Correspondence Principle

Equivalent forms

\hbar\to 0 \Rightarrow \text{classical mechanics}

Every good quantum theory must contain the old classical one hiding inside its large-n or ℏ→0 corner — a consistency demand, not an accident.

Symbol	Name	SI unit	Dimension	Range
$n$	Quantum number Large for the classical limit	1	1	1 – 40
$\Delta E$	Level spacingoutput Energy gap between adjacent levels	J	ML^2T^-2	—
$E_n$	Energy level nth energy (e.g. hydrogen −13.6/n² eV)	J	ML^2T^-2	—
$\hbar$	Reduced Planck constant h/2π, the quantum of action	J*s	ML^2T^-1	—

Where it holds

A heuristic/consistency requirement, not a precise formula. Fails to be simple when classical chaos or genuinely non-classical states (entanglement, tunneling) dominate.

Dimensional analysis

\Delta E \to [M*L^2*T^-2]

(energy)

E_{n+1}-E_n

→ energy

-

energy

Difference of energies is an energy; the ratio

\Delta E/E

is dimensionless and vanishes as n grows.

\checkmark

Discovery

Niels Bohr · 1920

Bohr articulated the principle around 1913–1920 while building his atomic model: the frequency of light emitted between high-n orbits matches the electron's classical orbital frequency. He used it as a bridge to guess quantum selection rules before a full theory existed, and named it explicitly in 1920.

Research status: stable

Real-world applications

Justifying classical mechanics for macroscopic bodies
Semiclassical (WKB) methods and quantum-classical modeling
Rydberg-atom and quantum-chaos studies

Common misconceptions

The classical limit is large action (S≫ $\hbar )$ or large n, not literally $\hbar =0$ in nature
Quantum→classical is not automatic for all states — Schrödinger-cat and entangled states resist it
It's a consistency guide, not a derivation of classical mechanics

Experimental verification

Rydberg atoms

(n\sim 100s)

behave like classical orbiting charges and radiate at the predicted classical frequencies; laser-driven Rydberg wave packets trace near-Keplerian orbits.

Derivation

For hydrogen,

E_n = -13.6\,\mathrm{eV/n}^{2}

, so the spacing

\Delta E = E_{n+1}-E_n \approx 27.2\,\mathrm{eV/n}^{3}

shrinks as

1/n^{3}

.

The emitted frequency

\Delta E/h

approaches the classical orbital frequency f_classical = (electron speed

)/(2\pi \times

orbit radius) as

n\to \infty

.

Ehrenfest's theorem provides the general version: expectation values ⟨x⟩,⟨p⟩ obey Newton's equations, so wave packets track classical trajectories when

\hbar is

negligible.

What if…

What if n is small

(n=1)

?

Levels are far apart relative to their energy — the system is deeply quantum and shows no classical resemblance.

What if you take

\hbar \to 0

formally?

The Schrödinger equation reduces (via WKB) to the classical Hamilton–Jacobi equation — classical trajectories reappear.

1

Vanishing relative spacing

Given · hydrogen, compare

n=2

and

n=100

Find · fractional level spacing

\Delta E/E

Steps

| $E_n| \propto 1/n^{2}$ , $\Delta E \propto 1/n^{3}$
$\Delta E/|E_n| \approx 2/n$
$n=2 \to 1.0$ (very quantum); $n=100 \to 0.02$ (nearly continuous)

Result ·

\Delta E/|E| \approx 2/n \to 1.0

at

n=2

but 0.02 at

n=100

bohr modelEhrenfest Theorem→wkb approximation