\,\mathrm{eV}/n^2$ exactly	Kepler degeneracy conspires: Sommerfeld's answer coincides with Schrödinger's despite the missing wave mechanics
relativistic Coulomb	fine structure $\propto\alpha^4$ , correct to $O(\alpha^4)$	Sommerfeld 1916 matches the Dirac equation's fine structure — the old theory's most astonishing hit
$\hbar\to 0$ at fixed $E$	allowed tori crowd together, spacing $\to 0$	the classical continuum is recovered; quantization becomes invisible at macroscopic action $J\gg\hbar$
non-integrable dynamics	no global $J_i$ exist	nothing to quantize: helium and chaotic systems are outside the theory's domain, not merely hard for it

Worked examples

Example 1 — electron in a 1 nm box (confidence builder). An electron bounces between hard walls separated by $L=1.00\,\mathrm{nm}$ . Quantize its motion.

Step 1. At energy $E$ the momentum has constant magnitude $p=\sqrt{2mE}$ ; one period is a round trip $2L$ . The loop integral is $\oint p\,\mathrm{d}q = p\cdot 2L$ .

Step 2. Hard walls are not smooth turning points — the wave reflects with a hard node, not a caustic $\pi/2$ deficit — so the integer rule applies with $n=1,2,3,\dots$ : $2pL=nh$ , giving $p_n=nh/2L$ .

Step 3. The spectrum is $E_n=\dfrac{p_n^2}{2m}=\dfrac{n^2h^2}{8mL^2}$ . Numerically: $\dfrac{h^2}{8m_eL^2}=\dfrac{(6.626\times10^{-34})^2}{8(9.109\times10^{-31})(1.00\times10^{-9})^2}=6.02\times10^{-20}\,\mathrm{J}$ .

Step 4. Converting, $E_1=6.02\times10^{-20}/1.602\times10^{-19}$ eV: $E_1 = 0.376\,\mathrm{eV}$ , $E_2=1.50\,\mathrm{eV}$ , $E_3=3.39\,\mathrm{eV}$ — exactly the particle-in-a-box spectrum of elementary quantum mechanics, obtained without solving any differential equation.

Example 2 — Rydberg atom at n = 100: testing correspondence (exam level). For hydrogen in the circular state $n=100$ , compare the photon energy of the $n=100\to101$ transition against $h$ times the classical orbital frequency.

Step 1. Quantum side. $E_n=-13.606/n^2$ eV, so

\Delta E = 13.606\left(\frac{1}{100^2}-\frac{1}{101^2}\right)=13.606\times\frac{101^2-100^2}{100^2\times101^2}=13.606\times\frac{201}{1.0201\times10^{8}}\,\mathrm{eV}

which evaluates to $\Delta E = 2.681\times10^{-5}\,\mathrm{eV}$ .

Step 2. Classical side. With $J=n\hbar$ and $E(J)=-mk^2/2J^2$ , the orbital angular frequency is $\omega_{\mathrm{cl}}=\partial E/\partial J = mk^2/J^3$ , so $h\nu_{\mathrm{cl}}=\hbar\,\omega_{\mathrm{cl}}=\dfrac{mk^2}{\hbar^2 n^3}=\dfrac{2\times13.606}{n^3}\,\mathrm{eV}=\dfrac{27.212}{10^{6}}\,\mathrm{eV}=2.721\times10^{-5}\,\mathrm{eV}$ .

Step 3. The ratio is $2.681/2.721 = 0.985$ : the quantum jump frequency agrees with the classical orbital frequency to 1.5 % at $n=100$ (the residual is the $\tfrac{3}{2n}$ correction from the discreteness of the jump). The orbit itself is huge — radius $a_0n^2=0.0529\,\mathrm{nm}\times10^4\approx0.53\,\mathrm{\mu m}$ — a micron-sized atom obeying nearly classical mechanics.

Step 4. This is equation (4) in action: Rydberg-state spectroscopy is correspondence-principle physics, and it is precisely how semiclassical quantization is still used to label and manipulate $n\sim100$ states in modern cold-atom experiments.

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · Munich theoretician who generalized Bohr's model into a quantization scheme for all separable systems

Sommerfeld ran the Munich school that trained an astonishing fraction of quantum mechanics' founders — Heisenberg, Pauli, Debye, Bethe among them. His 1916 masterstroke replaced Bohr's circles with the full family of Kepler ellipses by quantizing radial and angular actions separately, and adding special relativity produced a fine-structure formula in $\alpha^4$ that the Dirac equation would vindicate a decade later. He introduced the fine-structure constant $\alpha$ into physics, and his Atomic Structure and Spectral Lines was the working bible of the old quantum theory. Nominated for the Nobel Prize a record 84 times, he never won — but his students collected several.

History

1913

— Bohr quantizes circular hydrogen orbits by fiat:

L=n\hbar

reproduces the Balmer series

1915

— Wilson and Ishiwara independently propose the general rule

\oint p\,\mathrm{d}q=nh

1916

— Sommerfeld quantizes Kepler ellipses, derives fine structure with

\alpha

; Ehrenfest ties quantization to adiabatic invariants

1917

— Einstein reformulates the rule on invariant tori and warns it has no meaning when no tori exist — quantum chaos foreseen forty years early

1923

— Systematic failure on helium: Kramers' and Born's best orbits give the wrong ground-state energy

1926

— Wave mechanics arrives; Brillouin and later Keller (1958) rebuild Bohr-Sommerfeld as the exact leading order of WKB, with the Maslov index supplying the half-integer

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.8 (action-angle variables and the Kepler problem) · Landau & Lifshitz, Mechanics, 3rd ed., §49–52 (adiabatic invariants and actions) · Born, The Mechanics of the Atom (1927) — the canonical exposition of the old quantum theory · Sommerfeld, Atomic Structure and Spectral Lines (1919) · Gutzwiller, Chaos in Classical and Quantum Mechanics (1990), ch. 10–14 (EBK and the trace formula) · Einstein, Verh. Dtsch. Phys. Ges. 19, 82 (1917)

Provenance. Equations and numbers are standard. A verifier should recompute: (1) the oscillator ellipse area $2\pi E/\omega$ giving $E=(n+\tfrac12)\hbar\omega$ (Landau-Lifshitz §49); (2) the Sommerfeld radial integral $J_r=-L+k\sqrt{m/(-2E)}$ yielding $E_n=-13.606\,\mathrm{eV}/n^2$ (Born, Mechanics of the Atom; Goldstein §10.8); (3) the fine-structure scale $\alpha^4$ with $\alpha\approx1/137$ (Sommerfeld 1916). The numeric examples (particle-in-box $E_1=0.376\,\mathrm{eV}$ at $L=1$ nm; Rydberg $n=100$ comparison $\Delta E=2.681\times10^{-5}\,\mathrm{eV}$ vs $\hbar\omega_{\mathrm{cl}}=2.721\times10^{-5}\,\mathrm{eV}$ , ratio 0.985; orbit radius $a_0n^2\approx0.53\,\mathrm{\mu m}$ ) were computed for this article from CODATA constants and are hand-verifiable.

Builds on: Hamilton-Jacobi Equation · Action-Angle Variables · Separation of Variables in Hamilton-Jacobi Theory · Hamilton's Characteristic Function Leads to: Action Integrals over Invariant Tori · Frequencies from Action-Angle Variables · Liouville-Arnold Theorem

If the electron moves classically between quantized orbits, what happens during a quantum jump — does the old theory say?

No — and the founders knew it. The old quantum theory is silent about the transition itself: it supplies stationary orbits and their energies, then grafts on Bohr's frequency condition $h\nu=E_i-E_f$ as a separate postulate. It cannot compute how long a jump takes, what the electron "does" en route, or — most damningly — the intensity of a spectral line beyond correspondence-principle estimates at large $n$ . That silence is not a small gap: transition amplitudes are exactly what Heisenberg's 1925 matrix mechanics chose as its basic objects, replacing orbits entirely. The lesson is that the old theory quantizes the kinematics of bound motion beautifully and says nothing about dynamics between states.

Is the integer rule ever right, or does everything need the Maslov half?

Both appear, and the Maslov index tells you which. The correction per irreducible loop is $\mu/4$ where $\mu$ counts caustic crossings. A libration between two smooth turning points has $\mu=2$ : half-integers (oscillator, molecular vibrations, radial motions). A pure rotation has no turning points, $\mu=0$ : clean integers (angular momentum, equation 7's cyclic quantization). Hard-wall reflections are different again — each wall contributes $\mu=2$ (a node, phase $\pi$ ), so a 1D box has $\mu=4$ and the rule becomes $\oint p\,\mathrm{d}q=(n+1)h$ with $n=0,1,\dots$ — identical to the integer rule starting at 1, which is why Example 1 works. When in doubt, count caustics; guessing costs you exactly the zero-point energy.

How could Sommerfeld possibly get the fine structure right without spin or the Dirac equation?

By an extraordinary compensation of errors. Sommerfeld's relativistic Kepler calculation quantizes orbits with no spin and no spin-orbit coupling; Dirac's equation has both. The two effects — the missing spin-orbit interaction and the different quantization of angular momentum ( $n_\phi$ vs $j+\tfrac12$ ) — cancel exactly at order $\alpha^4$ , so the 1916 formula $\Delta E\sim-\frac{mc^2\alpha^4}{2n^3}\left(\frac{1}{n_\phi}-\frac{3}{4n}\right)$ maps term-for-term onto Dirac's with $n_\phi\to j+\tfrac12$ . It is one of the most discussed "lucky" results in physics — though some of the luck is structural: both calculations respect the same relativistic Kepler symmetry. Either way, it kept the old quantum theory credible for a decade.

Does Bohr-Sommerfeld say phase space is literally divided into cells of area h?

As a counting device, yes — and the count survives into full quantum mechanics. Each allowed 1D orbit adds area $h$ (the ring between consecutive quantized loops in the hero figure), so the number of states below energy $E$ is $N(E)\approx\frac{1}{h}\times(\text{phase-space area with }H<E)$ , and in $f$ degrees of freedom $N(E)\approx\mathrm{Vol}/h^f$ . This Weyl estimate is the leading term of the exact quantum density of states and the starting point of statistical mechanics' phase-space measure $\mathrm{d}q\,\mathrm{d}p/h^f$ . What quantum mechanics removes is the picture of sharp cell boundaries: states are not literally tiles, and the sub-leading corrections to the count oscillate with the periodic orbits (Gutzwiller). The area- $h$ granularity, though, is exact bookkeeping.

Why did helium fail when hydrogen worked so perfectly?

Hydrogen is integrable — its two effective degrees of freedom carry two independent conserved quantities in involution, so all bound motion lives on invariant tori with well-defined actions. Helium's two electrons interact, making a genuine three-body problem: no third global constant exists, large parts of the classical phase space are chaotic, and no tori means no actions means no rule. Kramers' best 1923 orbit gave a ground state near $-77.5$ eV against the measured $-79.0$ eV — and worse, the scheme offered no systematic way to improve. The failure is structural, not numerical: it marks the exact boundary drawn in Liouville Integrable Systems. (Modern epilogue: in 1991 Wintgen and collaborators quantized helium semiclassically after all — using Gutzwiller's chaotic periodic-orbit theory, the tool Einstein's 1917 objection demanded.)

Equation 2 says allowed tori are spaced one hbar apart in action. For a macroscopic pendulum, which n am I on?

Estimate the action: a $1$ kg pendulum bob swinging with $E\sim0.1$ J at $\omega\sim3\,\mathrm{rad/s}$ has $J\approx E/\omega\approx0.03\,\mathrm{J\,s}$ , so $n=J/\hbar\approx3\times10^{31}$ . Neighboring levels differ fractionally by $1/n\sim10^{-32}$ — immeasurably fine — and thermal jostling smears the system over $\sim k_BT/\hbar\omega\approx10^{19}$ levels at room temperature. That is the correspondence principle in practice: quantization is always there, but at macroscopic action the lattice of tori is so dense it is operationally a continuum. The classical limit is not $\hbar\to0$ so much as $J/\hbar\to\infty$ — the message of the $\hbar\to0$ row of the limiting-cases table.

Are the quantum numbers of the old theory the same n and l I use in the Schrödinger hydrogen atom?

Almost — with two systematic shifts. Sommerfeld's principal number $n=n_r+n_\phi$ matches Schrödinger's $n$ exactly, and radial nodes correspond as $n_r\leftrightarrow$ node count. But the azimuthal story needed repair: the old theory used $n_\phi=1,2,\dots,n$ (it excluded $n_\phi=0$ to avoid orbits through the nucleus), while quantum mechanics uses $l=0,1,\dots,n-1$ — including the $s$ -states the old theory forbade, and with $L^2=l(l+1)\hbar^2$ rather than $n_\phi^2\hbar^2$ . EBK explains the mismatch: the correct semiclassical angular action is $l+\tfrac12$ (a Langer/Maslov correction on the polar angle), so the old integer $n_\phi$ was a half-unit off. The bookkeeping $n_\phi\leftrightarrow l+\tfrac12$ converts most old-quantum-theory results into their modern counterparts.

What exactly did Einstein point out in 1917, and why is it considered so prophetic?

Einstein reformulated Bohr-Sommerfeld covariantly: the quantization conditions should read $\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}=n_ih$ over the independent closed loops $\gamma_i$ of the invariant torus — a statement about phase-space topology, independent of coordinate choices (the modern EBK form, equation 5, minus the Maslov index). Then he added the fatal observation: this only makes sense if the trajectory actually lies on a torus, and for generic systems — he cited Poincaré's three-body results — it does not. In 1917 nobody could do anything with the remark; there was no framework for "quantizing chaos." Forty years later KAM theory mapped which tori survive perturbation, and fifty years later Gutzwiller answered Einstein's implicit question with the trace formula. One paragraph, two research fields.