---
title: "The Old Quantum Theory: Bohr, Sommerfeld, and Correspondence"
module: Origins of the Quantum
moduleNumber: 1
lessonNumber: 4
order: 104
summary: >
  Between Bohr's 1913 atom and Schrödinger's 1926 equation, physics ran on a
  provisional recipe: keep classical orbits, but admit only those whose action
  integral is a whole multiple of Planck's constant. This lesson develops the
  Wilson-Sommerfeld phase-integral rule, applies it to the oscillator and to the
  elliptical Kepler orbits of hydrogen, derives Sommerfeld's relativistic fine
  structure and the quantization of orbit orientation, and shows how the
  correspondence principle fixed intensities and selection rules. The systematic
  failures — helium, line intensities, the anomalous Zeeman effect — mark exactly
  where a theory of orbits had to give way to a theory of waves.
topics: [Origins of the Quantum]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 4 — The Nuclear Atom; §4-3 The Bohr Model, Correspondence Principle, Fine-Structure Constant"
  - book: Griffiths & Schroeter
    ref: "Afterword; §4.2 The Hydrogen Atom (spectrum recap)"
  - book: Shankar
    ref: "Ch. 12 — Rotational Invariance and Angular Momentum (semiclassical history)"
draft: false
---

The [Bohr model](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen)
fixed the hydrogen spectrum by quantizing one quantity, the orbital angular
momentum $L = n\hbar$. That single rule is too narrow to survive contact with any
other system: it says nothing about an oscillator, a rotator, or an orbit that is
not a circle. What follows Bohr is not a theory in the modern sense but a
_quantization program_ — a prescription for selecting, out of the continuum of
classical motions a system can execute, the discrete subset nature actually
occupies. The program keeps Newtonian (and later relativistic) mechanics intact
and overlays a condition on the allowed orbits. It held from 1913 to 1925, made
several sharp predictions, and then failed in ways precise enough to point at its
replacement. This lesson treats it as what it was: a bridge, whose planks are
worth naming because each reappears, reinterpreted, in wave mechanics.

## The phase integral

Bohr's $L = n\hbar$ is a statement about one coordinate, the azimuthal angle
$\varphi$, whose conjugate momentum $p_\varphi = L$ is constant over the circular
orbit. Write the condition as an integral around one full revolution:

$$
\oint p_\varphi \,\d\varphi = \int_0^{2\pi} L \,\d\varphi = 2\pi L = 2\pi n\hbar = nh.
$$

The circular orbit hides the content, because $p_\varphi$ is constant and the
integral is trivial. The generalization, proposed independently by William Wilson
(1915) and Arnold Sommerfeld (1916), is to demand exactly this for _every_
coordinate of a periodic system, with the momentum allowed to vary around the
path.

> **Postulate (Wilson-Sommerfeld quantization).** For a mechanical system whose
> motion is periodic and separable into coordinates $q_i$ with conjugate momenta
> $p_i$, the physically realized orbits are those for which each phase integral,
> taken over one full period of that coordinate, is an integer multiple of
> Planck's constant:
> $$
> J_i \equiv \oint p_i \,\d q_i = n_i h, \qquad n_i = 0, 1, 2, \dots
> $$
> The integer $n_i$ is a quantum number, and $J_i$ is the **action variable** of
> the coordinate.

The integral $\oint p\,\d q$ is the area enclosed by the orbit's trace in the
$(q, p)$ phase plane. The rule therefore has an immediate geometric reading: the
allowed motions are those enclosing a phase-space area equal to a whole number of
quanta $h$. Classical mechanics fills the phase plane with a continuum of nested
orbits; the quantum condition keeps only a discrete ladder of them, each ring
larger than the one below it by one unit of action.

$$
% caption: Phase-plane orbits of a one-dimensional system. Classical mechanics
% allows every nested loop; the Wilson-Sommerfeld rule keeps only those enclosing
% area h, 2h, 3h, ... so the enclosed area increases by one quantum per level.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (-4.0,0) -- (4.0,0) node[anchor=north east] {position $q$};
  \draw[->, black] (0,-2.9) -- (0,2.9) node[anchor=south east] {momentum $p$};
  % nested quantized ellipses (area proportional to n): semi-axes scale as sqrt(n)
  \draw[acc, thick] (0,0) ellipse (1.3 and 0.75);
  \draw[acc, thick] (0,0) ellipse (1.84 and 1.06);
  \draw[acc, thick] (0,0) ellipse (2.25 and 1.30);
  \draw[acc, thick] (0,0) ellipse (2.60 and 1.50);
  \node[acc, anchor=west, font=\scriptsize] at (1.32,0.35) {$n=1$};
  \node[acc, anchor=west, font=\scriptsize] at (1.86,0.62) {$n=2$};
  \node[acc, anchor=west, font=\scriptsize] at (2.62,0.30) {$n=4$};
  % annulus shading hint: mark the area between two rings
  \draw[black, <->] (0,0.75) -- (0,1.06);
  \node[black, anchor=west, font=\scriptsize] at (0.1,2.05) {each ring encloses};
  \node[black, anchor=west, font=\scriptsize] at (0.1,1.75) {one more quantum $h$};
\end{tikzpicture}
$$

### The one-dimensional oscillator

A particle of mass $m$ in the potential $V(q) = \tfrac{1}{2}m\omega^2 q^2$ has
energy $E = p^2/2m + \tfrac{1}{2}m\omega^2 q^2$. The constant-energy curve in the
phase plane is the ellipse

$$
\frac{p^2}{2mE} + \frac{q^2}{2E/m\omega^2} = 1,
$$

with semi-axes $\sqrt{2mE}$ along $p$ and $\sqrt{2E/m\omega^2}$ along $q$. The
enclosed area is $\pi$ times the product of the semi-axes:

$$
J = \oint p\,\d q = \pi\,\sqrt{2mE}\,\sqrt{\frac{2E}{m\omega^2}}
= \frac{2\pi E}{\omega} = \frac{E}{\nu},
$$

using $\omega = 2\pi\nu$. Setting $J = nh$ gives the allowed energies directly:

$$
E_n = n h \nu = n\hbar\omega, \qquad n = 0, 1, 2, \dots
$$

This is Planck's oscillator spectrum, now derived rather than postulated. It is
also _almost_ right: the true spectrum is $(n + \tfrac{1}{2})\hbar\omega$. The
old quantum theory misses the zero-point energy $\tfrac{1}{2}\hbar\omega$ because
it quantizes the bare action $\oint p\,\d q = nh$, whereas the semiclassical
[WKB treatment](/quantum-mechanics/approximation-methods/the-wkb-approximation)
carries a turning-point correction that promotes this to
$\oint p\,\d q = (n + \tfrac{1}{2})h$. The half-integer will recur throughout the
program's near-misses.

> **Worked example.** A diatomic molecule vibrates at $\nu = 1.3 \times 10^{13}\
> \text{Hz}$. The old-quantum spacing between vibrational levels is
> $$
> \Delta E = h\nu = (4.14 \times 10^{-15}\ \text{eV}\cdot\text{s})(1.3 \times 10^{13}\ \text{s}^{-1})
> = 0.054\ \text{eV},
> $$
> about twice the room-temperature thermal quantum $kT \approx 0.025\ \text{eV}$,
> so the vibrational mode is only partly excited at $300\ \text{K}$ — the same
> freezing-out that starved the short-wavelength cavity modes in Planck's law.

### Adiabatic invariance: why the action

Nothing so far explains why the action $\oint p\,\d q$, and not some other
function of the orbit, is the quantity to set equal to $nh$. Paul Ehrenfest
supplied the principle. If a system's parameters are changed slowly compared with
its period — an oscillator whose spring is stiffened gradually, a pendulum whose
length is slowly shortened — certain combinations of the dynamical variables stay
fixed even as the energy and frequency drift. These are the **adiabatic
invariants**, and for a one-dimensional periodic system the action $J = \oint
p\,\d q$ is one of them.

> **Definition (Adiabatic invariant).** A quantity that remains constant under an
> arbitrarily slow change of a system's parameters, to all orders in the rate of
> change. For a periodic mechanical system the action $J = \oint p\,\d q$ is
> adiabatically invariant; for the harmonic oscillator this is the statement that
> $E/\nu$ is conserved as the frequency is slowly varied.

Ehrenfest's argument for quantizing the invariant is a consistency demand: a
quantum number, once assigned, should not change merely because the apparatus is
tuned slowly. Only an adiabatic invariant can carry such a label without
contradiction. This is the deepest justification the old theory offered, and it
is correct as far as it goes — the action variable becomes, in wave mechanics,
the quantity whose integer value counts the nodes of the stationary wavefunction.

## Multiply periodic motion and elliptical orbits

The hydrogen atom is more than a circle. An electron in the Coulomb potential
$V(r) = -kZe^2/r$ moves, in general, on an ellipse with the nucleus at one focus.
In plane polar coordinates $(r, \varphi)$ the motion separates: the azimuthal
momentum $p_\varphi = L$ is conserved, while the radial momentum $p_r = m\dot r$
oscillates between the orbit's perihelion and aphelion. Two coordinates, two
phase integrals, two quantum numbers.

The azimuthal integral repeats Bohr's condition,

$$
J_\varphi = \oint p_\varphi \,\d\varphi = 2\pi L = n_\varphi h
\;\Rightarrow\;
L = n_\varphi \hbar, \qquad n_\varphi = 1, 2, 3, \dots
$$

The integer $n_\varphi \equiv k$ is the **azimuthal quantum number**. The radial
integral is harder; carrying it out with the Coulomb energy
$E = p_r^2/2m + L^2/2mr^2 - kZe^2/r$ and the substitution $u = 1/r$ gives

$$
J_r = \oint p_r \,\d r
= 2\pi\!\left(\frac{kZe^2 \sqrt{m}}{\sqrt{-2E}} - L\right) = n_r h,
$$

where $n_r = 0, 1, 2, \dots$ is the **radial quantum number**. Solving for the
energy after inserting $L = k\hbar$ and $J_r = n_r h$,

$$
E = -\frac{mk^2 Z^2 e^4}{2\hbar^2}\,\frac{1}{(n_r + k)^2}.
$$

Define the **principal quantum number** $n = n_r + k$. The energy depends only on
this sum:

> **Theorem (Sommerfeld energy levels).** The energy of a hydrogen-like Kepler
> orbit under the phase-integral conditions is
> $$
> E_n = -\frac{mk^2 Z^2 e^4}{2\hbar^2}\,\frac{1}{n^2}, \qquad n = n_r + k,
> $$
> identical to Bohr's result. For fixed $n$ the azimuthal number runs over
> $k = 1, 2, \dots, n$ (so $n_r = n - k \geq 0$), giving $n$ distinct orbits of
> the same energy.

The nonrelativistic energy is degenerate: it cannot tell an eccentric orbit from
a circular one of the same major axis. The geometry behind the degeneracy is
Kepler's. An orbit with quantum numbers $(n, k)$ is an ellipse whose semi-major
axis $a = n^2 a_0/Z$ is fixed by $n$ alone, while its semi-minor axis obeys

$$
\frac{b}{a} = \frac{k}{n}.
$$

The circular orbit is $k = n$; decreasing $k$ at fixed $n$ produces ever more
eccentric ellipses, all sharing the same energy and the same major axis. The
case $k = 0$ is excluded — it would be a degenerate line through the nucleus —
and its absence is the old theory's version of the rule that the ground state
has one unit of angular momentum.

$$
% caption: Sommerfeld ellipses for principal quantum number n = 3. All three
% share the same major axis and the same energy; the azimuthal number k sets the
% eccentricity through b/a = k/n, with k = 3 the Bohr circle and k = 1 the most
% elongated orbit. The nucleus sits at the common focus.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=0.95]
  \definecolor{acc}{HTML}{4A6FA5}
  % common focus (nucleus) at origin; each ellipse centered at (-c,0), a=3
  % k=3 circle: b=3, c=0
  \draw[black, thick] (0,0) ellipse (3 and 3);
  % k=2: b=2, c=sqrt(9-4)=2.236
  \draw[acc, thick] (-2.236,0) ellipse (3 and 2);
  % k=1: b=1, c=sqrt(9-1)=2.828
  \draw[black!70, thick, densely dashed] (-2.828,0) ellipse (3 and 1);
  % nucleus
  \fill[black] (0,0) circle (2.6pt);
  \node[anchor=west, font=\scriptsize] at (0.15,-0.35) {nucleus};
  % labels
  \node[black, anchor=south, font=\scriptsize] at (0,3.05) {$k=3$};
  \node[acc, anchor=south, font=\scriptsize] at (-2.4,2.05) {$k=2$};
  \node[black!70, anchor=north, font=\scriptsize] at (-3.0,-1.05) {$k=1$};
  \node[anchor=west, font=\scriptsize] at (2.2,2.2) {$n=3$};
\end{tikzpicture}
$$

## Relativistic fine structure

The degeneracy is an artifact of the nonrelativistic energy. An electron on an
eccentric orbit runs fastest at perihelion, where it dips deep into the Coulomb
well, and slowest at aphelion. Its speed, in units of $c$, is set by the
**fine-structure constant**

$$
\alpha = \frac{ke^2}{\hbar c} = 7.297 \times 10^{-3} \approx \frac{1}{137.036},
$$

a pure number built from the constants of electromagnetism and quantum theory.[^codata]
Because the speed varies around an eccentric orbit, the relativistic mass
increase $m \to m/\sqrt{1 - v^2/c^2}$ varies too, and a $1/r^2$ term is added to
the effective radial force. A closed ellipse under a pure $1/r^2$ attraction does
not remain closed once this correction enters: the perihelion advances a little
each revolution, and the orbit traces a slowly rotating rosette.

$$
% caption: Relativistic precession of a Sommerfeld ellipse. The varying speed on
% an eccentric orbit makes the relativistic mass vary, advancing the perihelion
% by a small angle each revolution; the orbit is a slowly rotating rosette rather
% than a closed ellipse.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=0.95]
  \definecolor{acc}{HTML}{4A6FA5}
  % focus at origin, ellipse a=2.6, b=1.3, c=sqrt(6.76-1.69)=2.252, center (-2.252,0)
  \draw[acc, thick, rotate around={0:(0,0)}]   (-2.252,0) ellipse (2.6 and 1.3);
  \draw[acc, thick, rotate around={40:(0,0)}]  (-2.252,0) ellipse (2.6 and 1.3);
  \draw[acc, thick, rotate around={80:(0,0)}]  (-2.252,0) ellipse (2.6 and 1.3);
  \draw[black, thick, rotate around={120:(0,0)}] (-2.252,0) ellipse (2.6 and 1.3);
  \fill[black] (0,0) circle (2.6pt);
  \node[anchor=north, font=\scriptsize] at (0.0,-0.2) {nucleus};
  % precession arc + arrow
  \draw[->, black, thick] (2.4,0.2) arc (5:70:2.5);
  \node[black, anchor=west, font=\scriptsize] at (2.05,1.75) {perihelion advance};
\end{tikzpicture}
$$

Carrying the relativistic Kepler problem through the phase integrals, Sommerfeld
found that the energy now depends separately on $n$ and $k$:

> **Theorem (Sommerfeld fine-structure formula).** To leading order in
> $(Z\alpha)^2$, the relativistic phase-integral energy of a hydrogen-like atom
> is
> $$
> E_{n,k} = -\frac{mc^2 (Z\alpha)^2}{2n^2}
> \left[\,1 + \frac{(Z\alpha)^2}{n^2}\!\left(\frac{n}{k} - \frac{3}{4}\right) + \cdots\right],
> $$
> with $k = 1, 2, \dots, n$. The leading bracket term reproduces the Bohr
> spectrum; the $(Z\alpha)^2$ correction splits each level $n$ into $n$
> sublevels, lifting the degeneracy between orbits of different eccentricity.

The correction scales as $\alpha^2 \approx 5 \times 10^{-5}$ relative to the gross
structure, a splitting a few parts in $10^5$ of the level energy. That is the
observed **fine structure** of the hydrogen lines, and Sommerfeld's formula
matched the measured $H_\alpha$ splitting to the precision of 1916
spectroscopy — one of the most celebrated quantitative successes of the old
theory.

> **Worked example.** For hydrogen ($Z = 1$) the two extreme orbits at $n = 2$
> are $k = 1$ and $k = 2$. Their energy separation from the formula is
> $$
> \Delta E_{n=2} = \frac{mc^2 \alpha^4}{32}
> = \frac{(5.11 \times 10^5\ \text{eV})(7.297 \times 10^{-3})^4}{32}
> = 4.5 \times 10^{-5}\ \text{eV},
> $$
> equivalently $0.36\ \text{cm}^{-1}$ in wavenumber. The measured $n = 2$ fine
> splitting is $0.365\ \text{cm}^{-1}$; the numerical agreement is exact to this
> order, though the physical origin is not orbital eccentricity but electron
> [spin-orbit coupling](/quantum-mechanics/approximation-methods/fine-structure-and-the-real-hydrogen-atom),
> which the Dirac equation later showed conspires to give the same energies with
> $k \to j + \tfrac{1}{2}$.

The agreement is thus partly an accident: Sommerfeld's $k$ plays the algebraic
role that the total angular momentum $j$ plays in the correct theory, and the two
labelings happen to yield identical energies at this order. The old theory got
the right answer through a picture that later proved wrong — a warning that
predictive success does not certify a mechanism.

## Space quantization

A third phase integral appears when the orbit is allowed to tilt in three
dimensions. Sommerfeld argued that the plane of the orbit cannot point in an
arbitrary direction relative to an external axis — say the direction of a weak
magnetic field — but only along a discrete set of orientations. The component of
the angular momentum along the field axis is itself quantized:

> **Definition (Space quantization).** The projection of the orbital angular
> momentum on a chosen axis takes only the values
> $$
> L_z = m\hbar, \qquad m = 0, \pm 1, \pm 2, \dots, \pm k,
> $$
> where $m$ is the **magnetic quantum number**. The orbital plane is restricted
> to the $2k + 1$ orientations for which $L_z$ is an integer multiple of $\hbar$.

$$
% caption: Space quantization of an orbit with azimuthal number k = 2. The
% angular-momentum vector may point only along orientations whose projection on
% the field axis is an integer multiple of hbar; the allowed projections are
% shown as tick values in units of hbar, symmetric about zero.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % field axis
  \draw[->, black] (0,-2.6) -- (0,2.9) node[anchor=south] {$z$ axis};
  % vectors of fixed length R=2.4 at angles giving Lz = 2,1,0,-1,-2 (in units where R corresponds to sqrt(6))
  % cos = m/sqrt(k(k+1)) = m/sqrt(6): m=2 ->0.816, m=1->0.408, m=0->0
  % endpoints: (R sin, R cos)
  \foreach \m/\cs/\sn in {2/0.816/0.577, 1/0.408/0.913, 0/0.0/1.0}{
    \draw[->, acc, thick] (0,0) -- ({2.4*\sn},{2.4*\cs});
  }
  \foreach \m/\cs/\sn in {1/-0.408/0.913, 2/-0.816/0.577}{
    \draw[->, acc, thick] (0,0) -- ({2.4*\sn},{2.4*\cs});
  }
  % projection ticks on axis
  \foreach \y/\lab in {1.96/2, 0.98/1, 0.0/0}{
    \draw[black, dashed] (0,\y) -- (1.7,\y);
    \node[anchor=west, font=\scriptsize] at (1.75,\y) {$L_z = \lab$};
  }
  \node[acc, anchor=east, font=\scriptsize] at (-0.55,1.55) {$L$};
\end{tikzpicture}
$$

Space quantization was a startling claim — that an orbit "knows" about an axis
even before a field is applied strongly — and it was tested directly by the
[Stern-Gerlach experiment](/quantum-mechanics/spin/spin-half-pauli-matrices-and-stern-gerlach)
in 1922, which sent silver atoms through an inhomogeneous field and found the
beam split into a discrete set of spots rather than a continuous smear. The old
theory predicted the discreteness correctly but the _count_ wrongly: it allowed
$2k + 1$ orientations, always odd, whereas silver split into two. The missing
ingredient, again, was half-integer angular momentum — electron spin — which the
orbital picture had no room for.

## The correspondence principle as a computational tool

Bohr's third postulate, that quantum results must merge into classical ones at
large quantum numbers, is often quoted as a philosophical guideline. In the old
theory it was a working instrument that supplied two things the quantization
conditions alone could not: the _frequencies_ of emitted radiation in a regime
where they could be checked, and the _intensities_ and _selection rules_ that the
energy levels left entirely open.

Consider a transition $n \to n - \tau$ for integer $\tau$. The Bohr frequency is
$\nu_{\text{quantum}} = (E_n - E_{n-\tau})/h$. For a classical multiply periodic
motion, the position expands in a Fourier series over harmonics $\tau$ of the
fundamental orbital frequency $\nu_{\text{orb}}$, and the electron radiates at
each harmonic $\tau\,\nu_{\text{orb}}$. Bohr's principle equates the two in the
limit of large $n$:

$$
\nu_{\text{quantum}}(n \to n-\tau)
\;\xrightarrow{\;n \to \infty\;}\;
\tau\,\nu_{\text{orb}}(n).
$$

For the hydrogen $\tau = 1$ jump this is checked directly. The quantum ratio
between the $n \to n-1$ line frequency and the classical orbital frequency at
level $n$ is

$$
\frac{\nu_{\text{quantum}}}{\nu_{\text{orb}}}
= \frac{n(2n - 1)}{2(n-1)^2},
$$

which falls monotonically from $3$ at $n = 2$ toward $1$ as $n$ grows, confirming
the classical limit.

$$
% caption: Approach to the classical limit. The ratio of the n to n-1 quantum
% line frequency to the classical orbital frequency at level n falls from 3 at
% small n toward 1 (dashed) as n increases, the quantitative content of the
% correspondence principle.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[anchor=south east] {quantum number $n$};
  \draw[->, black] (0,0) -- (0,4.1) node[anchor=south east, align=left] {frequency\\ratio};
  % dashed classical limit at plot y = 0.6 (ratio 1)
  \draw[black, dashed, thick] (0,0.6) -- (7.0,0.6);
  \node[black, anchor=west, font=\scriptsize] at (4.2,0.85) {classical limit};
  \node[anchor=east, font=\scriptsize] at (-0.05,0.6) {$1$};
  \node[anchor=east, font=\scriptsize] at (-0.05,3.6) {$3$};
  % data points: (x, y) with y = (ratio-1)*1.5+0.6
  \draw[acc, thick, smooth] plot coordinates
    {(0.8,3.6)(1.6,1.91)(2.8,1.21)(4.0,0.86)(5.2,0.72)(6.4,0.65)};
  \foreach \p in {(0.8,3.6),(1.6,1.91),(2.8,1.21),(4.0,0.86),(5.2,0.72),(6.4,0.65)}
    \fill[acc] \p circle (1.7pt);
  \node[anchor=north, font=\scriptsize] at (0.8,-0.05) {$2$};
  \node[anchor=north, font=\scriptsize] at (2.8,-0.05) {$5$};
  \node[anchor=north, font=\scriptsize] at (5.2,-0.05) {$20$};
\end{tikzpicture}
$$

The intensities followed the same logic. The power a classical charge radiates
into the $\tau$-th harmonic is proportional to the squared amplitude of the
$\tau$-th Fourier coefficient of its motion. Bohr and Kramers took the quantum
line strength for the transition $n \to n-\tau$ to correspond, at large $n$, to
that classical coefficient — the first quantitative theory of spectral
intensities. The rule also generated **selection rules**: if a coordinate's
Fourier series contains no $\tau$-th harmonic, the corresponding transition does
not radiate.

- **Harmonic oscillator.** The motion $q(t) = A\cos\omega t$ is a single
  harmonic, $\tau = 1$ only. Transitions must change the quantum number by
  exactly one, $\Delta n = \pm 1$ — the selection rule later derived from the
  matrix elements of $x$ in the number basis.
- **Kepler orbit.** An eccentric ellipse has a full spectrum of Fourier
  harmonics, so hydrogen radiates on many $\Delta n$; but the _azimuthal_ motion
  restricts the change in $k$, giving $\Delta k = \pm 1$, the ancestor of the
  dipole rule $\Delta\ell = \pm 1$.

The same correspondence limit even fixes the _value_ of Planck's constant
relative to the spectroscopic Rydberg constant: demanding that the $n \to n-1$
frequency equal the classical orbital frequency at large $n$ forces the angular
momentum step to be exactly $\hbar$, so the constant in $L = n\hbar$ is not free.
This closure — quantization at small $n$, classical radiation at large $n$, and
the two stitched together by a single constant — is the intellectual high-water
mark of the old theory.

## Where the program broke

The failures were not vague dissatisfactions; each was a specific system the
method could not handle, and together they mapped the boundary of orbit-based
quantization.

| Failure | What the old theory gave | What experiment showed |
| --- | --- | --- |
| Ground-state hydrogen | $k = 1$, so $L = \hbar$ | $L = 0$ (an $s$ state) |
| Helium ground state | no stable, unique quantized orbit | $-79.0\ \text{eV}$, sharply defined |
| Line intensities | only the large-$n$ correspondence estimate | exact ratios at all $n$ |
| Anomalous Zeeman effect | $2k + 1$ even-count splitting unaccounted | half-integer inner quantum numbers |
| Molecular band spectra | integer rotational quantization | half-integer $(n + \tfrac{1}{2})$ fits |

The helium failure is the deepest, and it is structural rather than numerical.
The Wilson-Sommerfeld rule presupposes that the motion is **multiply periodic** —
separable into coordinates each executing its own independent period, so that a
well-defined action $\oint p_i\,\d q_i$ exists for each. The two-electron atom is
a three-body problem with electron-electron repulsion, and its classical motion
is in general **nonintegrable**: the trajectories are chaotic, no separation into
periodic coordinates exists, and there are simply no action variables to set
equal to $nh$.[^shankar] The method has nothing to quantize. Every attempt to
force helium through it — Bohr's crossed-orbit model, Kramers' and Van Vleck's
calculations — gave energies wrong by electron volts or configurations that were
mechanically unstable. A theory that cannot treat the second element in the
periodic table is not a theory of atoms.

The ground-state angular momentum was a quieter but equally fatal sign. The old
theory forbids $k = 0$ because a zero-angular-momentum orbit is a line through the
nucleus, a collision. Yet the true hydrogen ground state has exactly $\ell = 0$:
its electron has no orbital angular momentum and no orbit at all. The resolution
is that there is no trajectory — the electron is described by a
[standing matter wave](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction),
and the quantum number $n$ counts nodes of that wave, not revolutions of a
particle. De Broglie's 1924 reinterpretation of $L = n\hbar$ as the condition
that an integer number of electron wavelengths close around the orbit was the
hinge, and Schrödinger's 1926 equation replaced the orbits entirely.

The old quantum theory retains a permanent place, not as a stepping stone to be
discarded but because two of its constructions survive intact. The phase integral
$\oint p\,\d q = (n + \tfrac{1}{2})h$ is the
[WKB quantization condition](/quantum-mechanics/approximation-methods/the-wkb-approximation),
still the tool of choice for highly excited states and tunneling rates, and the
correspondence principle remains the standard check that any quantum result
reduces to classical mechanics in the appropriate limit. What the program lacked
was a wave equation to derive its conditions from rather than impose them by
hand — the subject of the modules that follow.

[^codata]: CODATA recommended value $\alpha^{-1} = 137.035999$ and
$\alpha = 7.2973525 \times 10^{-3}$: NIST/CODATA, _Fundamental Physical
Constants_, https://physics.nist.gov/cuu/Constants/. The combination
$\alpha = ke^2/\hbar c$ is dimensionless and frame-independent.
[^shankar]: **Shankar**, _Principles of Quantum Mechanics_, 2nd ed., Ch. 12
(introduction) — the semiclassical quantization of angular momentum and the
restriction of the old quantum theory to separable, multiply periodic systems;
the nonintegrability of the helium three-body problem as the obstruction to
extending the Bohr-Sommerfeld conditions. See also Tipler & Llewellyn, §4-3
(correspondence principle, fine-structure constant) and Griffiths & Schroeter,
Afterword (historical placement of the old quantum theory before wave mechanics).