| WKB approximation | |
|---|---|
| Name | WKB approximation |
| Field | Quantum mechanics |
| Introduced | 1920s |
| Inventors | Hermann Weyl; Gregory Breit; Eugene Wigner; Harold Jeffreys |
| Related | Semiclassical approximation, Wentzel–Kramers–Brillouin method |
WKB approximation
The WKB approximation is a semiclassical method for approximating solutions of linear differential equations with a slowly varying parameter, widely used in Quantum mechanics to estimate wavefunctions and energy levels in the limit of small Planck constant. It provides physically intuitive descriptions of phenomena such as tunneling, quantization conditions, and scattering phase shifts, bridging full quantum treatments and classical mechanics.
The WKB approximation (named after Gregory Breit, Eugene Wigner, Hermann Weyl, and independently Harold Jeffreys's early work) is central to semiclassical analysis in Quantum mechanics and mathematical physics. It interprets the quantum amplitude in terms of a rapidly varying phase given by the classical action from Hamiltonian mechanics and yields leading-order behavior in the limit ħ → 0. The method underpins conceptual connections between the Bohr model, Old quantum theory, and modern formulations such as the Gutzwiller trace formula. It is used extensively in atomic, molecular, and condensed matter physics, as well as in quantum chemistry and cosmology (e.g., in semiclassical treatments of quantum cosmology and instanton approximations).
The WKB ansatz assumes a wavefunction of the form ψ(x) = exp[(i/ħ)S(x)] with an asymptotic expansion S = S_0 + ħ S_1 + ħ^2 S_2 + ... . Substituting into the one-dimensional Schrödinger equation for a particle of mass m in potential V(x) yields a hierarchy of equations: the leading-order gives the Hamilton–Jacobi equation for S_0, and subsequent orders compute amplitude corrections. The approximation relies on the potential varying slowly on the scale of the local de Broglie wavelength λ(x) = 2πħ/p(x) where p(x)=√{2m(E−V(x))}. Matching conditions at classical turning points, where p(x) vanishes, require special treatment via connection formulas linked to Airy functions and the Stokes phenomenon from complex analysis. Rigorous justification employs techniques from asymptotic analysis and the theory of ordinary differential equations, and connects to microlocal analysis and the method of stationary phase.
WKB is a stationary-phase approximation to the quantum propagator and is closely related to the Feynman path integral semiclassical expansion. The classical action S_0 along trajectories appears in both formalisms; in path integrals the leading contribution comes from classical paths making the action stationary. WKB also underlies semiclassical quantization rules such as the Einstein–Brillouin–Keller quantization and contributes to trace formulae like the Gutzwiller trace formula linking classical periodic orbits to quantum energy spectra. In multidimensional systems, WKB solutions connect to the theory of Lagrangian manifolds in phase space and to tools developed in symplectic geometry and Maslov index theory, which track phase jumps associated with caustics.
WKB provides practical formulas for many quantum problems. For one-dimensional tunneling through a barrier it yields the exponential transmission coefficient T ≈ exp[−(2/ħ) ∫ |p(x)| dx] used in models of alpha decay and field emission; this derivation parallels the Gamow theory of alpha decay. For bound states, the quantization condition ∮ p(x) dx = 2πħ(n + 1/2) gives approximate energy levels in potentials like the harmonic oscillator, double-well, and potential wells studied in quantum chemistry and nuclear physics. In scattering theory, the WKB approximation produces semiclassical phase shifts and connection with the Born approximation for high-energy scattering. Applications span solid-state physics (e.g., tunneling in Josephson junctions and semiconductor heterostructures), molecular physics (vibrational levels), and mesoscopic physics.
Beyond leading order, successive terms S_1, S_2, ... give amplitude and phase corrections that improve quantitative accuracy; these are computed by solving linear transport equations. Matching across turning points uses Airy-function approximations to derive the standard WKB connection formulas and the Maslov phase of π/2 per turning point. Uniform approximations such as the Langer modification adjust centrifugal terms in radial problems to yield correct quantization for systems like the hydrogen atom. Higher-order WKB has been developed and applied in semiclassical quantization via the Dunham expansion and in resurgent analysis connecting perturbative series to nonperturbative effects, as seen in studies by Écalle and in applications to quantum tunnelling beyond leading exponential order.
WKB fails near turning points, at caustics, or when the potential varies on scales comparable to the wavelength. It can produce incorrect prefactors or miss interference effects when multiple classical paths contribute; in chaotic systems naive WKB must be replaced by more sophisticated semiclassical methods like the Gutzwiller trace formula. In radial problems naive WKB miscounts angular-momentum contributions, remedied by the Langer correction. Uniform approximations (e.g., via Airy, Bessel, or parabolic-cylinder functions) and connection with complex Wentzel–Kramers–Brillouin techniques extend validity through turning points and across Stokes lines. For rigorous control one may use the exact WKB method, Borel summation, and microlocal techniques developed in mathematical physics and by researchers at institutions such as Institut des Hautes Études Scientifiques and universities with strong analysis groups.
Category:Quantum mechanics Category:Semiclassical physics