The WKB (Semiclassical) Approximation

The WKB approximation (Wentzel–Kramers–Brillouin) is the systematic expansion of quantum mechanics in powers of $\hslash$, valid when the potential varies slowly over a de Broglie wavelength. It bridges quantum and classical mechanics — recovering Bohr–Sommerfeld quantization and the tunneling exponential — and applies precisely where perturbation theory has no small parameter but the system is nearly classical.

1. The Semiclassical Ansatz

Write the stationary wavefunction as an exponential of a phase and expand the phase in powers of $\hslash$:

$$\psi (x)=\exp \left[\frac{i}{\hslash }\right(S_0(x)+\hslash S_1(x)+\cdots \left)\right].$$

Substituting into the time-independent Schrödinger equation and matching orders of $\hslash$: the leading order gives the classical Hamilton–Jacobi equation $(S_0^{\prime }{)}^2=2m(E-V)=p(x{)}^2$, so $S_0=\pm \int pdx$ with the local momentum

$$p(x)=\sqrt{2m(E-V(x))}.$$

The next order fixes the amplitude, $S_1=\frac{i}{2}\ln p$, yielding the WKB wavefunction:

$$\boxed{\psi (x)\approx \frac{1}{\sqrt{p(x)}}\exp [\pm \frac{i}{\hslash }{\int }^{x}p(x^{\prime })d{x}^{\prime }].}$$

The $1/\sqrt{p}$ prefactor is conservation of probability current (the particle spends more time, and the amplitude is larger, where it moves slowly). Validity requires the wavelength to change slowly, $|d\lambda /dx|\ll 1$, i.e. $\hslash |p^{\prime }|/p^2\ll 1$ — which fails exactly at the classical turning points where $p(x)=0$.

2. Classically Forbidden Regions

Where $E<V$ (forbidden region), $p(x)$ is imaginary and the oscillatory solution becomes exponential:

$$\psi (x)\approx \frac{1}{\sqrt{|p(x)|}}\exp [\pm \frac{1}{\hslash }{\int }^x|p(x^{\prime })|d{x}^{\prime }].$$

This exponential decay is the WKB description of tunneling — the smooth-barrier generalization of the square-barrier result.

3. Connection Formulae

Because WKB breaks down at turning points, the oscillatory and exponential solutions on either side must be joined by connection formulae, obtained by solving the Schrödinger equation exactly near a turning point (where $V$ is locally linear, giving Airy functions) and matching asymptotics. The upshot is a definite phase relation: an allowed-region oscillation matched to a decaying exponential picks up a $\pi /4$ phase at each turning point.

4. Bohr–Sommerfeld Quantization

Applying the connection formulae at both turning points $a,b$ of a potential well and demanding single-valuedness gives the Bohr–Sommerfeld quantization condition with its characteristic Maslov correction:

$$\boxed{\oint pdx=2{\int }_a^b\sqrt{2m(E-V)}dx=(n+\frac{1}{2})2\pi \hslash ,n=0,1,2,\dots }$$

The classical action over one period is quantized in units of $h$, shifted by $\frac{1}{2}$ (two turning points $\times \frac{1}{4}$). This reproduces the exact spectra of the harmonic oscillator (where WKB is exact) and is an excellent approximation for smooth wells and high quantum numbers — the precise sense in which large-$n$ quantum mechanics becomes classical (Bohr's correspondence principle).

5. Tunneling Rates and the Gamow Factor

For a barrier between turning points $a,b$ with $E<V$, the connection formulae give the transmission probability as the exponential of the forbidden-region action:

$$T\approx \exp [-\frac{2}{\hslash }{\int }_a^b\sqrt{2m(V(x)-E)}dx].$$

This Gamow factor explains the enormous range of nuclear $\alpha$-decay lifetimes (the Geiger–Nuttall law), cold field emission of electrons, and reaction rates in stellar nucleosynthesis — all set by the exponential sensitivity of $T$ to the barrier's height, width, and the particle mass.

See also

• Wave mechanics — the $\psi =\sqrt{\rho }{e}^{iS/\hslash }$ form and classical limit.
• Piecewise potentials — the exact square-barrier tunneling.
• Variational method — the other non-perturbative technique.
• Path integral — the stationary-phase view of the classical limit.