When a potential changes little over one wavelength, the wavefunction looks locally like a plane wave with the local momentum $p(x)=\sqrt{2m(E-V)}$. Its phase advances by $\int p\,dx/\hbar$, and its amplitude changes too: probability piles up where the particle moves slowly, so the amplitude goes as $1/\sqrt{p}$. This is the WKB approximation, named after Wentzel, Kramers and Brillouin. It is the bridge between quantum and classical mechanics, and it fails exactly where a classical particle would turn back.

A car travelling through a town spends more time on the slow streets than on the fast ones; a photograph taken at a random moment most likely shows it on a slow street. The WKB amplitude is that photograph’s probability.

A WKB wave in a potential that falls steadily, $V=-x$ at $E=0$ with $\hbar=m=1$: as the particle speeds up its wavelength shortens, and its amplitude, following , shrinks — it spends less time where it moves faster.

Where $E>V(x)$, the local momentum is $p(x)=\sqrt{2m(E-V)}$. When changes slowly on the scale of a wavelength, the solutions are approximately the two below; where , becomes imaginary and they grow or decay.

• Validity: the local wavelength $\lambda=2\pi\hbar/p$ changes little over one wavelength, $|d\lambda/dx|\ll1$.
• $|\psi|^{2}\propto1/p\propto1/v$: the probability density follows the time a classical particle spends at each place.

Write the wavefunction as an exponential of a phase and put it into the Schrödinger equation. Order by order in $\hbar$: the leading order says the phase’s slope is the classical momentum; the next says how the amplitude must change to conserve probability.

Put $\psi=e^{iS/\hbar}$ into $-\frac{\hbar^{2}}{2m}\psi''+V\psi=E\psi$.

The terms without $\hbar$: the phase advances by the classical action.

The terms with one power of $\hbar$.

Integrate and exponentiate.

Keeping the first two orders; the next is small when $\hbar|p'|\ll p^{2}$.

The WKB amplitude goes as one over the square root of the local momentum, so the probability density goes as one over the speed: large where the particle is slow.

Where the energy exceeds the potential, the particle has a local momentum $p(x)=\sqrt{2m(E-V)}$, and a local wavelength .

At a classical turning point the local momentum is zero, so the WKB amplitude $1/\sqrt{p}$ blows up. The approximation fails there and must be patched.

Since the density goes as $1/p$, a place where the particle is twice as fast has half the density.

Great work! You now know how to:

• derive the WKB wavefunction from an expansion in $\hbar$
• relate its amplitude to the classical time spent
• judge where the approximation holds and where it fails