WKB approximation

The WKB approximation is an asymptotic method for solving differential equations whose coefficients vary slowly. In quantum mechanics, it approximates wavefunctions using the local momentum and is especially useful away from turning points.

Eikonal approximation: An approximation for wave propagation in which phase varies rapidly while amplitude varies slowly. WKB uses the same separation of rapidly varying phase from slowly varying amplitude.
Schrödinger equation: The quantum-mechanical equation governing how a system's wavefunction changes in space or time. The one-dimensional stationary equation is the standard setting for quantum WKB.
Quantum tunneling: The passage of a quantum particle through a potential barrier that classical mechanics forbids. WKB estimates the exponentially small transmission through a classically forbidden region.
Exact solution: A solution that satisfies a mathematical equation without approximation. WKB is useful when exact solutions are unavailable, but its accuracy depends on scale separation.
Gregory Wentzel: A German physicist who independently developed a semiclassical approximation to wave equations in 1926. His work supplied the W in the method's name.
Semiclassical approximation: A family of methods that approximate quantum systems in the limit of small Planck constant. WKB is a central semiclassical method, organized around an expansion in Planck's constant.
de Broglie wavelength: The wavelength associated with a particle, equal to Planck's constant divided by its momentum. WKB is reliable when this local wavelength changes little over one wavelength.
Bohr–Sommerfeld quantization: A semiclassical rule that restricts the action around a bound orbit to discrete values. Integrating the WKB phase between turning points yields this approximate energy condition.
Perturbation theory: Methods that approximate solutions by expanding in a small parameter around a solvable problem. Ordinary perturbation theory expands around a reference solution; WKB instead tracks a varying local phase.
Hendrik Anthony Kramers: A Dutch physicist whose 1926 work developed a semiclassical treatment of quantum wave equations. His contribution supplied the K and helped establish the approximation's quantum use.

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