Quantum theory Solvable models and approximation methods Theorem

Bohr–Sommerfeld quantization condition

Builds on Schrödinger operator Discrete spectrum WKB approximation

Statement

Let $V \colon \R \to \R$ be smooth, let $m > 0$ , and for $\hbar > 0$ let $P_\hbar = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x)$ on $L^2(\R)$ . Let $a < b$ be such that $\liminf_{\abs{x} \to \infty} V(x) > b$ , and, for every $E \in [a, b]$ , the set $\set{x : V(x) \le E}$ is a closed interval $[x_-(E), x_+(E)]$ with $V'(x_\pm(E)) \ne 0$ . For $E \in [a, b]$ let $A(E) = 2 \int_{x_-(E)}^{x_+(E)} \sqrt{2m (E - V(x))} \, dx$ be the area enclosed by the classical orbit $p^2/2m + V(x) = E$ in the phase plane. Then, for small $\hbar$ , the eigenvalues of $P_\hbar$ in $[a, b]$ are simple. They can be labelled $E_n(\hbar)$ , with $n$ running over some integers $n \ge 0$ , so that $A(E_n(\hbar)) = 2\pi\hbar \left( n + \tfrac12 \right) + O(\hbar^2) \quad \text{as } \hbar \to 0.$ The bound $O(\hbar^2)$ is uniform in $n$ .

Remarks

In the old quantum theory of Bohr and Sommerfeld the rule was a postulate, written $\oint p \, dq = nh$ with $h = 2\pi\hbar$ , without the term $\frac12$ . The modern form follows from the WKB approximation. The WKB solutions decaying on each side are glued across the two turning points. Each turning point contributes a phase of $\pi/2$ , the Maslov correction, and this gives the $\frac12$ .

Example: for the harmonic oscillator $V(x) = \frac12 m\omega^2 x^2$ the orbit at energy $E$ is an ellipse of area $A(E) = 2\pi E/\omega$ . The condition gives $E = \hbar\omega(n + \frac12)$ , which is the exact spectrum.

The hypothesis that $E$ is not a critical value of $V$ is needed. Near the top of a barrier or at the bottom of a degenerate well, the turning points merge and the rule must be corrected.

Going deeper

For a student

The rule is the simplest bridge between classical and quantum mechanics: it computes the discrete spectrum of a one-dimensional Schrödinger operator from the classical action alone. It also shows that the number of eigenvalues below $E$ is about $A(E)/(2\pi\hbar)$ , a one-dimensional case of Weyl's law. In several degrees of freedom it generalises, for integrable systems, to the Einstein–Brillouin–Keller conditions on invariant tori.

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Asymptotic analysis concept
Discrete spectrum concept
Ordinary differential equation concept
Schrödinger operator concept
WKB approximation definition
Bohr–Sommerfeld quantization condition statement

Tag	`6MX64VZHYF`
URI	`m2://6MX64VZHYF`
Kind	statement, theorem
Writing	W0 stub
MSC	81Q20, 35P20
Status	This copy of the site has no build data. Run `m2 status 6MX64VZHYF`.
About	Schrödinger operator
Uses	Schrödinger operator, Discrete spectrum, WKB approximation
Changed	3 November 2026