WKB and the Gamow exponent

Nothing is approximated by writing ψ(x) = exp(iS(x)/ħ); S is simply a new unknown, complex in general. Substituting into ψ″ = −(p²/ħ²)ψ, with p(x) = √(2m(E − V)), gives the exact nonlinear equation (S′)² − iħS″ = p². Now expand S = S₀ + ħS₁ + ħ²S₂ + … and collect powers of ħ. At order ħ⁰, (S₀′)² = p², so S₀ = ±∫p dx′ — the classical action, which is where the word semiclassical comes from. At order ħ¹, 2S₀′S₁′ = iS₀″, so S₁ = (i/2) ln p. Exponentiating both terms gives ψ ≈ p^(−1/2) exp(±(i/ħ)∫p dx′). The amplitude is not decoration: |ψ|² ∝ 1/p ∝ 1/v is the classical dwell-time density, and it is exactly the factor that holds the probability current j = (ħ/m) Im(ψ*ψ′) constant along x, as a stationary state in one dimension demands.

Truncating after S₁ is legitimate only if ħ|S₀″| ≪ (S₀′)², that is ħ|p′| ≪ p². Write λ̄ = ħ/p for the reduced local wavelength and the same statement reads |dλ̄/dx| ≪ 1: the wavelength must change by little over one wavelength. Expanded, it is ħ m |V′| / |p|³ ≪ 1, with |p| = ħκ under a barrier. Read what that does and does not say. It is not “ħ is small” — ħ is a constant. It is a joint claim about the gradient of V and the momentum at that energy, so a step in the potential breaks WKB at any energy, while a smooth one supports it even for a light particle. Take the field-emission barrier of the medium example: at the metal surface ħ m eF / |p|³ = 0.031, comfortably small, but it grows without bound as p → 0, passing 1 where V − E has fallen to 0.44 eV — the last 0.15 nm of a 1.50 nm barrier, beside the outer turning point. WKB is excellent over ninety per cent of the region and worthless in the rest.

At x = a where V(a) = E the momentum vanishes, the amplitude p^(−1/2) diverges, and the criterion fails outright. The cure is local. Over a small neighbourhood V(x) ≈ E + V′(a)(x − a); taking the forbidden side to the right, V′(a) > 0, the Schrödinger equation becomes ψ″ = α³(x − a)ψ with α = (2m|V′(a)|/ħ²)¹⁄³: the Airy equation, whose bounded solution is Ai(α(x − a)). Its scale 1/α sets the patch width — for an electron on a 3.0 V nm⁻¹ ramp, (ħ²/2m eF)¹⁄³ = 0.23 nm. Away from a, Ai has known asymptotics: a decaying exponential on the forbidden side, and on the allowed side an oscillation carrying an extra −π/4 of phase. Matching those asymptotics onto the two WKB branches in the overlap window — far enough from a for Airy's asymptotics, near enough that the linearisation still holds — is the entire content of the connection formulas. Where no such window exists, there is nothing to connect.

Under a barrier the two solutions are exp(+∫κ dx) and exp(−∫κ dx), and they differ by the vast factor e^(2γ). Start on the allowed side and propagate inward and an error of one part in e^(2γ) in your coefficients seeds the growing branch, which then swamps the decaying one you wanted; impose decay where decay is required and carry the solution outward instead, and no such contamination arises. That is why the safe statement is (1/√κ) exp(−∫ₐˣ κ dx′) → (2/√k) cos(∫ₓᵃ k dx′ − π/4), read from the forbidden side out. The same asymmetry appears numerically as the transfer matrix's e^(2κL) condition number. It is also the reason WKB delivers a reliable barrier exponent and an unreliable prefactor: the exponent is just the integral of κ, while the prefactor is precisely the coefficient bookkeeping that the growing branch corrupts.

Send a wave in from the left, demand a purely outgoing wave on the right, connect at both turning points keeping the branch that decays from a to b, and the transmitted amplitude emerges suppressed by e^(−γ) with γ = ∫ₐᵇ κ dx between the roots of V(x) = E. Squaring gives T ≈ e^(−2γ). Two cautions follow. First, a and b move with E, yet because κ vanishes at both, the moving limits add nothing to dγ/dE: the whole energy dependence is κ shrinking, dγ/dE = −(m/ħ²)∫ₐᵇ dx/κ, weighted towards the ends where κ is small. Second, the prefactor. For a 4.00 eV rectangular barrier 0.600 nm wide at E = 1.00 eV, WKB returns 2γ = 10.65 and T = 2.4 × 10⁻⁵, while the exact opaque result is 7.1 × 10⁻⁵ — a factor 16E(V₀ − E)/V₀² = 3.00 that WKB cannot see, because vertical walls are not linear turning points and no Airy patch applies to them. Kemble's uniform formula T = [1 + e^(2γ)]⁻¹ repairs the near-top behaviour and is exact for a parabolic barrier.

For a well bounded by two linear turning points a and b, demanding decay outside both and consistency between them gives ∫ₐᵇ p dx = (n + ½)πħ with n = 0, 1, 2, … — Bohr–Sommerfeld, with the half Bohr's original rule lacked. The ½ is phase bookkeeping: each soft turning point costs π/4. Change the boundary condition and the count changes. A hard wall forces a node, costing π/2, so the infinite square well obeys ∫p dx = nπħ and returns Eₙ = n²π²ħ²/2mL² exactly; for the harmonic oscillator the soft rule is exact too, giving (n + ½)ħω. In Python this is a quadrature inside a root find: locate a and b with brentq on V(x) − E, form φ(E) = ∫ₐᵇ p dx/ħ − (n + ½)π with scipy.integrate.quad, then brentq on φ. It costs milliseconds against a full diagonalisation and, for a smooth well and n ≥ 2, is usually right to a fraction of a per cent.