WKB Approximation

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:

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

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

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

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

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 $V(x) = V_0|x/a|^\alpha$:

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

• Applying WKB at a turning point. The approximation diverges at $x_0$ where $p(x_0) = 0$. Connection formulas using Airy functions are necessary to match solutions across turning points.
• Forgetting the Maslov index. For bound states, each smooth turning point contributes a phase of $\pi/4$, giving $\pi/2$ total (the $1/2$ in $n+1/2$). Hard walls give different phases.
• Using WKB for rapidly varying potentials. The condition $|d\lambda/dx| \ll 1$ (where $\lambda = h/p$) must hold; WKB fails at sharp potential steps or barriers.
• Assuming WKB works for all $n$ in bound states. The approximation improves for large $n$ (highly excited states), but is poor for $n=0$ ground states in shallow wells.

• Nuclear physics: Alpha decay is the canonical example of WKB tunnelling. The Geiger-Nuttall law $T_{1/2} \propto \exp(1/\sqrt{E})$ follows directly from the WKB transmission probability.
• Quantum chemistry: Tunnelling corrections in reaction rate theory use WKB transmission coefficients for barrier crossing, important for proton transfer and enzyme kinetics.
• Semiconductor physics: Field emission (Fowler-Nordheim tunnelling) from metal surfaces and tunnelling in MOSFETs are described by WKB transmission through triangular barriers.
• Cosmology: The WKB method is used in inflationary cosmology to compute the spectrum of primordial density perturbations generated by quantum fluctuations during inflation.

Problem. Estimate the ground state energy of the quartic oscillator $V(x) = \frac{1}{4}m\omega^2 a^2 (x/a)^4$ using the WKB quantisation condition.

Solution. For $V(x) = \alpha x^4$ with $\alpha = m\omega^2/(4a^2)$, the turning points are at $x = \pm (E/\alpha)^{1/4}$. The WKB integral: $\int_{-A}^{A} \sqrt{2m(E - \alpha x^4)}\,dx = (n + 1/2)\pi\hbar$.

For $n = 0$: $\int_{-A}^{A} \sqrt{2m\alpha(A^4 - x^4)}\,dx = \pi\hbar/2$ where $A = (E/\alpha)^{1/4}$.

The integral evaluates to $\frac{4}{3}A^2\sqrt{2m\alpha A^4} \cdot \frac{\Gamma(1/4)^2}{4\sqrt{\pi}} \approx 1.748 A^2\sqrt{2m\alpha A^2}$.

Solving: $E_0 \approx \left(\frac{3\pi\hbar}{8} \frac{\Gamma(1/4)^2}{\sqrt{\pi}} \sqrt{\frac{\alpha}{2m}}\right)^{4/3} \approx 1.022 \hbar \sqrt{\alpha/m} a^2$.

The exact result is $E_0 = 1.060 \hbar \sqrt{\alpha/m} a^2$, so WKB is within 4% for the ground state and improves for higher $n$. $\blacksquare$