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. Describe Sommerfeld's elliptical orbits and how relativity yields fine structure (qualitatively).
5. Enumerate the failures of the old quantum theory and explain why each demanded a new framework.

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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.1.3 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.1.3):

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

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.1.3 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 and fine structure (qualitative)

The general Coulomb bound orbit is an ellipse, needing two conditions: $\oint p_\varphi\,d\varphi = n_\varphi h$ and $\oint p_r\,dr = n_rh$. Sommerfeld (1916) found the energy depends only on $n = n_r + n_\varphi$ — all ellipses of the same $n$ are degenerate with Bohr's circle — while $n_\varphi = 1,\dots,n$ sets the eccentricity: the ancestor of the quantum numbers $n$ and $\ell$ of P.6.2. Adding special relativity, eccentric orbits (fastest near the nucleus) shift slightly: the degeneracy splits by relative amounts of order $\alpha^2 \approx 5.3\times10^{-5}$ — the observed fine structure of hydrogen lines, and the origin of $\alpha$'s name. Eerily, Sommerfeld's formula agrees with the exact Dirac result — a celebrated coincidence of compensating errors (no spin, wrong orbits, right answer).

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).

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.

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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.1.3.

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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."

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 $n$.

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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.

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