| Wentzel–Kramers–Brillouin approximation | |
|---|---|
| Name | Wentzel–Kramers–Brillouin approximation |
| Caption | Semiclassical trajectory illustration |
| Introduced | 1926 |
| Inventors | Gregory Wentzel, H. A. Kramers, Léon Brillouin |
| Field | Quantum mechanics |
| Related | WKB, Semi-classical approximation, Maslov index |
Wentzel–Kramers–Brillouin approximation
The Wentzel–Kramers–Brillouin approximation (commonly abbreviated WKB or JWKB) is a semi‑classical method for approximating solutions of the Schrödinger equation in regions where the potential varies slowly compared to the particle wavelength. It provides asymptotic expansions that connect quantum wave behavior to classical mechanics and is widely used to estimate tunneling probabilities, bound state quantization, and scattering phase shifts in atomic, molecular, and condensed matter systems.
The WKB approximation was independently developed by Gregory Wentzel, Hendrik Anthony Kramers, and Léon Brillouin in 1926 to address problems in early quantum theory such as quantization rules and barrier penetration. Physically, WKB captures the transition between oscillatory and evanescent wave behavior across regions determined by the classical energy relative to the potential, i.e., classically allowed and forbidden regions. The method links the quantum phase to the classical action, thereby underpinning connections to the Hamilton–Jacobi equation, the Bohr–Sommerfeld quantization condition, and modern semiclassical techniques used in atomic physics, nuclear physics, and mesoscopic physics.
The WKB procedure starts from the one‑dimensional time‑independent Schrödinger equation for a particle of mass m and energy E in potential V(x). One inserts the ansatz psi(x)=exp[(i/ħ)S(x)] and expands the action S(x) in powers of Planck's constant ħ: S = S_0 + (ħ/i)S_1 + .... At leading order S_0 satisfies the Hamilton–Jacobi equation, giving momentum p(x)=±sqrt{2m(E−V(x))}. Successive orders yield amplitude corrections and transport equations that enforce probability current conservation. The resulting WKB solutions in the classically allowed region are locally psi(x) ≈ A(x) e^{(i/ħ) ∫ p dx} + B(x) e^{-(i/ħ) ∫ p dx}, with amplitude A(x)∝|p(x)|^{-1/2}. The first rigorous formulations and error estimates were analyzed in mathematical studies of singular perturbation and asymptotic analysis by researchers working on Sturm–Liouville theory and semiclassical spectral asymptotics.
WKB makes explicit the correspondence principle articulated by Niels Bohr: in the limit ħ→0 quantum quantities reduce to classical ones. The leading WKB phase equals the classical action S_cl = ∫ p dx, and quantization arises from constructive interference of phases along closed classical orbits. This yields the Bohr–Sommerfeld quantization rule and corrections via the Maslov index which accounts for phase shifts at turning points. In multidimensional systems, semiclassical propagation is associated with classical trajectories governed by Hamiltonian mechanics and is formalized in approaches like the Gutzwiller trace formula and semiclassical propagator methods employed in chemical physics and quantum chaos studies.
WKB is extensively applied to compute tunnel transmission through potential barriers, yielding the exponential Gamow factor used in alpha decay models and field emission approximations. For bound states in smoothly varying potentials, WKB gives energy quantization conditions and approximate eigenfunctions for systems such as the one‑dimensional harmonic oscillator (away from turning points) and anharmonic wells. In scattering theory, WKB provides phase shifts and semiclassical approximations to the S‑matrix for slowly varying potentials, connecting to the Born approximation in complementary regimes. Practical applications span quantum tunneling in Josephson junctions in superconducting circuits, semiclassical models of molecular vibrations, and electron transport in semiconductor heterostructures.
The WKB approximation requires that the relative change in the local de Broglie wavelength be small over one wavelength: |dλ/dx| ≪ 1, or equivalently |ħ d p/dx| ≪ p^2. It breaks down near turning points where E≈V(x) and p→0, and in regions with abrupt potential changes or strong coupling between modes. To construct global solutions one uses connection formulas that match oscillatory and exponential WKB solutions across turning points; these matching conditions were formalized by R. E. Langer and others and incorporate phase shifts (¼π per simple turning point) leading to modified quantization rules. Rigorous justifications employ matched asymptotic expansions and complex WKB analysis based on analytic continuation in the complex plane and Stokes phenomena.
Variants and extensions of WKB include the JWKB approximation (Jeffreys–Wentzel–Kramers–Brillouin), uniform approximations using Airy functions near turning points, and the incorporation of the Maslov index to account for caustics in higher dimensions. Multidimensional semiclassical methods generalize WKB via the method of stationary phase, Lagrangian manifolds, and the Maslov canonical operator developed in microlocal analysis and by researchers like Vladimir Maslov. Related techniques include the semiclassical Green's function, Wigner–Weyl phase‑space methods, and path integral stationary‑phase approximations pioneered by Richard Feynman. Modern computational semiclassical approaches are employed in ab initio molecular dynamics, semiclassical initial value representations, and semiclassical quantization of chaotic systems in the context of quantum chaos and Gutzwiller trace formula studies.
Category:Quantum mechanics Category:Asymptotic analysis Category:Semiclassical physics