The WKB Approximation

The variational method is best when a system has a single dominant state to estimate. The WKB approximation (Wentzel–Kramers–Brillouin) is a complementary tool: it builds approximate wavefunctions and energies whenever the potential varies slowly compared with the local wavelength. It is the bridge between quantum mechanics and classical mechanics — a semiclassical method.

The idea: a slowly varying wavelength

For a particle of energy $E$ in a potential $V(x)$ , define the local momentum

p(x) = \sqrt{\,2m\,[\,E - V(x)\,]\,}.

Where $E > V$ the particle is classically allowed and $p(x)$ is real; where $E < V$ it is classically forbidden and $p(x)$ becomes imaginary. The corresponding local de Broglie wavelength is $\lambda(x) = 2\pi\hbar / p(x)$ .

WKB assumes this wavelength changes only a little over its own length — the potential is smooth on the scale of $\lambda$ . Then the wavefunction looks locally like a plane wave whose wavelength and amplitude drift slowly with position.

The WKB wavefunction

Writing $\psi(x) = e^{iS(x)/\hbar}$ and expanding $S$ in powers of $\hbar$ , the leading terms give the WKB wavefunction. In a classically allowed region ( $E > V$ ) it is oscillatory:

\psi(x) \approx \frac{C}{\sqrt{p(x)}}\, \exp\!\left(\pm \frac{i}{\hbar}\int^{x} p(x')\,dx'\right).

In a classically forbidden region ( $E < V$ ), $p$ is imaginary, $p(x) = i\,|p(x)|$ , and the exponential becomes a real growing or decaying function:

\psi(x) \approx \frac{C}{\sqrt{|p(x)|}}\, \exp\!\left(\pm \frac{1}{\hbar}\int^{x} |p(x')|\,dx'\right).

Two features deserve attention:

The phase accumulates the action. The oscillatory exponent $\tfrac{1}{\hbar}\int p\,dx$ is the classical action divided by $\hbar$ , counting how many radians of phase the wave sweeps out.
The $1/\sqrt{p}$ amplitude conserves probability flux. Where the particle moves fast ( $p$ large) the amplitude is small, and where it moves slowly the amplitude is large — exactly the classical statement that a particle is more likely to be found where it lingers.

Where it breaks: turning points

At a classical turning point $x_0$ the energy equals the potential, $E = V(x_0)$ , so $p(x_0) = 0$ . The wavelength blows up and the $1/\sqrt{p}$ amplitude diverges — the slowly-varying assumption fails. The oscillatory solution on the allowed side and the exponential solution on the forbidden side must be stitched together across this region, which the connection formulas accomplish.

Why it matters

The WKB wavefunction reproduces the classical picture in the limit $\hbar \to 0$ while still carrying genuine quantum phase. From it follow two of the most useful semiclassical results: the Bohr–Sommerfeld quantization rule for bound-state energies, and an exponential estimate for barrier tunneling. The next lessons derive each in turn.