WKB Quantization

Purpose

WKB quantization estimates bound-state energies from the classical action. For a one-dimensional well with two isolated smooth turning points $x_1(E)<x_2(E)$, define

$$p(x;E) = \sqrt{2m\bigl[E-V(x)\bigr]}$$

in the allowed interval. The leading quantization condition is

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

The $1/2$ is the phase contribution of two simple turning points. It must be changed for hard walls, singular endpoints, higher-order turning points, or other boundary conditions.

The matching derivation belongs to Bohr–Sommerfeld Quantization and Turning Points and Connection Formulas. This card is organized around choosing and applying the correct action rule.

At a glance

Let

$$I(E) = \int_{x_1(E)}^{x_2(E)} p(x;E)\,dx$$

denote the one-way action across the allowed interval, and let

$$J(E)=\oint p\,dx=2I(E)$$

denote the action around the full one-dimensional classical orbit.

Situation	Leading condition
Two smooth turning points	$I(E_n)=\pi\hbar(n+1/2)$
Closed orbit with Maslov index $\mu$	$J(E_n)=2\pi\hbar(n+\mu/4)$
One smooth endpoint, one Dirichlet wall	$I(E_n)=\pi\hbar(n+3/4)$
One smooth endpoint, one Neumann wall	$I(E_n)=\pi\hbar(n+1/4)$
Two Dirichlet walls	$I(E_n)=\pi\hbar(n+1)$
Two Neumann walls	$I(E_n)=\pi\hbar n$
Local validity away from endpoints	$\hbar\lvert p'\rvert/p^2\ll1$
High-level spacing	$\Delta E\simeq2\pi\hbar/T_{\mathrm{cl}}(E)$

Here $n=0,1,2,\ldots$ in every row. The hard-wall rows assume ideal energy-independent Dirichlet or Neumann conditions. Robin conditions and finite discontinuities generally produce an energy-dependent reflection phase instead of a universal fractional shift.

Action convention and factor of two

In one dimension, the right-moving branch contributes

$$\int_{x_1}^{x_2}p\,dx=I,$$

and the left-moving return branch contributes another $I$: both $p$ and $dx$ change sign, so $p\,dx$ remains positive along the oriented orbit. Therefore

$$\oint p\,dx=2I.$$

The equivalent smooth-well formulas are

$$I(E_n) = \pi\hbar \left(n+\frac12\right)$$

and

$$J(E_n) = 2\pi\hbar \left(n+\frac12\right).$$

Mixing the half-orbit integral with the closed-orbit right-hand side creates a factor-of-two error. State which action convention is being used before doing the integral.

Where the turning-point shift comes from

Suppose both exterior regions are forbidden and normalizability selects decaying tails. Away from the left simple turning point, the corresponding allowed-region standing wave is

$$\psi_L(x) \propto \frac{1}{\sqrt{p(x)}} \sin\left[ \frac{1}{\hbar} \int_{x_1}^{x}p(x')\,dx' + \frac{\pi}{4} \right].$$

Matching the decaying tail at the right turning point gives

$$\psi_R(x) \propto \frac{1}{\sqrt{p(x)}} \sin\left[ \frac{1}{\hbar} \int_x^{x_2}p(x')\,dx' + \frac{\pi}{4} \right].$$

The two standing-wave forms are compatible only when

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

The naive WKB amplitudes diverge where $p=0$, but the exact local wavefunction does not. Linearizing a simple turning point reduces the local equation to the Airy equation; Airy matching supplies each $\pi/4$ phase. The leading WKB forms should not be evaluated inside that turning-point layer.

Endpoint-phase form

For a single allowed interval, a useful bookkeeping form is

$$I(E_n) = \pi\hbar \bigl(n+\alpha_L+\alpha_R\bigr).$$

With the state label beginning at $n=0$, the standard endpoint contributions are

Endpoint	Contribution
Simple smooth turning point with a decaying exterior	$\alpha=1/4$
Dirichlet hard wall	$\alpha=1/2$
Neumann hard wall	$\alpha=0$

This table packages familiar one-dimensional cases; it is not a replacement for boundary matching. At a finite potential step, the reflection coefficient has an energy-dependent phase. At a singular endpoint, the admissible local solution determines the phase. If the endpoint cannot be classified by one of the rows, derive its reflection phase rather than guessing an index.

Maslov and EBK form

For a smooth one-dimensional closed orbit, the quantization rule may be written

$$\frac{1}{\hbar}\oint p\,dx - \frac{\pi}{2}\mu = 2\pi n,$$

or

$$\oint p\,dx = 2\pi\hbar \left(n+\frac{\mu}{4}\right).$$

Two ordinary turning points give $\mu=2$. The sign convention for a Maslov index varies across the literature, so the phase equation above should accompany any quoted value of $\mu$.

For an integrable system with independent cycles $\gamma_i$, EBK quantization generalizes the rule to

$$J_i = \oint_{\gamma_i}p\cdot dq = 2\pi\hbar \left(n_i+\frac{\mu_i}{4}\right).$$

The action convention here does not include a factor of $1/(2\pi)$. Some texts define the action variable as $J_i/(2\pi)$, so translate the definition before comparing formulas. The geometric meaning of the index is developed at Maslov Index and EBK Quantization.

Solving for an energy

For two smooth turning points:

1. Solve $V(x)=E$ for the relevant real roots $x_1(E)$ and $x_2(E)$.
2. Verify that the interval is classically allowed and each endpoint is simple: $V'(x_i)\ne0$.
3. Evaluate $I(E)$ with $p$ real and nonnegative between the roots.
4. Solve $I(E_n)=\pi\hbar(n+1/2)$ for each desired integer $n$.
5. Check the local WKB parameter away from the endpoint layers and compare low-lying levels with an independent method.

The square-root endpoint behavior is integrable. For numerical quadrature, an endpoint-smoothing substitution is often more stable than sampling the raw integrand at the roots. A root solver should track one connected classical well rather than jumping between distinct allowed intervals.

For a fixed connected well,

$$\frac{dI}{dE} = \int_{x_1}^{x_2} \frac{m}{p(x;E)}\,dx = \frac{T_{\mathrm{cl}}(E)}{2}.$$

Boundary terms vanish because $p(x_i;E)=0$. Hence $I(E)$ increases with $E$, and adjacent high-lying levels satisfy

$$\Delta E \simeq \frac{\pi\hbar}{dI/dE} = \frac{2\pi\hbar}{T_{\mathrm{cl}}(E)}.$$

Equivalently, the semiclassical density of states in that well is

$$\frac{dn}{dE} \simeq \frac{T_{\mathrm{cl}}(E)}{2\pi\hbar}.$$

These relations are valuable checks on both the action normalization and the trend of the calculated spectrum.

Check: harmonic oscillator

For

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

the classical closed action is

$$J(E)=\frac{2\pi E}{\omega},$$

so

$$I(E)=\frac{\pi E}{\omega}.$$

The two-soft-turning-point rule gives

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

and therefore

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

Leading WKB happens to reproduce the exact oscillator energies. This special exactness checks the phase and factor of two; it does not make leading WKB exact for a generic smooth potential.

Check: infinite square well

Inside an ideal well of width $L$, $p=\sqrt{2mE}$ is constant, but the endpoints are Dirichlet hard walls rather than smooth turning points. The correct endpoint rule is

$$pL=\pi\hbar(n+1), \qquad n=0,1,2,\ldots,$$

which gives

$$E_n = \frac{\pi^2\hbar^2}{2mL^2}(n+1)^2.$$

If the physical levels are instead labeled $N=1,2,\ldots$, this is the usual $E_N=\pi^2\hbar^2N^2/(2mL^2)$. Substituting $n+1/2$ would incorrectly treat the abrupt walls as Airy turning points.

Radial problems and the Langer correction

The reduced radial equation contains the singular centrifugal term

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

Applying the ordinary one-dimensional WKB rule directly at $r=0$ generally gives the wrong phase. In the standard radial semiclassical treatment, use the Langer replacement

$$\ell(\ell+1) \longrightarrow \left(\ell+\frac12\right)^2$$

inside the effective radial momentum:

$$p_r(r;E) = \sqrt{ 2m\bigl[E-V(r)\bigr] - \frac{\hbar^2(\ell+1/2)^2}{r^2} }.$$

When this momentum has two appropriate simple radial turning points,

$$\int_{r_1}^{r_2}p_r(r;E)\,dr = \pi\hbar \left(n_r+\frac12\right).$$

This formula assumes the Langer-transformed radial problem and its standard regularity condition. Other dimensions, singular potentials, and self-adjoint boundary conditions require a fresh endpoint analysis.

Validity audit

Away from turning points, a common leading-order diagnostic is

$$\epsilon_{\mathrm{WKB}}(x) = \hbar \frac{\lvert p'(x)\rvert}{p(x)^2} \ll1.$$

It measures the fractional change of the local de Broglie scale over one wavelength, up to convention-dependent factors. The condition inevitably fails as a simple turning point is approached; the calculation remains useful only because an overlapping Airy approximation repairs that local region.

Quantization is usually best when

$$J(E_n)\gg\hbar,$$

or equivalently when many oscillations fit inside the allowed interval. Low-lying states may still be accurate for special potentials, but that is not the generic asymptotic guarantee.

Leading WKB is not a variational method. Its energies are neither general upper bounds nor general lower bounds, and the neglected corrections do not have a universal sign. Higher-order WKB or independent numerical calculations are needed to estimate the error.

When the simple rule fails

Higher-order or coalescing turning points

Airy matching assumes

$$V(x_t)=E, \qquad V'(x_t)\ne0.$$

Near a barrier top, turning points can merge and the Airy neighborhoods overlap. A uniform approximation adapted to the merged structure is then required.

Multiple wells

If an energy intersects the potential in four or more turning points, each allowed well may first have its own local action condition. Tunneling through the intervening barriers couples those local states and produces avoided crossings or exponentially small splittings. Independent two-turning-point rules estimate the uncoupled level centers, not the complete split spectrum.

Discontinuous or singular endpoints

Hard walls, finite steps, inverse-square singularities, and radial origins carry boundary information not contained in the universal $\pi/4$ Airy phase. Use the exact local boundary condition or an appropriate uniform approximation.

Nonintegrable multidimensional motion

EBK quantization requires invariant tori and independent action cycles. Generic chaotic motion has no global set of such actions, so a direct component-by-component EBK rule is unavailable.

Calculation workflow

1. Draw or analyze $V(x)$ at the target energy and count the allowed intervals.
2. Classify every endpoint as soft, hard, singular, discontinuous, or coalescing.
3. Define explicitly whether the action is one-way $I$ or closed-orbit $J$.
4. Attach endpoint phases or a stated Maslov convention.
5. Evaluate the action with consistent units and solve the resulting scalar equation for $E_n$.
6. Differentiate the action or inspect $T_{\mathrm{cl}}$ to check the level spacing.
7. Audit $\epsilon_{\mathrm{WKB}}$ away from matched endpoint layers.
8. Benchmark low states, near-degeneracies, and suspected tunnel splittings independently.

Common mistakes

• Using $\oint p\,dx$ with the half-orbit right-hand side.
• Omitting the two-turning-point $1/2$ shift.
• Counting every zero of $V'(x)$ as a turning point; a turning point requires $V(x)=E$.
• Integrating $\sqrt{2m(E-V)}$ as a real momentum through a forbidden region.
• Assigning the Airy $\pi/4$ phase to a hard wall or singular endpoint.
• Applying the ordinary one-dimensional rule to a radial equation without handling the origin and Langer correction.
• Using a separate two-turning-point condition for each side of a double well and then claiming the tunnel splitting.
• Treating a numerical root of the WKB equation as an exact eigenvalue or a rigorous spectral bound.
• Assuming the harmonic oscillator’s exact leading-WKB spectrum is typical.
• Comparing action variables across sources without checking whether a factor of $2\pi$ is included in their definition.

Exercises

1. Starting from $J(E)=2\pi E/\omega$ for the harmonic oscillator, derive the exact leading-WKB spectrum and the classical-spacing formula.

Solution

Two smooth turning points give

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

so

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

The period is

$$T_{\mathrm{cl}} = \frac{dJ}{dE} = \frac{2\pi}{\omega}.$$

Therefore

$$\Delta E \simeq \frac{2\pi\hbar}{T_{\mathrm{cl}}} = \hbar\omega,$$

which is also exact for this spectrum.

2. Use the endpoint table to recover the spectrum of a particle between two Dirichlet walls separated by $L$.

Solution

Each Dirichlet wall contributes $1/2$, so

$$I(E_n)=pL=\pi\hbar(n+1).$$

Thus

$$p_n=\frac{\pi\hbar}{L}(n+1)$$

and

$$E_n = \frac{p_n^2}{2m} = \frac{\pi^2\hbar^2}{2mL^2}(n+1)^2.$$

There is no soft-turning-point $1/2$ shift.

3. Restrict a harmonic oscillator to $x\ge0$. Show that a Dirichlet condition at the origin selects the odd full-line levels, while a Neumann condition selects the even levels.

Solution

The one-way action from the wall at $0$ to the positive turning point is half the full-line half-orbit action:

$$I_+(E)=\frac{\pi E}{2\omega}.$$

For a Dirichlet wall and one soft turning point,

$$\frac{\pi E_n}{2\omega} = \pi\hbar \left(n+\frac34\right),$$

so

$$E_n = \hbar\omega \left(2n+\frac32\right).$$

These are the odd full-line levels. For a Neumann wall,

$$\frac{\pi E_n}{2\omega} = \pi\hbar \left(n+\frac14\right),$$

and therefore

$$E_n = \hbar\omega \left(2n+\frac12\right),$$

the even full-line levels.

4. For $V(x)=a\lvert x\rvert^s$ with $a>0$ and $s>0$, use scaling to find the large-$n$ power law of the WKB energies.

Solution

The turning point scales as

$$x_t\sim\left(\frac{E}{a}\right)^{1/s}.$$

The half-orbit action therefore scales as

$$I(E) \sim x_t\sqrt{mE} \sim \sqrt{m}\, a^{-1/s} E^{1/2+1/s}.$$

Quantization requires $I(E_n)\propto\hbar(n+1/2)$, so

$$E_n \propto \left(n+\frac12\right)^{2s/(s+2)}.$$

For $s=2$ this is linear oscillator spacing; for $s=4$ it gives $E_n\propto(n+1/2)^{4/3}$. The scaling does not determine the dimensionless coefficient, which comes from the full action integral.

Canonical links

• WKB Approximation
• Turning Points and Connection Formulas
• Bohr–Sommerfeld Quantization
• Maslov Index
• EBK Quantization
• WKB Bound States in a Smooth Potential

References

• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981, Secs. 46–50.
• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer, 1999, Ch. 10.
• J. L. Dunham, “The Wentzel–Brillouin–Kramers Method of Solving the Wave Equation,” Physical Review 41, 713–720 (1932), doi:10.1103/PhysRev.41.713.
• M. V. Berry and K. E. Mount, “Semiclassical Approximations in Wave Mechanics,” Reports on Progress in Physics 35, 315–397 (1972), doi:10.1088/0034-4885/35/1/306.
• N. Fröman and P. O. Fröman, JWKB Approximation: Contributions to the Theory, North-Holland, 1965.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003, Chs. 1–2.