The Old Quantum Theory: Bohr, Sommerfeld, and Correspondence

The Bohr model fixed the hydrogen spectrum by quantizing one quantity, the orbital angular momentum $L=n\hslash$. 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 $L=n\hslash$ is a statement about one coordinate, the azimuthal angle $\phi$, whose conjugate momentum $p_{\phi }=L$ is constant over the circular orbit. Write the condition as an integral around one full revolution:

The circular orbit hides the content, because $p_{\phi }$ 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.

The integral $\oint p\mathbf{d}q$ 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.

The one-dimensional oscillator

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

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:

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

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

Adiabatic invariance: why the action

Nothing so far explains why the action $\oint p\mathbf{d}q$, 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 $J=\oint p\mathbf{d}q$ is one of them.

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 $V(r)=-kZ{e}^2/r$ moves, in general, on an ellipse with the nucleus at one focus. In plane polar coordinates $(r,\phi )$ the motion separates: the azimuthal momentum $p_{\phi }=L$ 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,

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

where $n_r=0,1,2,\dots$ is the radial quantum number. Solving for the energy after inserting $L=k\hslash$ and $J_r=n_{r}h$,

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

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

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.

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

a pure number built from the constants of electromagnetism and quantum theory.¹ 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.

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

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.

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:

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 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 $n$:

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

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

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 $\hslash$, so the constant in $L=n\hslash$ 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.

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 $\oint p_i\mathbf{d}{q}_i$ 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 $nh$.² 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, and the quantum number $n$ counts nodes of that wave, not revolutions of a particle. De Broglie's 1924 reinterpretation of $L=n\hslash$ 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 $\oint p\mathbf{d}q=(n+\frac{1}{2})h$ is the WKB quantization condition, 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.