Correspondence & the Limits of the Old Quantum Theory

If quantum rules govern atoms and Newton governs planets, the two regimes must join smoothly somewhere in between. Bohr's correspondence principle makes that seam precise, and the Wilson–Sommerfeld rule $\oint p\,dq = nh$ generalizes his quantization to any periodic system — the old quantum theory at its most powerful. This lesson is its victory lap and its obituary: we tally what it gets right, then the failures that demanded a genuinely new mechanics. The old theory kept landing on its feet without ever knowing why; physics needed to know why.

Learning Objectives

After this lesson you will be able to:

1. State Bohr's correspondence principle precisely and explain what it does — and does not — assert.
2. Derive the $1/n^3$ scaling of both the hydrogen transition frequency and the classical orbital frequency, and show their ratio tends to 1.
3. Apply Wilson–Sommerfeld quantization $\oint p\,dq = nh$ to the particle in a box, the harmonic oscillator, and circular orbits, interpreting the loop integral as a phase-space area.
4. Apply Sommerfeld's three quantum conditions to the Coulomb problem: the degeneracy in $n$, space quantization, the old-theory state count, and the relativistic fine-structure formula.
5. Enumerate the failures of the old quantum theory and explain why each demanded a new framework.
6. Describe how Heisenberg's matrix mechanics and Schrödinger's wave mechanics replaced the old theory, and why they turned out to be one theory.

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Intuition

Planck's constant is tiny: $h = 6.626\times10^{-34}$ J·s. A pendulum with energy 0.01 J and frequency 1 Hz sits, by Planck's rule, on rung $n = E/h\nu \approx 1.5\times10^{31}$ of its energy ladder. The rungs are there, but their relative spacing is $10^{-31}$ — utterly invisible. Quantization matters only where actions are comparable to $h$; where they are enormous, quantum predictions must blur into classical ones. That is the correspondence principle, and Bohr wielded it as a design constraint: any proposed quantum rule must reproduce classical physics at large $n$ — the same logic by which relativity must reduce to Newton for $v \ll c$.

The second idea generalizes Bohr's one rule for one system. Wilson and Sommerfeld noticed what $L = n\hbar$ really says: the area enclosed by the orbit in phase space — the loop integral $\oint p\,dq$ of P.1.4 Hamiltonian Mechanics — comes in whole multiples of $h$. Phase space is rationed in cells of area $h$, one quantum state per cell; that single prescription quantizes any periodic system in one line.

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Theory

The correspondence principle, stated precisely

Correspondence principle (Bohr, 1920). In the limit of large quantum numbers ($n\to\infty$, equivalently actions $\gg h$), the predictions of the quantum theory — frequencies, intensities, dynamics of expectation values — must agree asymptotically with the predictions of classical physics for the same system.

Note the wording: it is a statement about the limit of predictions, not a claim that quantum systems "turn classical." Bohr also used it constructively: demanding agreement at large $n$ selects the correct quantum rule at small $n$.

Hydrogen: transition frequency vs orbital frequency

Classically, an electron on a circular orbit radiates at its orbital revolution frequency. Quantum mechanically, hydrogen radiates at $\nu = \Delta E/h$. Correspondence demands these merge at large $n$. Write $E_n = -E_R/n^2$ with $E_R = 13.6$ eV, so $E_R/h = 3.29\times10^{15}$ Hz.

Transition frequency for the smallest jump, $n+1\to n$:

$$\nu_{trans} = \frac{E_{n+1}-E_n}{h} = \frac{E_R}{h}\left(\frac{1}{n^2}-\frac{1}{(n+1)^2}\right) = \frac{E_R}{h}\,\frac{2n+1}{n^2(n+1)^2} \;\xrightarrow{\ n\gg1\ }\;\frac{2E_R}{h\,n^3} .$$

Orbital frequency of Bohr orbit $n$: from P.2.2, $v_n = \alpha c/n$ and $r_n = n^2a_0$, so

$$\nu_{orb} = \frac{v_n}{2\pi r_n} = \frac{\alpha c}{2\pi a_0}\,\frac{1}{n^3} , \qquad\text{and}\qquad \frac{\alpha c}{2\pi a_0} = \frac{me^4}{(4\pi\varepsilon_0)^2\,2\pi\hbar^3} = \frac{2E_R}{h} = 6.58\times10^{15}\ \mathrm{Hz}$$

(substitute $a_0 = 4\pi\varepsilon_0\hbar^2/me^2$ and $\alpha = e^2/4\pi\varepsilon_0\hbar c$) — the same prefactor. Both frequencies scale as $1/n^3$, and their ratio is exactly

$$\frac{\nu_{trans}}{\nu_{orb}} = \frac{2n+1}{n^2(n+1)^2}\cdot\frac{n^3}{2} = \frac{n(2n+1)}{2(n+1)^2} = 1 - \frac{3}{2n} + O\!\left(\frac{1}{n^2}\right)\;\longrightarrow\;1 .$$

At $n=10$ the quantum line is 14% below the classical prediction; at $n=10^4$ (Rydberg atoms — made in the lab, and antennas in radio astronomy) they agree to 0.015%. Bohr ran the argument backwards in 1913: demanding $\nu_{trans}\to\nu_{orb}$ is one way to derive $L = n\hbar$.

Wilson–Sommerfeld quantization: rationing phase space

For any coordinate $q$ periodic in time, with conjugate momentum $p$ (in the sense of P.1.4):

$$\boxed{\,\oint p\,dq = nh, \qquad n = 1,2,3,\dots\,}$$

The integral runs over one period; geometrically it is the area enclosed by the phase-space orbit — the action variable $J$ of Hamiltonian mechanics. Allowed orbits enclose an integer number of phase-space cells of area $h$. And the action-angle machinery of P.1.4 gives correspondence for free: since the classical frequency is $\nu_{cl} = dE/dJ$, at large $n$

$$\nu_{trans} = \frac{E_n - E_{n-1}}{h} \approx \frac{1}{h}\frac{dE}{dn} = \frac{dE}{dJ} = \nu_{cl}$$

for every Wilson–Sommerfeld system, not just hydrogen.

(a) Particle in a box. A particle bounces between walls at $x=0$ and $x=L$ with $p = +\sqrt{2mE}$ rightward, $-\sqrt{2mE}$ leftward: the phase orbit is a rectangle of width $L$ and height $2\sqrt{2mE}$,

$$\oint p\,dq = \int_0^L\!\sqrt{2mE}\,dx + \int_L^0\!(-\sqrt{2mE})\,dx = 2L\sqrt{2mE} = nh \quad\Longrightarrow\quad E_n = \frac{n^2h^2}{8mL^2} .$$

This $E_n \propto n^2$ is exactly the true quantum result for the infinite well (P.5.2 Step, Well & Barrier).

(b) Harmonic oscillator. For $H = \frac{p^2}{2m} + \frac12m\omega^2x^2 = E$ the phase orbit is an ellipse with semi-axes $x_{max} = \sqrt{2E/m\omega^2}$ and $p_{max} = \sqrt{2mE}$, of area

$$\oint p\,dq = \pi\,x_{max}\,p_{max} = \pi\sqrt{\frac{2E}{m\omega^2}}\sqrt{2mE} = \frac{2\pi E}{\omega} = \frac{E}{\nu} = nh \quad\Longrightarrow\quad E_n = nh\nu .$$

This is precisely Planck's blackbody assumption from P.2.1 — now derived rather than postulated. But it misses the zero-point energy: the true spectrum is $E_n = (n+\tfrac12)h\nu$, as the full quantum treatment shows in P.5.3 The Harmonic Oscillator. A quantum oscillator's ground state is not at rest — a fact with no old-quantum explanation (and the reason liquid helium never freezes at atmospheric pressure).

(c) Circular orbits. Take $q = \varphi$ with conjugate momentum $p_\varphi = L$, constant on the orbit:

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

Bohr's postulate is the special case of Wilson–Sommerfeld for the angular coordinate.

Sommerfeld's ellipses, space quantization and fine structure

Bohr's circles are a special case: the general Coulomb bound orbit is an ellipse, and in three dimensions it can lie in any plane. Sommerfeld (1916) quantized the full problem. Write the electron's motion around a nucleus of charge $Ze$ in spherical coordinates, with $k_C \equiv Ze^2/4\pi\varepsilon_0$. The problem separates, each coordinate is periodic, and each gets its own Wilson–Sommerfeld condition. The three action integrals are

$$J_\varphi = \oint p_\varphi\,d\varphi = 2\pi\lvert L_z\rvert = n_\varphi h, \qquad J_\theta = \oint p_\theta\,d\theta = 2\pi\big(L - \lvert L_z\rvert\big) = n_\theta h, \qquad J_r = \oint p_r\,dr = n_r h ,$$

with $L$ the magnitude of the angular momentum and $n_\varphi, n_\theta, n_r = 0, 1, 2, \dots$ The first is immediate, since $p_\varphi = L_z$ is constant. The other two are standard integrals of Kepler-problem action-angle theory ([Gold] §10.8); the radial one evaluates to $J_r = -(J_\theta + J_\varphi) + \pi k_C\sqrt{2m/\lvert E\rvert}$. Adding the three conditions,

$$J_r + J_\theta + J_\varphi = \pi k_C\sqrt{\frac{2m}{\lvert E\rvert}} = nh, \qquad n \equiv n_r + n_\theta + n_\varphi \quad\Longrightarrow\quad E_n = -\frac{mk_C^2}{2\hbar^2n^2} = -\frac{13.6\ \mathrm{eV}\;Z^2}{n^2}.$$

The Bohr energies again, now for every ellipse: the energy depends only on the total action, so all orbits with the same $n$, of any shape or tilt, are degenerate. The other quantum numbers fix the geometry instead:

• The azimuthal number $k \equiv n_\theta + n_\varphi$ sets the angular momentum, $L = k\hbar$, and the shape: the ellipse has semi-minor to semi-major axis ratio $b/a = k/n$. Sommerfeld allowed $k = 1, \dots, n$; $k = 0$ was excluded because that "ellipse" is a line through the nucleus. $k = n$ is Bohr's circle.
• The magnetic number $m = \pm n_\varphi$, with $\lvert m\rvert \le k$, sets $L_z = m\hbar$. So the orbital plane can only take discrete tilts relative to the $z$-axis, $\cos\alpha = L_z/L = m/k$. This is space quantization, and it is what Stern and Gerlach set out to test in 1922 (P.6.3).

These are the ancestors of the quantum numbers $n$, $\ell$, $m$ of P.6.2, but the bookkeeping is wrong. Counting every $\lvert m\rvert \le k$ for $k = 1, \dots, n$ gives $\sum_{k=1}^{n}(2k + 1) = n(n+2)$ states per level, where quantum mechanics has $\ell = 0, \dots, n-1$ and $n^2$ states. For $n = 1$ the old theory has three states with $L = \hbar$; the truth is a single state with $L = 0$. History added an irony: for silver's ground state Bohr excluded $m = 0$, so the old theory predicted two Stern–Gerlach beams, and two were seen. The count was right for the wrong reason, as P.6.3 explains.

Fine structure. In hydrogen the electron moves at $v/c \sim \alpha/n$ (with $\alpha = e^2/4\pi\varepsilon_0\hbar c \approx 1/137$, the fine-structure constant), so special relativity matters at the level of $(v/c)^2 \sim 10^{-5}$. Eccentric orbits dive closer to the nucleus and move faster there, so their relativistic correction differs from that of round ones, and the degeneracy in $k$ splits. Sommerfeld's result, expanded to leading order, is

$$E_{n,k} \approx -\frac{mc^2(Z\alpha)^2}{2n^2}\left[1 + \frac{(Z\alpha)^2}{n^2}\left(\frac{n}{k} - \frac34\right)\right].$$

Each Bohr level splits into $n$ sublevels, spread by a relative amount of order $(Z\alpha)^2$; this is the fine structure of the spectral lines, and the reason for $\alpha$'s name. Uncannily, Dirac's relativistic quantum theory (1928) gives exactly this formula with $k$ replaced by $j + \tfrac12$, where $j$ is the total angular momentum including spin. Sommerfeld had no spin, wrong orbits and a wrong count of states, and still got the right energies: a famous coincidence of compensating errors. Worked Example 3 puts numbers on it.

The honest failure list

Push the old quantum theory beyond hydrogen-like periodicity and it breaks:

• Helium and every multi-electron atom. The classical three-body problem has no periodic orbits to quantize; every attempted helium calculation failed badly. The old theory could not do chemistry's second element.
• Intensities and selection rules. It predicts where lines are, but is silent on how bright — no transition dynamics, only allowed states and ad-hoc jumps.
• Aperiodic systems. $\oint p\,dq$ requires a closed orbit. Scattering, ionization, anything unbound: no periodicity, no rule, no prediction.
• Half-integer angular momenta. The anomalous Zeeman effect and the two-spot result of the Stern–Gerlach experiment point to angular momenta of $\hbar/2$ — impossible for any $\oint p\,dq = nh$ rule. This is spin, the qubit's physical prototype, told in P.6.3 Magnetic Moments, Stern–Gerlach & Spin.
• No self-consistent dynamics. The deepest flaw: quantization rules are bolted onto classical orbits that, by the theory's own first postulate, the electron does not classically traverse (it doesn't radiate!). No equation of motion, no explanation of stationarity, no story for what happens during a jump. A scaffold, not a building.

The way out is a genuinely new mechanics in which waves are fundamental: light quanta first (P.3.1 Light as Particles), then matter waves and the equation that governs them (P.4.1 The Schrödinger Equation & the Born Rule).

Two ways out: matrix mechanics and wave mechanics

The new mechanics arrived twice, within a year, from opposite directions.

Matrix mechanics (1925). Heisenberg drew the lesson from the failure list: stop quantizing orbits nobody can observe, and build the theory from what spectroscopy measures, which is line frequencies $\nu_{nm} = (E_n - E_m)/h$ and line intensities. Classically, a periodic coordinate is a Fourier series $x(t) = \sum_k X_k e^{ik\omega t}$, one amplitude per harmonic. Heisenberg replaced it by an array of amplitudes $X_{nm}e^{2\pi i\nu_{nm}t}$, one per pair of states, that is, per spectral line. To multiply two such quantities consistently with the Ritz combination principle ($\nu_{nk} + \nu_{km} = \nu_{nm}$, P.2.2), he needed the rule $(XY)_{nm} = \sum_k X_{nk}Y_{km}$. Born recognized it as matrix multiplication, which does not commute. Born and Jordan then showed that the quantum condition takes the form

$$XP - PX = i\hbar\,I \qquad (I \text{ the identity matrix}),$$

and with Heisenberg built the full theory. Dirac independently traced the commutator back to the Poisson bracket (P.1.4), and by early 1926 Pauli had derived the hydrogen spectrum from matrices alone. No orbits, no pictures: only observables and their algebra.

Wave mechanics (1926). Schrödinger started from de Broglie's matter waves (P.3.2) and found a wave equation whose stationary solutions exist only at discrete energies, so quantization became an eigenvalue problem, like the modes of a string (P.4.1, P.6.2).

One theory. Within weeks Schrödinger showed that the integrals $X_{nm} = \int\psi_n^*\,x\,\psi_m\,dx$ built from his eigenfunctions satisfy every rule of Heisenberg's matrices. In modern language, Heisenberg's matrices are operators written in the basis of energy eigenstates, and Schrödinger's wavefunctions are state vectors written in the basis of positions. Same theory, different basis, a change of basis in the sense of 0.1.4. Dirac's and Jordan's transformation theory, and then von Neumann's Hilbert-space formulation, made this the standard framework, and it is where Term 1 begins. One observable in both dresses, the $z$-component of angular momentum:

$$\text{matrix mechanics: } L_z = X P_y - Y P_x, \qquad \text{wave mechanics: } \hat L_z = -i\hbar\left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right).$$

The ladder-operator treatment of the oscillator in P.5.3 is matrix mechanics in its native form.

What survives

Correspondence outlives the old theory. In modern QM it reappears as the classical limit: the Ehrenfest theorem shows that $\langle x\rangle$ and $\langle p\rangle$ obey (nearly) Newtonian equations (P.4.3 Wave Packets, Uncertainty & the Momentum Operator), and expectation-value dynamics is developed in the main program in 1.3.2 Expectation & Uncertainty. Wilson–Sommerfeld survives too, refined into the WKB approximation, where the half-integer correction $(n+\tfrac12)h$ emerges honestly.

Caution. Correspondence does not say "quantum systems become classical at large $n$." A superposition of two large-$n$ states — a Rydberg electron in $(\lvert n{=}100\rangle + \lvert n{=}101\rangle)/\sqrt2$, or a cat state of a big oscillator — is as nonclassical as physics gets, however large $n$ is. The principle only asserts that certain predictions (transition frequencies, expectation values for well-localized packets) approach classical values. Classicality is about which states a system occupies and how superpositions decohere, not about the size of $n$.

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Worked Examples

Example 1 — Correspondence at $n = 100$, with numbers

A hydrogen atom decays from $n = 101$ to $n = 100$.

Quantum: $\nu_{trans} = \frac{E_R}{h}\frac{2n+1}{n^2(n+1)^2} = 3.29\times10^{15}\times\frac{201}{10^4\times1.0201\times10^4} = 6.48\times10^{9}$ Hz.

Classical: $\nu_{orb} = \frac{2E_R}{hn^3} = \frac{6.58\times10^{15}}{10^6} = 6.58\times10^{9}$ Hz.

Ratio $0.985$ — agreement to 1.5%, which is $\frac{3}{2n} = 1.5\%$ on the nose. Both are ~6.5 GHz: microwave photons from an atom the size of a bacterium ($r_{100} = 10^4a_0 \approx 0.53\ \mu$m). Radio astronomers detect exactly such "radio recombination lines" from interstellar hydrogen.

Example 2 — Why your pendulum looks continuous

A 100 g pendulum swings at $\nu = 1$ Hz with $E = 0.01$ J (a few centimeters of amplitude). Wilson–Sommerfeld gives $E = nh\nu$, so

$$n = \frac{E}{h\nu} = \frac{0.01}{6.626\times10^{-34}\times1} = 1.5\times10^{31},$$

and the fractional level spacing is $h\nu/E \approx 7\times10^{-32}$. No conceivable measurement resolves rung $n$ from rung $n+1$: the ladder is exact and perfectly invisible. This is correspondence in reverse — why nobody noticed quantization in three centuries of classical mechanics.

Example 3 — The fine structure of hydrogen's $n = 2$ level

For hydrogen ($Z = 1$) the fine-structure term in Sommerfeld's formula is $-\frac{mc^2\alpha^4}{2n^4}\big(\frac{n}{k} - \frac34\big)$, with $mc^2\alpha^4 = (0.511\ \mathrm{MeV})(1/137.04)^4 = 1.449\times10^{-3}$ eV. For $n = 2$ the prefactor is $1.449\times10^{-3}/32 = 4.53\times10^{-5}$ eV, and

$$k = 1:\ \ -4.53\times10^{-5}\times\tfrac54 = -5.66\times10^{-5}\ \mathrm{eV}, \qquad k = 2:\ \ -4.53\times10^{-5}\times\tfrac14 = -1.13\times10^{-5}\ \mathrm{eV}.$$

The level splits by $\Delta E = 4.53\times10^{-5}$ eV, a frequency $\Delta E/h = 4.53\times10^{-5}/4.136\times10^{-15} = 1.09\times10^{10}$ Hz, i.e. 10.9 GHz or $0.365\ \mathrm{cm^{-1}}$: about four parts in a million of the 10.2 eV Lyman-α energy, which is why it took interferometers to see. The measured splitting of the $n = 2$ level, between the states modern notation calls $2p_{1/2}$ and $2p_{3/2}$, is 10.97 GHz. Dirac's theory gives the same 10.9 GHz, with $j + \tfrac12 = 1, 2$ in place of $k = 1, 2$. The one thing neither theory predicts is a further 1.06 GHz split between $2s_{1/2}$ and $2p_{1/2}$, measured by Lamb and Retherford in 1947: the Lamb shift, whose explanation launched quantum electrodynamics.

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Hands-on (Python)

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad

# --- 1. Correspondence check: nu_trans / nu_orb -> 1 ----------------------
n = np.arange(1, 201)
ratio = n * (2 * n + 1) / (2.0 * (n + 1) ** 2)     # exact ratio from Theory

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.5))
ax1.plot(n, ratio, ".", ms=3); ax1.axhline(1, color="gray", ls="--")
ax1.set_xlabel("n"); ax1.set_ylabel(r"$\nu_{trans}/\nu_{orb}$")
ax2.loglog(n, 1 - ratio, ".", ms=3, label="1 - ratio")
ax2.loglog(n, 3 / (2 * n), "--", label=r"$3/2n$"); ax2.set_xlabel("n"); ax2.legend()
plt.tight_layout(); plt.show()
print(ratio[[0, 9, 99, 199]])
# Expected: [0.375 0.868 0.985 0.992] -> deviation falls off exactly as 3/2n.

# --- 2. Wilson-Sommerfeld numerically: is the loop integral n*h? ----------
h = 6.626e-34                                      # J s
m = 9.109e-31                                      # kg (an electron)

# (a) Harmonic oscillator, nu = 1e15 Hz, at old-theory energies E_n = n h nu
nu = 1e15
omega = 2 * np.pi * nu
for n in [1, 2, 5, 10]:
    E = n * h * nu
    A = np.sqrt(2 * E / (m * omega**2))            # turning point
    p = lambda x: np.sqrt(2 * m * (E - 0.5 * m * omega**2 * x**2))
    upper, _ = quad(p, -A, A)                      # upper half of the ellipse
    print(f"oscillator n = {n:2d}:  loop/h = {2 * upper / h:.6f}")
# Expected: 1.000000, 2.000000, 5.000000, 10.000000  ->  area = n h exactly.

# (b) Particle in a box, L = 1 nm, at E_n = n^2 h^2 / 8 m L^2
L = 1e-9
for n in [1, 2, 5, 10]:
    E = n**2 * h**2 / (8 * m * L**2)
    print(f"box        n = {n:2d}:  loop/h = {2 * L * np.sqrt(2*m*E) / h:.6f}")
# Expected: again exactly n. Phase space is rationed in cells of area h.

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Exercises

E1 (easy). Treat Earth's orbit (mass $5.97\times10^{24}$ kg, speed $2.98\times10^{4}\ \mathrm{m\,s^{-1}}$, radius $1.50\times10^{11}$ m) as a Bohr orbit. What is $n$, and what does the answer say about detecting quantization of planetary motion?

Solution

$L = mvr = 2.66\times10^{40}\ \mathrm{J\,s}$, so $n = L/\hbar = 2.66\times10^{40}/1.055\times10^{-34} = 2.5\times10^{74}$. Adjacent orbits differ fractionally by $\sim1/n\sim10^{-74}$ — no conceivable astronomy detects the discreteness. Correspondence: at such $n$, classical mechanics is exact for all practical purposes.

E2 (easy). Show algebraically that $\frac{\nu_{trans}}{\nu_{orb}} = \frac{n(2n+1)}{2(n+1)^2} = 1 - \frac{3}{2n} + O(1/n^2)$.

Solution

$\frac{n(2n+1)}{2(n+1)^2} = \frac{2n^2+n}{2n^2+4n+2} = 1 - \frac{3n+2}{2n^2+4n+2}$, and $\frac{3n+2}{2n^2+4n+2} = \frac{3}{2n}\cdot\frac{1+2/3n}{1+2/n+1/n^2} = \frac{3}{2n} + O(1/n^2)$. The deviation is 15% at $n=10$, 0.15% at $n=1000$.

E3 (medium). Quantize a ball of mass $m$ bouncing elastically on a floor under gravity ($V = mgz$, $z\ge0$), showing $E_n = \left(\frac{9n^2h^2mg^2}{32}\right)^{1/3}$. Evaluate $E_1$ for a neutron ($m = 1.675\times10^{-27}$ kg).

Solution

Turning point $z_{max} = E/mg$; the loop doubles the upward path:

$$\oint p\,dz = 2\int_0^{E/mg}\!\!\sqrt{2m(E-mgz)}\,dz = 2\sqrt{2m}\cdot\frac{2E^{3/2}}{3mg} = \frac{4\sqrt{2m}}{3mg}E^{3/2} = nh .$$

So $E^{3/2} = \frac{3nhmg}{4\sqrt{2m}}$, and squaring inside the $2/3$ power, $E_n = \left(\frac{9n^2h^2m^2g^2}{32m}\right)^{1/3} = \left(\frac{9n^2h^2mg^2}{32}\right)^{1/3}$. For a neutron, $E_1 = 2.7\times10^{-31}\ \mathrm J \approx 1.7$ peV. The true quantum value (Airy functions) is 1.41 peV — and it has been measured: ultracold neutrons bouncing on a mirror (the qBounce experiments) exhibit these gravitational quantum states directly.

E4 (medium). Show $(v_n/c)^2 = \alpha^2/n^2$ and hence that fine-structure splittings are of order $\alpha^2|E_n| \sim 10^{-4}$ eV. Why did the old theory need elliptical orbits to see a line splitting rather than a mere shift?

Solution

From P.2.2 E3, $v_n = \alpha c/n$, so $(v_n/c)^2 = \alpha^2/n^2 = 5.3\times10^{-5}/n^2$. Relativistic corrections are of relative order $(v/c)^2$: $\Delta E \sim \alpha^2\times13.6\ \mathrm{eV} \approx 7\times10^{-4}$ eV at $n=1$ — the observed fine-structure scale. A splitting needs different shifts for states of the same $n$: only Sommerfeld's ellipses give orbits of equal $n$ but different eccentricity ($n_\varphi$) different relativistic corrections, splitting one Bohr level — and one line — into a multiplet.

E5 (hard). A rigid rotor (moment of inertia $I$) rotates freely in a plane: $E = L^2/2I$. (a) Quantize it with Wilson–Sommerfeld. (b) Compare $\nu_{trans}(n\to n-1)$ with the classical rotation frequency $\nu_{rot}$ in state $n$. (c) Confirm the general theorem $\nu_{trans} \approx dE/dJ$ for this system.

Solution

(a) $\oint p_\varphi\,d\varphi = 2\pi L = nh \Rightarrow L = n\hbar$, so $E_n = n^2\hbar^2/2I$. (b) $\nu_{trans} = \frac{E_n-E_{n-1}}{h} = \frac{(2n-1)\hbar}{4\pi I}$, while classically $\omega = L/I$ gives $\nu_{rot} = \frac{n\hbar}{2\pi I}$; the ratio $\frac{2n-1}{2n}\to1$. ✓ (c) Here $J = 2\pi L$, so $E = J^2/8\pi^2I$ and $\frac{dE}{dJ} = \frac{J}{4\pi^2I} = \frac{L}{2\pi I} = \nu_{rot}$, while $\nu_{trans} = \frac{\Delta E}{h} \approx \frac1h\frac{dE}{dn} = \frac{dE}{dJ}$ since $J = nh$. Correspondence is automatic for any Wilson–Sommerfeld system — the discrete shadow of the action-angle relation $\nu_{cl} = dE/dJ$ from P.1.4.

E6 (medium). Hydrogen's $n = 3$ level in the old and new theories. (a) List Sommerfeld's allowed $(k, m)$, count the states, and compare with the Schrödinger count, with and without spin. (b) Use the fine-structure formula to find the splittings between the $k = 1, 2, 3$ sublevels in GHz. (c) Dirac's formula is the same with $k \to j + \tfrac12$. Which states $n\ell_j$ share each sublevel?

Solution

(a) $k = 1$: $m = -1, 0, 1$; $k = 2$: $m = -2, \dots, 2$; $k = 3$: $m = -3, \dots, 3$. That is $3 + 5 + 7 = 15 = n(n+2)$ states. Schrödinger: $\ell = 0, 1, 2$ with $1 + 3 + 5 = 9 = n^2$ states, or 18 with the two spin states. The old count matches neither. (b) The prefactor is $mc^2\alpha^4/2n^4 = 1.449\times10^{-3}/162 = 8.94\times10^{-6}$ eV, and $n/k - \tfrac34 = 2.25,\ 0.75,\ 0.25$ for $k = 1, 2, 3$. The splittings are $8.94\times10^{-6}\times1.5 = 1.34\times10^{-5}$ eV between $k = 1$ and 2, i.e. 3.24 GHz, and $8.94\times10^{-6}\times0.5 = 4.47\times10^{-6}$ eV between $k = 2$ and 3, i.e. 1.08 GHz. (c) $k = 1 \leftrightarrow j = \tfrac12$: $3s_{1/2}$ and $3p_{1/2}$. $k = 2 \leftrightarrow j = \tfrac32$: $3p_{3/2}$ and $3d_{3/2}$. $k = 3 \leftrightarrow j = \tfrac52$: $3d_{5/2}$. The measured $3p_{1/2}$–$3p_{3/2}$ and $3d_{3/2}$–$3d_{5/2}$ splittings are 3.25 and 1.08 GHz. Sommerfeld's numbers are right; his labels are not, since each of his sublevels is really a pair of states with different $\ell$.

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Checkpoint

1. State the correspondence principle. What exactly converges, and in what limit?
2. Both $\nu_{trans}$ and $\nu_{orb}$ in hydrogen scale as $1/n^3$. Why is the matching prefactor the nontrivial content of the check?
3. Apply $\oint p\,dq = nh$ to the harmonic oscillator from memory. What does the true quantum spectrum add, and where in this program is it derived?
4. Give three distinct failures of the old quantum theory and, for each, the missing ingredient modern QM supplies.
5. Explain the error in: "By correspondence, a Rydberg atom with $n = 500$ behaves classically."
6. In Sommerfeld's theory, what does each of $n$, $k$ and $m$ determine, and what does the theory get right and wrong about hydrogen's $n = 2$ level?
7. In what precise sense are matrix mechanics and wave mechanics the same theory?

Answers

1. As $n\to\infty$ (actions $\gg h$), quantum predictions — emitted frequencies, expectation values, relative level spacings — converge to classical predictions for the same system. It constrains the theory's limit, not the nature of its states.
2. Two $1/n^3$ power laws trivially have a constant ratio; correspondence requires that ratio to be exactly 1, i.e. the prefactor $2E_R/h$ appearing on both sides. That requirement fixes the constant in the quantization rule — Bohr used it to justify $L = n\hbar$.
3. The phase orbit is an ellipse of area $2\pi E/\omega = E/\nu$; setting $E/\nu = nh$ gives $E_n = nh\nu$. True QM adds the zero-point energy, $E_n = (n+\tfrac12)h\nu$, derived in P.5.3.
4. Helium/multi-electron atoms — needs a wave equation valid beyond periodic orbits; line intensities — needs transition amplitudes/matrix elements; half-integer angular momenta (anomalous Zeeman, Stern–Gerlach) — needs spin, an internal degree of freedom with no orbit. (Also acceptable: aperiodic/scattering systems; no self-consistent equations of motion.)
5. Large $n$ does not make the state classical — superpositions of large-$n$ states interfere and are maximally nonclassical. Only specific predictions (transition frequencies, wave-packet expectation values) approach classical values; classical behavior depends on which state the system is in and on decoherence, not on $n$.
6. $n$ (the total action) fixes the energy; $k$ fixes $L = k\hbar$ and the ellipse's shape ($b/a = k/n$); $m$ fixes $L_z = m\hbar$, i.e. the orbit's tilt, $\cos\alpha = m/k$. For $n = 2$ it gets the 10.9 GHz fine-structure splitting right (it coincides with Dirac's formula), but the states wrong: $k = 1, 2$ with $L = \hbar, 2\hbar$ and 8 states, against $\ell = 0, 1$ and 4 orbital states.
7. They describe the same states and operators in different bases. Heisenberg's matrices are the operators' matrix elements between energy eigenstates, $X_{nm} = \int\psi_n^*x\,\psi_m\,dx$, which Schrödinger showed obey Heisenberg's algebra; Schrödinger's wavefunctions are the states written in the position basis.

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Further Reading

• [ER] Eisberg & Resnick, §4.11–4.13 — Wilson–Sommerfeld rules, the correspondence principle, and the critique of the old quantum theory.
• [Gold] Goldstein, Poole & Safko, Ch. 10 — action-angle variables; the classical machinery behind $\oint p\,dq$ and $\nu = dE/dJ$.
• [Gri] Griffiths & Schroeter, Ch. 9 (WKB) — Wilson–Sommerfeld reborn with its honest $\tfrac12$ correction.
• [Gri] Griffiths & Schroeter, §7.3 — the fine structure of hydrogen done properly, with relativity and spin–orbit coupling as perturbations.
• [Gold] Goldstein, Poole & Safko, §10.8 — the Kepler problem in action-angle variables: the three action integrals of Sommerfeld's quantization.

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