---
title: Limits of the Old Quantum Theory and the WKB Bridge
module: Early Atomic Models and the Old Quantum Theory
moduleNumber: 1
lessonNumber: 5
order: 105
summary: >
  The old quantum theory works only where the classical motion is separable into
  independent periodic coordinates. It fails for helium, forbids the correct
  zero angular momentum of the hydrogen ground state, and misses the half-integer
  in the oscillator and in molecular spectra. The WKB quantization condition,
  derived from the Schrodinger equation, is the modern descendant of the
  Sommerfeld rule and repairs the half-integer through the Maslov correction.
topics: [Early Atomic Models and the Old Quantum Theory]
sources:
  - book: Bransden & Joachain
    ref: "Ch. 2 — The Old Quantum Theory; §2.7 Limitations and the Correspondence Principle"
  - book: Griffiths & Schroeter
    ref: "Ch. 9 — The WKB Approximation; §9.1 The Classical Region, §9.3 Connection Formulas"
  - book: Foot
    ref: "Ch. 1 — Early Atomic Physics; the correspondence principle"
draft: false
---

The Bohr-Sommerfeld theory quantizes the action of each separable coordinate. Its
reach ends where that separation ends. A survey of what the theory gets wrong
locates the boundary precisely, and a semiclassical derivation from the
Schrödinger equation shows what the action-quantization rule becomes once wave
mechanics is in place: the same $\oint p\,\d q$ condition, corrected by a
half-integer whose origin is the phase a wave loses at a classical turning point.

## The correspondence principle as a design rule

Bohr's correspondence principle is more than a consistency check. In the old
quantum theory it functions as a quantitative tool for fixing quantities the
quantization rule leaves open. The principle states that quantum predictions must
merge into the classical ones in the limit of large quantum numbers, where
neighbouring levels are closely spaced.[^foot-corr]

For hydrogen, the frequency radiated in the transition $n \to n - 1$ at large $n$
must equal the classical orbital frequency $f_{\text{cl}} = v/2\pi r$ of the
electron. Both evaluate to

$$
f = \frac{Z^2 m\kappa^2}{2\pi\hbar^3}\frac{1}{n^3}, \qquad \kappa = \frac{Ze^2}{4\pi\epsilon_0},
$$

which fixes the constant in the Rydberg formula without appeal to spectroscopic
data. The agreement is exact only as $n \to \infty$; at small $n$ the quantum
transition frequency and the classical orbital frequency diverge, and the discrete
spectrum is genuinely non-classical.

$$
% caption: The quantum transition frequency for the jump n to n minus 1 and the
% classical orbital frequency at level n both fall as the inverse cube of n and
% converge at large n; at small n they differ, marking the region where the
% correspondence principle gives no guidance.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.4,0) node[right, font=\scriptsize] {$n$};
\draw[->, black] (0,0) -- (0,4.4) node[above, font=\scriptsize] {frequency};
% classical ~ 1/n^3, scaled
\draw[acc, very thick] plot[smooth] coordinates
  {(1,4.0)(1.5,2.2)(2,1.3)(2.5,0.85)(3,0.6)(4,0.35)(5,0.24)(6,0.18)};
\node[acc, font=\scriptsize, anchor=west] at (2.4,1.55) {classical};
% quantum transition, slightly higher at small n, merging
\draw[black, very thick, dashed] plot[smooth] coordinates
  {(1,2.7)(1.5,1.75)(2,1.15)(2.5,0.8)(3,0.58)(4,0.34)(5,0.235)(6,0.178)};
\node[black, font=\scriptsize, anchor=west] at (3.3,0.9) {quantum jump};
\node[black, font=\scriptsize, anchor=west] at (4.6,0.55) {converge};
\end{tikzpicture}
$$

The principle also predicts selection rules. A classical orbit of frequency
$f_{\text{cl}}$ radiates at $f_{\text{cl}}$ and its harmonics $2f_{\text{cl}},
3f_{\text{cl}}, \dots$, with amplitudes set by the Fourier components of the
motion. For a nearly circular orbit only the fundamental is present, so at large
$n$ the atom radiates only in transitions that change the azimuthal number by one,
$\Delta k = \pm 1$. This is the correspondence-principle origin of the dipole
selection rule carried over into the quantum theory of
[radiative transitions](/atomic-physics/radiative-transitions-and-line-shapes/selection-rules-forbidden-transitions).

## Where the old quantum theory fails

The theory succeeds for hydrogen and hydrogen-like ions and fails everywhere the
motion is not separable into independent periodic coordinates.[^bj-limits]

- **Non-separable systems.** Helium has two electrons and a mutual repulsion; the
  classical three-body motion is not integrable, has no complete set of action
  variables, and cannot be quantized by the Sommerfeld rule. Every attempt to
  compute the helium ground-state energy from quantized orbits failed, giving
  values in gross disagreement with experiment.
- **The ground-state angular momentum.** The Sommerfeld ground state is $n = 1$,
  which forces $k = 1$ and $L = \hbar$. The measured hydrogen ground state has
  zero orbital angular momentum, $\ell = 0$. The old theory cannot represent an
  $L = 0$ state without invoking the excluded collision orbit through the nucleus.
- **Line intensities.** The theory places spectral lines but offers no way to
  compute how bright each is; intensities require transition amplitudes, which a
  classical orbit does not supply except through the correspondence limit.
- **The missing half-integer.** The action rule gives the oscillator $E_n =
  n\hbar\omega$, with no zero-point energy. Molecular vibrational and rotational
  spectra, and the specific heats that depend on them, require the levels
  $(n + \tfrac12)\hbar\omega$. The old rule is systematically wrong by half a
  quantum.
- **Half-integer angular momentum.** The Stern-Gerlach splitting into two beams,
  the anomalous Zeeman pattern, and the doublet structure of the alkali spectra
  all demand angular-momentum quantum numbers in half-integer steps, impossible in
  a theory built on closed orbits.

These are not defects to be patched by more careful bookkeeping. They mark the
point where the picture of a particle on a classical orbit must be abandoned for a
wave.

## The semiclassical wavefunction

The WKB approximation extracts the semiclassical limit of the Schrödinger equation
and, in doing so, recovers and corrects the Sommerfeld rule.[^gs-wkb] Write the
one-dimensional stationary wavefunction as an amplitude and a phase,

$$
\psi(x) = A(x)\,\mathrm{e}^{\,\mathrm{i} S(x)/\hbar},
$$

and substitute into $-\dfrac{\hbar^2}{2m}\psi'' + V\psi = E\psi$. Collecting powers
of $\hbar$, the leading order gives the classical Hamilton-Jacobi relation for the
phase,

$$
\left(\frac{\d S}{\d x}\right)^2 = 2m\bigl(E - V(x)\bigr) = p(x)^2,
$$

so $S(x) = \pm\displaystyle\int p(x)\,\d x$ with the local classical momentum
$p(x) = \sqrt{2m(E - V(x))}$. Collecting the terms of first order in $\hbar$ gives
the **transport equation** for the amplitude,

$$
2A'S' + A S'' = 0
\quad\Longrightarrow\quad
\frac{A'}{A} = -\frac{S''}{2S'} = -\frac{p'}{2p}
\quad\Longrightarrow\quad
A(x) \propto \frac{1}{\sqrt{p(x)}}.
$$

The amplitude scales as $p^{-1/2}$, which keeps the probability current
$\lvert\psi\rvert^2 v \propto \lvert A\rvert^2 p$ constant along $x$: the particle
is more likely to be found where it moves slowly. In the classically
allowed region $E > V$ the wavefunction oscillates,

$$
\psi(x) \approx \frac{1}{\sqrt{p(x)}}
\exp\!\left(\pm\frac{\mathrm{i}}{\hbar}\int p(x)\,\d x\right),
$$

and in the forbidden region $E < V$ it grows or decays with the imaginary momentum
$\lvert p\rvert = \sqrt{2m(V - E)}$. The approximation holds where the de Broglie
wavelength changes slowly, $\lvert \d\lambda/\d x\rvert \ll 1$, which breaks down
exactly at the **turning points** where $E = V(x)$ and $p \to 0$.

$$
% caption: A particle of energy E is bound between the turning points x1 and x2
% where the energy line meets the potential; the wavefunction oscillates in the
% classically allowed region and decays into the forbidden regions on either side.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% potential well (parabola-ish)
\draw[black, very thick] plot[smooth, domain=-2.7:2.7, samples=60]
  ({\x}, {0.42*\x*\x});
\node[black, font=\scriptsize, anchor=west] at (2.3,2.6) {$V(x)$};
% energy line
\draw[black, thick] (-2.2,2.0) -- (2.2,2.0);
\node[black, font=\scriptsize, anchor=east] at (-2.2,2.0) {$E$};
% turning points
\fill[black] (-2.18,2.0) circle (1.6pt);
\fill[black] (2.18,2.0) circle (1.6pt);
\node[font=\scriptsize, below] at (-2.18,1.85) {$x_1$};
\node[font=\scriptsize, below] at (2.18,1.85) {$x_2$};
% oscillatory wavefunction inside
\draw[acc, thick] plot[smooth, domain=-2.1:2.1, samples=90]
  ({\x}, {2.0 + 0.5*sin(360*\x/0.9)*exp(-0.05*\x*\x)});
\node[acc, font=\scriptsize, anchor=south] at (0,3.15) {oscillates};
\node[black, font=\scriptsize] at (0,0.75) {allowed};
\end{tikzpicture}
$$

## The quantization condition and the Maslov correction

The oscillatory interior solution must connect smoothly to the decaying solutions
on both sides. Near each turning point the potential is approximately linear, and
the exact solution there is an Airy function whose asymptotic forms supply the
**connection formulas**. For a turning point at $x_2$ with the allowed region on
its left, the physically admissible (decaying) exterior solution matches the
interior wave as[^gs-connect]

$$
\frac{2}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\int_x^{x_2} p\,\d x' - \frac{\pi}{4}\right)
\;\longleftrightarrow\;
\frac{1}{\sqrt{\lvert p\rvert}}\exp\!\left(-\frac{1}{\hbar}\int_{x_2}^{x}\lvert p\rvert\,\d x'\right),
$$

and a mirror-image formula holds at the left turning point $x_1$. Each match
introduces a phase shift of $\pi/4$. Requiring the interior wave to be consistent
with the decaying solutions at both ends forces the accumulated phase to be a
half-integer multiple of $\pi$,

$$
\frac{1}{\hbar}\int_{x_1}^{x_2} p(x)\,\d x
= \left(n + \tfrac12\right)\pi, \qquad n = 0, 1, 2, \dots
$$

Written as a full-period loop this is the corrected action rule.

> **Definition (WKB quantization condition).** A particle bound between two soft
> turning points has energies fixed by
> $$
> \oint p\,\d q = 2\int_{x_1}^{x_2} p(x)\,\d x = \left(n + \tfrac12\right)h,
> \qquad n = 0, 1, 2, \dots
> $$
> The Sommerfeld rule $\oint p\,\d q = nh$ is the same condition without the
> half-integer.

The extra $\tfrac12$ is the **Maslov correction**. Its general form attaches a
quarter to each turning point,

$$
\oint p\,\d q = \left(n + \frac{\mu}{4}\right)h,
$$

where the **Maslov index** $\mu$ counts the encounters with a caustic: a soft
turning point where the potential is smooth contributes $1$, so a bound state
between two soft turning points has $\mu = 2$ and the half-integer follows. A hard
wall, where the wavefunction must vanish, contributes a different phase and shifts
$\mu$ accordingly. The old quantum theory took $\mu = 0$ and missed exactly this
term.

$$
% caption: The bound motion traces a closed loop in the phase plane between the
% turning points x1 and x2; the WKB condition admits only the loops whose enclosed
% area equals n plus one half times h, shifting every Sommerfeld level up by half
% a quantum.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (-3.4,0) -- (3.7,0) node[right, font=\scriptsize] {$q$};
\draw[->, black] (0,-2.2) -- (0,2.4) node[above, font=\scriptsize] {$p$};
\draw[acc, very thick, fill=acc!10] (0,0) ellipse [x radius=2.7, y radius=1.5];
\fill[black] (-2.7,0) circle (1.8pt);
\fill[black] (2.7,0) circle (1.8pt);
\node[font=\scriptsize, below] at (-2.7,-0.05) {$x_1$};
\node[font=\scriptsize, below] at (2.7,-0.05) {$x_2$};
\node[acc, font=\scriptsize] at (0.9,0.85) {area};
\end{tikzpicture}
$$

> **Worked example (The harmonic oscillator, exactly).** For
> $V(x) = \tfrac12 m\omega^2 x^2$ the turning points are $x_{1,2} = \mp A$ with
> $A = \sqrt{2E/m\omega^2}$. The action integral is
> $$
> \int_{-A}^{A}\sqrt{2m\!\left(E - \tfrac12 m\omega^2 x^2\right)}\,\d x
> = \sqrt{2mE}\int_{-A}^{A}\sqrt{1 - \frac{x^2}{A^2}}\,\d x
> = \sqrt{2mE}\cdot\frac{\pi A}{2} = \frac{\pi E}{\omega}.
> $$
> The WKB condition $\int_{x_1}^{x_2} p\,\d x = (n + \tfrac12)\pi\hbar$ then gives
> $$
> \frac{\pi E}{\omega} = \left(n + \tfrac12\right)\pi\hbar
> \quad\Longrightarrow\quad
> E_n = \left(n + \tfrac12\right)\hbar\omega,
> $$
> the exact spectrum, zero-point energy included. WKB is exact here because the
> connection-formula phase is exact for a quadratic potential.

$$
% caption: The WKB spectrum of the harmonic oscillator is a ladder of equally
% spaced levels at energies n plus one half times h-bar omega; the lowest level
% sits half a quantum above the potential minimum, the zero-point energy the old
% quantum theory could not produce.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, very thick] plot[smooth, domain=-2.6:2.6, samples=60]
  ({\x}, {0.5*\x*\x});
\node[black, font=\scriptsize, anchor=west] at (2.2,2.9) {$V(x)$};
% levels at E = (n+1/2), spacing 0.75 in this scale, starting 0.375
\foreach \n/\y in {1/1.17, 2/1.92, 3/2.67} {
  \pgfmathsetmacro{\xr}{sqrt(2*\y)}
  \draw[black, thick] ({-\xr},\y) -- ({\xr},\y);
}
\pgfmathsetmacro{\xr}{sqrt(2*0.42)}
\draw[acc, thick] ({-\xr},0.42) -- ({\xr},0.42);
\node[acc, font=\scriptsize, anchor=west] at (0.95,0.42) {$n=0$};
\node[black, font=\scriptsize, anchor=west] at (1.6,1.17) {$n=1$};
\draw[black, dashed] (-2.8,0) -- (2.8,0);
\node[black, font=\scriptsize, anchor=west] at (-2.8,-0.28) {zero-point gap};
\end{tikzpicture}
$$

## The Langer correction and the radial problem

Applying WKB to the hydrogen atom exposes a subtlety at the origin. The radial
equation for $u(r) = r R(r)$ carries the centrifugal term,

$$
-\frac{\hbar^2}{2m}\frac{\d^2 u}{\d r^2}
+ \left[V(r) + \frac{\ell(\ell + 1)\hbar^2}{2mr^2}\right]u = E u,
$$

so the radial momentum is
$p_r(r) = \sqrt{2m(E - V) - \ell(\ell+1)\hbar^2/r^2}$. Quantizing
$\int p_r\,\d r = (n_r + \tfrac12)\pi\hbar$ with this momentum gives the wrong
levels: the half-space $r > 0$ has a hard inner boundary at $r = 0$, where the
linear-potential connection formula does not apply, and the bare centrifugal term
behaves incorrectly there.[^bj-langer] Langer showed that the substitution
$r = \mathrm{e}^{s}$ maps the radial problem onto a full-line problem in $s$ with
two soft turning points, for which the $\tfrac12$ correction is valid, provided the
centrifugal coefficient is replaced,

> **Definition (Langer correction).** In the radial WKB integral the centrifugal
> term is replaced by
> $$
> \ell(\ell + 1) \;\longrightarrow\; \left(\ell + \tfrac12\right)^2,
> $$
> after which $\displaystyle\int_{r_1}^{r_2} p_r\,\d r = (n_r + \tfrac12)\pi\hbar$
> reproduces the exact Coulomb and isotropic-oscillator spectra.

With the replacement, the radial WKB integral for the Coulomb potential yields
$E_n = -Z^2(13.6\,\text{eV})/n^2$ with $n = n_r + \ell + 1$, the exact hydrogen
spectrum. The effective angular-momentum quantum number is $\ell + \tfrac12$, the
same half-integer shift that turned Sommerfeld's $k$ into $j + \tfrac12$ to match
the [Dirac levels](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen).
The old theory's $k = \ell + 1$ and the semiclassical $\ell + \tfrac12$ differ by
exactly the half-quantum that a careful turning-point analysis supplies.

## Tunnelling and the escape of a bound particle

The WKB wavefunction is nonzero in the classically forbidden region, so a particle
can pass through a barrier that classical mechanics forbids. For a barrier between
$a$ and $b$ where $V > E$, the amplitude decays by the exponential of the action
accumulated with the imaginary momentum $\lvert p\rvert = \sqrt{2m(V - E)}$, and
the transmission probability is the square,

$$
T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b \sqrt{2m(V(x) - E)}\,\d x\right).
$$

> **Worked example (Rectangular barrier).** For a barrier of height $V_0$ and
> width $L$ with $E < V_0$, the momentum is constant,
> $\lvert p\rvert = \sqrt{2m(V_0 - E)}$, and the integral is $\lvert p\rvert L$, so
> $$
> T \approx \exp\!\left(-\frac{2L}{\hbar}\sqrt{2m(V_0 - E)}\right).
> $$
> An electron ($m = 9.1\times10^{-31}\,\text{kg}$) meeting a $V_0 - E = 1\,\text{eV}$
> barrier of width $L = 0.5\,\text{nm}$ has
> $\lvert p\rvert = \sqrt{2m\cdot1.6\times10^{-19}\,\text{J}}
> = 5.4\times10^{-25}\,\text{kg m s}^{-1}$, giving exponent
> $2L\lvert p\rvert/\hbar \approx 5.1$ and $T \approx 6\times10^{-3}$. The
> probability drops by an order of magnitude for each additional $0.2\,\text{nm}$
> of width — the steep width dependence behind scanning tunnelling microscopy and
> the [alpha decay](/nuclear-physics/alpha-decay/alpha-decay-gamow-theory) of
> nuclei.

## What survives

The WKB condition explains both why the Sommerfeld rule worked and why it needed
correcting. The rule worked because $\oint p\,\d q$ is the leading-order phase of a
semiclassical wave, so quantizing it selects the standing waves that close on
themselves. It needed the half-integer because a real wave slips a quarter-cycle
in phase at each turning point, a purely wave-mechanical effect with no place in
the orbit picture. The same reasoning extends beyond the old theory to processes it
could not describe at all. A particle can leak through a classically forbidden
barrier, since the WKB wavefunction is nonzero there; the tunnelling probability is
$\exp\!\bigl(-\tfrac{2}{\hbar}\int\lvert p\rvert\,\d x\bigr)$, the mechanism behind
alpha decay. And an $\ell = 0$ ground state is now allowed, because the wave has no
orbit to collide with the nucleus.

$$
% caption: The chain from the Bohr model to wave mechanics; the correspondence
% principle and the action rule of 1913 to 1916 are absorbed into the Schrodinger
% and Heisenberg theories of 1925 to 1926, with WKB as the semiclassical bridge
% between them.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, thick] (0,0) -- (12,0);
\foreach \x in {0,3,6,9,12} \draw[black] (\x,0.12) -- (\x,-0.12);
\node[black, font=\scriptsize, align=center, anchor=south] at (0,0.2) {Bohr\\1913};
\node[black, font=\scriptsize, align=center, anchor=south] at (3,0.2) {Sommerfeld\\1916};
\node[black, font=\scriptsize, align=center, anchor=south] at (6,0.2) {de Broglie\\1924};
\node[black, font=\scriptsize, align=center, anchor=south] at (9,0.2) {Heisenberg\\1925};
\node[black, font=\scriptsize, align=center, anchor=south] at (12,0.2) {Schrodinger\\1926};
\node[black, font=\scriptsize, anchor=north] at (0,-0.2) {stationary states};
\node[black, font=\scriptsize, anchor=north] at (3,-0.2) {action rule};
\node[black, font=\scriptsize, anchor=north] at (6,-0.2) {matter waves};
\node[black, font=\scriptsize, anchor=north] at (9,-0.2) {matrix mechanics};
\node[black, font=\scriptsize, anchor=north] at (12,-0.2) {wave equation};
\end{tikzpicture}
$$

The old quantum theory is a scaffold. It gave the right hydrogen spectrum, named
the quantum numbers that survive, and taught the correspondence principle that
still guides the classical limit of any quantum theory, but it could not stand as
a mechanics. The [Schrödinger
equation](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
supplies the wave that the action rule was silently approximating, and the
[matter-wave](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction) condition
that a standing electron wave close on itself turns Bohr's $L = n\hbar$ from a
postulate into a consequence.

[^foot-corr]: **Foot**, _Atomic Physics_, Ch. 1 — the correspondence principle and the matching of the large-$n$ transition frequency to the classical orbital frequency. <https://global.oup.com/academic/product/atomic-physics-9780198506959>
[^bj-limits]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §2.7 — the failures of the old quantum theory: non-separable systems, the helium ground state, line intensities, and the missing zero-point and half-integer effects. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>
[^gs-wkb]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_, 3rd ed., §9.1 — the semiclassical expansion, the classical momentum $p(x) = \sqrt{2m(E-V)}$, and the amplitude $p^{-1/2}$. <https://www.cambridge.org/highereducation/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF0897775375>
[^gs-connect]: **Griffiths & Schroeter**, §9.3 — the Airy-function connection formulas, the $\pi/4$ turning-point phase, and the quantization condition $\int p\,\d x = (n+\tfrac12)\pi\hbar$; the Maslov index generalizes the $\tfrac12$. See also **Bransden & Joachain**, §2.7.
[^bj-langer]: **Bransden & Joachain**, §2.7, and **Griffiths & Schroeter**, §9 (problems) — the failure of naive radial WKB at $r = 0$ and the Langer replacement $\ell(\ell+1) \to (\ell+\tfrac12)^2$ that recovers the exact Coulomb spectrum. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>