Contents
Introduction The Method Turning Points Bound-State Quantization Tunneling Misconceptions FAQ Sources Introduction
The Wentzel-Kramers-Brillouin (WKB) approximation, developed in 1926 by three physicists working independently, gives a semiclassical solution of the Schrödinger equation for slowly varying potentials. It bridges quantum and classical mechanics, providing analytic approximations to bound-state energies, tunneling rates, and scattering phases. WKB is one of the most-used techniques in quantum mechanics.
The Method
Write the wave function as ψ(x) = exp(iS(x)/ℏ) and expand S in powers of ℏ:
S = S₀ + ℏS₁ + ℏ²S₂ + ...
Substituting into the Schrödinger equation and keeping leading orders gives:
ψ(x) ≈ (1/√p(x)) exp(±(i/ℏ)∫p(x')dx')
where p(x) = √(2m(E−V(x))) is the classical momentum. Valid when the potential varies slowly compared to the local de Broglie wavelength [1 ].
Validity
The WKB approximation is good when |dλ/dx| ≪ 1, where λ = 2πℏ/p is the local de Broglie wavelength. It breaks down near classical turning points where p → 0.
Turning Points
At classical turning points (E = V), p = 0 and the WKB formula has a singularity. Special "connection formulas" (Airy functions) handle the transition between the oscillatory region (E > V) and the exponentially decaying region (E < V).
The standard connection: in the classically allowed region, ψ ∼ cos(∫p dx/ℏ − π/4). In the forbidden region, ψ ∼ exp(−∫|p| dx/ℏ). The −π/4 phase shift comes from the Airy-function asymptotics [2 ].
Bound-State Quantization
For a bound state with two turning points x₁ and x₂, the WKB quantization condition (Bohr-Sommerfeld) is:
∫x₁ x₂ p(x) dx = (n + 1/2)πℏ
where n = 0, 1, 2, ... is the quantum number. This generalizes Bohr's old quantization condition with a 1/2 correction from the Maslov index (each turning point contributes π/2 phase loss).
Examples
Harmonic oscillator: WKB gives exact result En = ℏω(n + 1/2).
Hydrogen: WKB gives the correct Rydberg formula (with some care about the angular momentum centrifugal term).
Anharmonic potentials: WKB gives surprisingly accurate energy levels for moderate-to-high quantum numbers.
Tunneling
The WKB transmission coefficient through a barrier is:
T ≈ exp[−(2/ℏ)∫x₁ x₂ √(2m(V(x)−E)) dx]
This is the standard formula for tunneling probability. Used in:
Alpha decay (Gamow 1928).
Fusion in stars (Gamow peak).
Scanning tunneling microscopy.
Field-ionization rates in strong electric fields.
Vacuum decay in field theory.
See the dedicated quantum tunneling article for details.
Common Misconceptions
"WKB is exact"
It's an approximation, exact only in the ℏ → 0 limit. For some special potentials (harmonic oscillator), WKB gives exact answers, but generally there are corrections.
"WKB always works for any potential"
It fails near turning points (handled by Airy functions) and for sharply varying potentials. Smooth potentials are best.
"WKB is just an approximation"
It's foundational. The leading approximation can be systematically extended; resummation gives even more accurate results. WKB techniques generalize to multi-dimensional problems and field theory.
"WKB gives classical mechanics"
It bridges quantum and classical, giving leading quantum corrections to classical results. The ℏ → 0 limit recovers classical mechanics; higher orders give quantum corrections.
FAQ
Who is "W," "K," "B"?
Wentzel, Kramers, Brillouin — three physicists who developed the method in 1926. Sometimes called "JWKB" including Jeffreys, who developed similar methods earlier for differential equations.
How accurate is WKB for the harmonic oscillator?
Exact! WKB gives En = ℏω(n + 1/2) — same as the exact Schrödinger solution. This is a coincidence (or rather, reflects the special structure of the oscillator).
What is the Maslov index?
A topological correction to the WKB phase counting turning points. Each soft turning point contributes π/2 phase. The 1/2 in the Bohr-Sommerfeld formula comes from two turning points contributing π total.
Can WKB handle multidimensional problems?
Yes — the multidimensional WKB has rich structure (caustics, focal points). The "Maslov-Arnold theory" generalizes the 1D Maslov index.
Sources
Griffiths, D. J. (2018). Introduction to Quantum Mechanics , 3rd ed.
Bender, C. M., Orszag, S. A. (1999). Advanced Mathematical Methods for Scientists and Engineers . Springer.
Sakurai, J. J., Napolitano, J. (2017). Modern Quantum Mechanics , 2nd ed.
Kramers, H. A. (1926). "Wellenmechanik und halbzahlige Quantisierung." Zeitschrift für Physik , 39(10-11), 828–840.