The Bohr-Sommerfeld Old Quantum Theory

Bohr's condition $L=n\hslash$ quantizes one degree of freedom, the angle swept by a circular orbit. A general bound motion has more degrees of freedom, and a Coulomb orbit is generically an ellipse, not a circle. In 1916 Sommerfeld supplied the missing generalization: quantize the action of every separable coordinate independently. The prescription reproduces the Bohr energies, introduces a second quantum number that labels the shape of the orbit, quantizes the orientation of the orbital plane in space, and — with the relativistic correction to the electron mass — splits each Bohr level into a fine-structure multiplet. This is the high-water mark of the old quantum theory, the last atomic model built from classical orbits before wave mechanics replaced them.

The action-integral quantization rule

For a periodic coordinate $q$ with conjugate momentum $p$, the action variable is the phase-space area enclosed over one period,

Sommerfeld and Wilson postulated that each such action is an integer multiple of Planck's constant.¹

The rule is dimensionally forced: $p\mathbf{d}q$ has units of action, the same as $h$, and no other combination of the orbital constants is dimensionless when divided by $h$. Its deeper justification is Ehrenfest's adiabatic principle, taken up at the end of this lesson: the action integrals coincide with the mechanical quantities that stay constant when the system is deformed slowly, so they are the only quantities that can carry an unchanging integer label.

The circular orbit recovered

For the hydrogen atom, use plane polar coordinates $(r,\phi )$ in the orbital plane. The Lagrangian $L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\phi }}^2)+\kappa /r$, with the Coulomb strength $\kappa \equiv Z{e}^2/4\pi {ϵ}_0$, gives the conjugate momenta

Because $\phi$ is cyclic, $p_{\phi }$ is a constant of the motion — it is the orbital angular momentum $L$. Its action integral over one revolution is immediate:

Quantizing with the azimuthal integer $k$ gives $2\pi L=kh$, that is $L=k\hslash$. For a circular orbit $p_r\equiv 0$, so the radial action vanishes and $k$ alone labels the state. Identifying $k$ with Bohr's $n$ reproduces $L=n\hslash$ and every result of the Bohr model. The content of Sommerfeld's extension is what happens when $p_r$ does not vanish.

Elliptical orbits and the radial quantum number

An orbit with $p_r\ne 0$ is an ellipse with the nucleus at one focus. The radial momentum follows from energy conservation. Writing the energy with the centrifugal term,

The motion oscillates between the perihelion $r_{\min }$ and aphelion $r_{\max }$, the two roots of the radicand, where $p_r=0$. The radial action is the integral over one in-and-out excursion,²

This is a standard integral of the form $\oint \sqrt{A+\frac{2B}{r}-\frac{C}{r^2}}\mathbf{d}r$ with $A=2mE<0$, $B=m\kappa$, and $C=L^2$. For a bound orbit ($A<0$) its value is

Quantizing $J_r=n_{r}h=2\pi n_r\hslash$ and inserting $L=k\hslash$ gives

Define the principal quantum number $n\equiv n_r+k$. Squaring isolates the energy,

The energy is exactly Bohr's, but the state is now labelled by two integers. The radial quantum number $n_r=0,1,2,\dots$ counts the radial oscillations, and the azimuthal quantum number $k=1,2,\dots ,n$ fixes the angular momentum $L=k\hslash$. The value $k=0$ is excluded: it describes a degenerate straight line through the nucleus, a collision orbit with no angular momentum, which the old theory rejects. For each $n$ there are $n$ allowed values of $k$, all degenerate in energy.

The shape of the orbit

The eccentricity of the ellipse is set by the ratio $k/n$. The angular momentum of a Kepler ellipse of semi-major axis $a$ and eccentricity $\epsilon$ is $L=\sqrt{m\kappa a(1-{\epsilon }^2)}$, while $a=\kappa /2|E|$ fixes the semi-major axis from the energy alone. Both the circular orbit ($k=n$) and the ellipse of the same $n$ share this $a$, so

where $b$ is the semi-minor axis and $L_{\text{circ}}=n\hslash$ is the angular momentum of the circle of the same energy. The orbit is roundest when $k=n$ and most elongated when $k=1$; the perihelion of the $k=1$ ellipse dips close to the nucleus, which is exactly where the relativistic correction below will bite hardest.

Space quantization

In three dimensions the orbital plane can tilt, adding a third degree of freedom. Working in spherical coordinates $(r,\theta ,\phi )$ with the polar axis fixed by an external field, the azimuthal angle $\phi$ about that axis is cyclic, and its conjugate momentum is the projection $L_z$ of the angular momentum on the axis. Its action integral gives a third condition,³

with the magnetic quantum number $m_{\ell }$ an integer. The polar action then constrains the total angular momentum magnitude to $L=k\hslash$ as before, with the requirement $|m_{\ell }|\le k$. The orbital plane cannot point in an arbitrary direction: only the $2k+1$ orientations with integer $L_z/\hslash$ are allowed. This space quantization is the old-theory ancestor of the directional quantization confirmed by the Stern-Gerlach experiment, treated with electron spin.

Counting the states of a given $n$ reproduces a result the full quantum theory recovers exactly. For each azimuthal number $k$ there are $2k-1$ allowed orientations (the values $m_{\ell }=-(k-1),\dots ,k-1$), and $k$ runs from $1$ to $n$, so the total is

Identifying the Sommerfeld azimuthal number with the modern orbital number through $k=\ell +1$ turns this into the familiar hydrogenic count: $\ell$ runs from $0$ to $n-1$, each with $2\ell +1$ values of $m_{\ell }$, again summing to $n^2$. The old theory already carries the degeneracy structure of the quantum hydrogen atom, though it assigns the states to orbits rather than to wavefunctions.

The relativistic fine-structure correction

The Bohr-Sommerfeld levels are exactly degenerate in $k$: every ellipse of a given $n$ has the same energy. Experiment shows a small splitting. Sommerfeld found its origin by keeping the relativistic dependence of the electron mass on speed. The elliptical orbit runs fastest at perihelion, where the mass is largest, so the orbit no longer closes: the perihelion precesses and the ellipse traces a slowly rotating rosette.⁴

Evaluating the action integrals with the relativistic momentum, Sommerfeld obtained a closed form for the energy that keeps $Z\alpha$ to all orders, where $\alpha =e^2/4\pi {ϵ}_0\hslash c\approx 1/137$ is the fine-structure constant:

Expanding in powers of $(Z\alpha {)}^2$ and writing $n=n_r+k$ gives the form used to compare with spectra,

The leading bracketed term is the Bohr energy; the correction is smaller by a factor $(Z\alpha {)}^2\sim 10^{-5}$ for hydrogen. It depends on $k$, so it lifts the degeneracy: within a level $n$, the small-$k$ (eccentric) orbits are shifted down more than the large-$k$ (round) ones, because the eccentric orbit spends part of its period deep in the potential where the electron moves relativistically.

The numerical agreement was a triumph. The success is partly a coincidence: the Sommerfeld formula agrees with the exact Dirac result once the azimuthal number is reinterpreted as $k\to j+\frac{1}{2}$, with $j$ the total angular momentum. Sommerfeld got the right levels from the wrong physics: the splitting is dominated by electron spin, absent from his purely orbital calculation, and two errors compensate to give the correct answer.

Adiabatic invariance

The action integrals were quantized without justification beyond dimensional consistency. Ehrenfest supplied the principle that singles them out. A quantity is an adiabatic invariant if it is unchanged when the parameters of the Hamiltonian are varied slowly compared with the orbital period. For a one-dimensional periodic system the action $J=\oint p\mathbf{d}q$ is such an invariant to all orders in the rate of change.⁵

The relevance is this. A quantum number must be a discrete label that cannot change under a smooth, slow deformation of the system, since a slow deformation excites no transition. Only an adiabatic invariant qualifies. The action integrals are the adiabatic invariants of a separable bound motion, so they are the correct quantities to set equal to $nh$. For the harmonic oscillator, slowly changing $\omega$ leaves $E/\omega =J/2\pi$ fixed, the classical shadow of the quantum statement that the integer $n$ does not change when a trap is squeezed gently. This principle carries into the wave theory as the invariance of the quantum number under adiabatic changes of a Hamiltonian.

The Bohr-Sommerfeld theory is the furthest the orbit picture reaches. It handles one-electron atoms, fixes the level count, and produces a fine structure of the right magnitude, but it fails the moment a second electron makes the motion non-separable, and it cannot supply the half-integers that a correct theory requires. Those limits, and the semiclassical rule that survives them, are the subject of the next lesson.