Introduction: When ℏ is Small
The WKB approximation (Wentzel-Kramers-Brillouin) bridges quantum and classical mechanics. It's valid when the potential varies slowly compared to the wavelength—the semiclassical regime.
The WKB Ansatz
Write \(\psi(x) = A(x)e^{iS(x)/\hbar}\) and expand in powers of \(\hbar\).
To leading order in allowed regions (\(E > V\)):
\[\psi(x) \approx \frac{C}{\sqrt{p(x)}}e^{\pm i\int p(x')\,dx'/\hbar}\]where \(p(x) = \sqrt{2m(E - V(x))}\) is the local classical momentum.
In Forbidden Regions
When \(E < V(x)\), \(p(x)\) becomes imaginary. The WKB solution is:
\[\psi(x) \approx \frac{C}{\sqrt{|p(x)|}}e^{\pm\int |p(x')|\,dx'/\hbar}\]Exponential decay or growth, as expected.
Validity Condition
WKB is valid when the wavelength changes slowly:
\[\left|\frac{d\lambda}{dx}\right| \ll 1 \quad \text{or} \quad \left|\frac{dp/dx}{p^2/\hbar}\right| \ll 1\]The Quantum Connection
WKB reveals quantum mechanics as "classical mechanics plus wave effects." The phase \(\int p\,dx/\hbar\) is the classical action. WKB fails at turning points (where \(p = 0\)) but connection formulas patch solutions across these points. It's widely used in atomic, molecular, and nuclear physics.