Limits of the Old Quantum Theory and the WKB Bridge

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 $\oint p\mathbf{d}q$ 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.¹

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

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.

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.

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.²

• 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 $L=\hslash$. 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 $E_n=n\hslash \omega$, with no zero-point energy. Molecular vibrational and rotational spectra, and the specific heats that depend on them, require the levels $(n+\frac{1}{2})\hslash \omega$. 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.³ Write the one-dimensional stationary wavefunction as an amplitude and a phase,

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

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

The amplitude scales as $p^{-1/2}$, which keeps the probability current $|\psi {|}^{2}v\propto |A{|}^{2}p$ 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,

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

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 as⁴

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 $\pi$,

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

The extra $\frac{1}{2}$ is the Maslov correction. Its general form attaches a quarter to each turning point,

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.

The Langer correction and the radial problem

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

so the radial momentum is $p_r(r)=\sqrt{2m(E-V)-\ell (\ell +1){\hslash }^2/r^2}$. Quantizing $\int p_r\mathbf{d}r=(n_r+\frac{1}{2})\pi \hslash$ 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.⁵ 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 $\frac{1}{2}$ correction is valid, provided the centrifugal coefficient is replaced,

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 $\ell +\frac{1}{2}$, the same half-integer shift that turned Sommerfeld's $k$ into $j+\frac{1}{2}$ to match the Dirac levels. The old theory's $k=\ell +1$ and the semiclassical $\ell +\frac{1}{2}$ 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 $|p|=\sqrt{2m(V-E)}$, and the transmission probability is the square,

What survives

The WKB condition explains both why the Sommerfeld rule worked and why it needed correcting. The rule worked because $\oint p\mathbf{d}q$ 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 $\exp (-\frac{2}{\hslash }\int |p|\mathbf{d}x)$, 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.

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 supplies the wave that the action rule was silently approximating, and the matter-wave condition that a standing electron wave close on itself turns Bohr's $L=n\hslash$ from a postulate into a consequence.