Bohr–Sommerfeld Quantization

Bohr–Sommerfeld quantization is the WKB rule for bound-state energies in a one-dimensional smooth potential well. It says that the classical action over one closed orbit is quantized in units of $2\pi\hbar$, with a turning-point correction. For a complete application with an analytic action and a converged spectral benchmark, see WKB Bound States in a Smooth Potential.

Required background. Turning-point connection formulas supply the two boundary phases.

For a bound orbit between two isolated simple smooth turning points $x_1$ and $x_2$,

$$p(x) = \sqrt{2m\left(E-V(x)\right)}, \qquad V(x_1)=V(x_2)=E,$$

the WKB quantization condition is

$$\int_{x_1}^{x_2}p(x)\,dx = \pi\hbar \left( n+\frac12 \right), \qquad n=0,1,2,\ldots.$$

Equivalently,

$$\oint p\,dx = 2\pi\hbar \left( n+\frac12 \right).$$

The closed integral is over a full classical period in phase space.

Bound-State Matching

Inside the well, the WKB wavefunction is oscillatory:

$$\psi(x) \approx \frac{1}{\sqrt{p(x)}} \left[ C_+e^{(i/\hbar)\int^x p(x')\,dx'} + C_-e^{-(i/\hbar)\int^x p(x')\,dx'} \right].$$

Outside the well, normalizability selects decaying forbidden-region tails. Each tail connects through a turning point to a sine-like allowed-region wave with a $\pi/4$ phase shift. Compatibility of the left and right matched forms requires the phase accumulated across the well to satisfy

$$\frac{1}{\hbar} \int_{x_1}^{x_2}p(x)\,dx + \frac{\pi}{2} = (n+1)\pi.$$

This is equivalent to

$$\int_{x_1}^{x_2}p(x)\,dx = \pi\hbar \left( n+\frac12 \right).$$

Turning-Point Correction

The $1/2$ shift is not put in by hand. It comes from the two smooth turning points. Each contributes a phase of $\pi/4$, so together they shift the standing-wave condition by $\pi/2$.

This is why the WKB condition differs from the old rule

$$\oint p\,dx=nh.$$

For smooth one-dimensional wells, the modern WKB rule includes the Maslov correction associated with the turning points.

Harmonic Oscillator Check

For

$$V(x) = \frac12m\omega^2x^2,$$

the turning points are

$$x_\pm = \pm \sqrt{\frac{2E}{m\omega^2}}.$$

The closed classical action is

$$\oint p\,dx = \frac{2\pi E}{\omega}.$$

Using the WKB rule gives

$$\frac{2\pi E}{\omega} = 2\pi\hbar \left( n+\frac12 \right),$$

so

$$E_n = \hbar\omega \left( n+\frac12 \right).$$

For the harmonic oscillator, WKB gives the exact energy spectrum. This exactness is special; most potentials receive higher-order corrections.

Large-Quantum-Number Spectra

WKB is usually most accurate for highly excited states because the action is large compared with $\hbar$. The rule estimates the zero-based level index associated with energy $E$ as

$$n(E) \approx \frac{1}{\pi\hbar} \int_{x_1(E)}^{x_2(E)}p(x;E)\,dx - \frac12.$$

The number of levels below $E$ is a staircase, not this index. Away from approximate eigenvalues and assuming a single sequence indexed from zero, leading WKB predicts

$$N_{<}(E)\approx\left\lfloor\frac{1}{\pi\hbar} \int_{x_1}^{x_2}p\,dx+\frac12\right\rfloor.$$

At a threshold, distinguish strict from non-strict counting; WKB errors can move a step. A smooth leading density discards this order-one staircase ambiguity.

The spacing of high-lying levels is controlled by the classical period. Since

$$\frac{d}{dE} \oint p\,dx = \tau(E),$$

where $\tau(E)$ is the classical period, the energy spacing is approximately

$$\Delta E \approx \frac{2\pi\hbar}{\tau(E)}.$$

This connects the density of quantum levels to classical motion.

For the undergraduate finite-well counting version of this idea, see Bound-State Counting.

Boundary Caveats

The formula above assumes two smooth turning points. It should not be blindly applied to hard-wall boxes, singular potentials, or discontinuous potentials. The infinite square well, for example, is better understood through exact boundary conditions; its walls are not smooth WKB turning points.

When endpoints are singular or when motion occurs on a multidimensional invariant torus, the phase correction must be modified. Those generalizations keep the same action-quantization spirit but require more geometry than the one-dimensional smooth-well rule.

Exercises

1. A particle moves in $V(x)=Fx$ for $x>0$ and obeys $\psi(0)=0$, with $F>0$. Find its leading quantization rule and energies.

Solution

There is one smooth turning point at $E/F$ and one hard wall. The decaying right tail gives $\sin[\hbar^{-1}\int_x^{E/F}p\,dx+\pi/4]$. At the wall it must vanish, so

$$\int_0^{E/F}p\,dx=\pi\hbar(n+3/4),\qquad n=0,1,\ldots.$$

The integral is $2\sqrt{2m}E^{3/2}/(3F)$, hence

$$E_n^{\mathrm{WKB}}= \left[\frac{3\pi\hbar F}{2\sqrt{2m}}(n+3/4)\right]^{2/3}.$$

This is the large-index asymptotic spectrum of the negative zeros of $\operatorname{Ai}$, not the two-smooth-turning-point $n+1/2$ rule.

2. Derive $dJ/dE=\tau$ for $J(E)=2\int_{x_1(E)}^{x_2(E)}p\,dx$.

Solution

The endpoint terms vanish because $p(x_i)=0$. Since $\partial_Ep=m/p=1/v$, differentiation gives $dJ/dE=2\int_{x_1}^{x_2}dx/v=\tau$. For simple turning points the integrable endpoint singularities cause no divergence. Near a separatrix the period may diverge and the usual smooth level-spacing expansion needs extra care.

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.