WKB Approximation

Why WKB matters

The Schrödinger equation can be solved exactly for the hydrogen atom, the harmonic oscillator, the particle in a box, and not much else. For everything else — real molecules, real barriers, real wells — we use perturbation theory, numerical diagonalization, or a semi-classical approximation. WKB is the gold-standard semi-classical method: it builds quantum wave functions from classical trajectories, with corrections in powers of ℏ. When ℏ is small compared to the action of the relevant motion (high quantum numbers, slow potentials), WKB becomes asymptotically exact.

• Alpha decay. Gamow (1928) used WKB to compute tunneling rates for alpha particles escaping nuclear potentials. His formula gives half-lives spanning 30 orders of magnitude, from Polonium-212 (μs) to Thorium-232 (10¹⁰ yr) — all from a single tunneling integral.
• Scanning tunneling microscope. Tip-sample current ∝ exp(−2κd) where κ = √(2m·V/ℏ²) and d is the gap. Reducing d by 1 Å changes current by a factor of ~10 — the source of STM's atomic resolution.
• Field emission (Fowler-Nordheim). Cold cathode emission, vacuum tubes, X-ray sources: electrons tunnel out of metal under strong electric fields. Same WKB form, with a triangular barrier.
• Cosmology and false vacuum decay. Bubble nucleation rates in inflation and vacuum metastability are WKB tunneling integrals through scalar-field potentials — Coleman-De Luccia instantons are formally WKB in Euclidean time.
• Bound-state spectra. Bohr-Sommerfeld quantization ∮p dx = (n+½)h gives surprisingly accurate energy levels for many smooth potentials, including the QHO (exact!) and the hydrogen atom (exact for the radial problem). For double wells, instanton-mediated tunneling splits degenerate levels by exponentially-small amounts.

The semi-classical expansion

Substitute ψ(x) = exp(iS(x)/ℏ) into the time-independent Schrödinger equation. The result is a non-linear equation for S(x): (S'(x))² − iℏ S''(x) = 2m(E − V(x)).

Expand S in powers of ℏ: S = S_0 + ℏ S_1 + ℏ² S_2 + .... Collecting orders:

Order ℏ⁰:  (S_0')² = 2m(E − V)
            → S_0' = ±p(x) = ±√(2m(E − V))
            → S_0(x) = ±∫ p(x') dx'    (classical action)

Order ℏ¹:  2 S_0' S_1' − i S_0'' = 0
            → S_1' = i S_0'' / (2 S_0') = i p' / (2p) = (i/2) d[ln p]/dx
            → S_1 = (i/2) ln p

So ψ ≈ exp(iS/ℏ) ≈ exp(i S_0 / ℏ) · exp(i S_1)
       = (1/√p) · exp(±i ∫ p dx / ℏ)

The 1/√p prefactor is the order-ℏ correction; the phase ∫p dx/ℏ is the classical action divided by Planck's constant. Higher-order corrections in ℏ exist (Maslov phases, S_2 contributions) but are usually subdominant.

Classically allowed vs forbidden

In the allowed region (E > V), p is real and ψ oscillates:

ψ_allowed(x) ≈ (A / √p(x)) · cos(∫ p(x') dx' / ℏ + φ)

Local de Broglie wavelength λ(x) = h/p(x) shrinks where p is large (deep in the well) and grows where p is small (near turning points). Amplitude 1/√p grows where p is small — quantum probability is highest where the classical particle moves slowest, recovering the classical 1/v distribution.

In the forbidden region (E < V), p² < 0 and we write p = i|p| with |p| = √(2m(V−E)):

ψ_forbidden(x) ≈ (B / √|p|) · exp(−∫ |p(x')| dx' / ℏ)
             + (C / √|p|) · exp(+∫ |p(x')| dx' / ℏ)

For a particle approaching a barrier, only the decaying exponential survives on the far side — giving the Gamow factor for the transmission probability.

Tunneling probability — the Gamow factor

For a barrier with V(x) > E between turning points x_a and x_b:

T ≈ exp(−2 ∫_{x_a}^{x_b} |p(x)| dx / ℏ)

  = exp(−2 ∫ √(2m(V − E)) dx / ℏ)

The exponential dependence makes T wildly sensitive to barrier height and width. Doubling barrier width can drop T by ten orders of magnitude. This sensitivity is what makes alpha decay rates span 30 orders of magnitude, and what gives STM its sub-Ångström resolution.

Worked example — square barrier

Electron with E = 1 eV facing a barrier of height V = 5 eV, width 1 nm. Then V − E = 4 eV = 6.4 × 10⁻¹⁹ J.

|p| = √(2 · 9.11 × 10⁻³¹ · 6.4 × 10⁻¹⁹)
    = √(1.17 × 10⁻⁴⁸)
    = 1.08 × 10⁻²⁴ kg·m/s

∫|p|dx = |p| · L = 1.08 × 10⁻²⁴ · 10⁻⁹ = 1.08 × 10⁻³³ J·s

2 ∫|p|dx / ℏ = 2 · 1.08 × 10⁻³³ / 1.055 × 10⁻³⁴
            = 20.5

T ≈ exp(−20.5) = 1.3 × 10⁻⁹

One electron in a billion tunnels through. Shrink barrier width to 0.5 nm: T ≈ exp(−10.2) = 3.7 × 10⁻⁵, ten thousand times higher. Halve the height: T grows further. Exponential sensitivity makes STM control gaps to picometer precision.

Bohr-Sommerfeld quantization

For a bound state with two turning points x_1, x_2 (E = V at both), the WKB wave function must be a single oscillating function inside, matched to decaying exponentials outside via Airy connection formulas. The matching conditions give:

∮ p(x) dx = (n + ½) · h,    n = 0, 1, 2, ...

(where ∮ is one full classical period — forward then back)

For the harmonic oscillator V = ½mω²x²: ∮p dx = πE/ω · 2 = 2πE/ω. Setting this equal to (n+½)h gives E_n = ℏω(n + ½) — the exact spectrum, not an approximation.

Common mistakes

• Trusting WKB near turning points. The 1/√p prefactor diverges as p → 0. Near a turning point you must switch to Airy-function patching; far from it WKB is fine again.
• Forgetting the factor of 2 in the tunneling formula. T = |ψ_far/ψ_near|² = exp(−2∫|p|dx/ℏ). Single integral inside the absolute square, but with the doubling from squaring.
• Treating WKB as universally valid. WKB is the leading semi-classical term; corrections in ℏ exist. Near caustics, in regions of strong reflection, in true deep-quantum problems (small n), WKB underperforms numerics.
• Wrong phase shift in Bohr-Sommerfeld. The +½ in (n + ½) h is the Maslov phase from two turning points (¼ each). For a half-well bounded by a hard wall on one side, replace +½ with +¼ (one turning point + one hard wall). For two hard walls: +0, giving Eₙ = n²π²ℏ²/(2mL²) — the exact particle-in-a-box.
• Forgetting the prefactor for current. Without 1/√p, the WKB ansatz violates probability current conservation. Some textbooks drop the prefactor in simplified treatments; track it carefully for quantitative work.
• Using WKB for sharp potentials. Step potentials and delta wells have V varying on scales much smaller than λ — WKB fails outright. Exact reflection-coefficient calculations are needed there.