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 the Core-level semiclassical orientation, see Semiclassical Limit Overview. For a complete application with an analytic action and a converged spectral benchmark, see WKB Bound States in a Smooth Potential.

For a bound orbit between two 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 $\hbar$. A useful way to read the rule is

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

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.

Common Mistakes

• Omitting the $1/2$ turning-point shift for a smooth well.
• Applying the smooth-turning-point formula to hard walls without adjusting the phase.
• Using WKB for the lowest few states without checking accuracy.
• Confusing the half-orbit integral $\int_{x_1}^{x_2}p\,dx$ with the closed-orbit integral $\oint p\,dx$.
• Treating the old quantum rule as equivalent to WKB without the Maslov correction.

Exercises

1. Starting from the closed-orbit rule, derive the half-orbit form for a one-dimensional bound state.

Solution

In one dimension, the closed phase-space orbit goes from $x_1$ to $x_2$ with positive momentum and returns from $x_2$ to $x_1$ with negative momentum. The action over the closed orbit is twice the positive half-orbit integral:

$$\oint p\,dx = 2 \int_{x_1}^{x_2}p(x)\,dx.$$

Therefore

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

is equivalent to

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

2. Use the harmonic oscillator action $\oint p\,dx=2\pi E/\omega$ to recover the oscillator spectrum.

Solution

Substitute into the WKB rule:

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

Solving for $E$ gives

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

3. Why does the smooth-well WKB rule not directly explain the exact infinite-square-well spectrum?

Solution

The infinite square well has hard walls, not smooth turning points where $p(x)$ vanishes linearly. The phase correction comes from boundary conditions at abrupt endpoints rather than Airy matching near smooth turning points. The action idea remains useful, but the turning-point phase is different.

References

• 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.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.