WKB Approximation

13.1 The WKB Method

The WKB (Wentzel—Kramers—Brillouin) method provides approximate solutions to the one-dimensional Schrodinger equation when the potential varies slowly compared to the de Broglie wavelength.

The ansatz $\psi(x) = A(x)e^{iS(x)/\hbar}$ substituted into $-\frac{\hbar^2}{2m}\psi"' + V\psi = E\psi$ gives, to leading order in $\hbar$:

$S(x) = \pm\int^x p(x')\,dx', \quad p(x) = \sqrt{2m[E - V(x)]}$

The WKB wavefunctions:

$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\!\left(\pm\frac{i}{\hbar}\int^x p(x')\,dx'\right) \quad \text{(classically allowed, E > V\text{)}}$

$\psi(x) \approx \frac{C}{\sqrt{|p(x)|}}\exp\!\left(\pm\frac{1}{\hbar}\int^x |p(x')|\,dx'\right) \quad \text{(classically forbidden, E < V\text{)}}$

13.2 Connection Formulas

At a classical turning point ($E = V(x_0)$), the WKB approximation breaks down. The Airy function connects the oscillating and decaying solutions:

$\frac{2}{\sqrt{p(x)}}\cos\!\left(\frac{1}{\hbar}\int_x^{x_0} p(x')\,dx' - \frac{\pi}{4}\right) \longleftrightarrow \frac{1}{\sqrt{|p(x)|}}\exp\!\left(-\frac{1}{\hbar}\int_{x_0}^x |p(x')|\,dx'\right)$

13.3 Bohr—Sommerfeld Quantisation

The WKB quantisation condition for a bound state in a potential well with turning points $a$ and $b$:

$\int_a^b p(x)\,dx = \left(n + \frac{1}{2}\right)\pi\hbar, \quad n = 0, 1, 2, \ldots$

The factor of $1/2$ (Maslov index) accounts for the phase loss at each turning point.

Application: Harmonic oscillator. $V(x) = \frac{1}{2}m\omega^2 x^2$. Turning points at $x = \pm\sqrt{2E/(m\omega^2)}$.

$\int_{-A}^{A}\sqrt{2mE - m^2\omega^2 x^2}\,dx = \frac{\pi E}{\omega} = \left(n + \frac{1}{2}\right)\pi\hbar$

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega$

The WKB gives the exact result for the harmonic oscillator --- a fortunate coincidence due to the quadratic potential.

Application: Power-law potential. For $V(x) = V_0|x/a|^\alpha$:

$E_n \propto \left(n + \frac{1}{2}\right)^{2\alpha/(\alpha+2)}$