For a large-parameter second-order ODE, WKB separates a rapidly varying exponential phase from a slowly varying transport amplitude.
The Liouville transformation of a second-order equation with gives . Neglecting this correction yields amplitude and phase or exponent . Its smallness measures a local slowly-varying-coefficient approximation, away from zeros of .
An oscillator with frequency leaves the WKB approximation when becomes order one. The shifted variable changes its leading undamped equation to . With this is the order-zero Bessel differential equation. Large- Bessel functions match the earlier oscillations, while the small- constant and logarithmic solutions become a constant and a linear function of late time.
For , , , set . The exact equation becomes . Let , using the Kummer function. It is regular at and tends to one. The Wronskian obeys , with . Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in , so and . This provides a check on a WKB approximation even near phases where its leading predicted slope vanishes.
For a separated duct mode with local width and nonzero axial wavenumber , conservation of modal flux gives amplitude proportional to . The phase is .
Near a simple turning point, rescaling the independent variable reduces a second-order equation to the Airy equation. The decaying Airy solution on the forbidden side matches an oscillatory solution with phase shift on the allowed side.
For a one-dimensional energy and potential , a classical turning point satisfies . The classically allowed region has and supports an oscillatory leading-order WKB solution.
For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of with the turning-point phase correction.
The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.
To classify infinity, set and classify . For a second-order equation in normal form, a transformed coefficient with a pole of order greater than two makes infinity an irregular singular point.
Writing in produces the Riccati equationExpanding in inverse powers of determines the exponential phase and algebraic prefactor recursively.
For constant nonzero , putting in cancels the quadratic derivative terms and gives . The logarithmic derivative is defined on intervals where , and multiplying by a nonzero scalar does not change . For variable , the derivative coefficient becomes .
At a regular point of a second-order linear ordinary differential equation, a nontrivial solution cannot have both and zero, by initial-value uniqueness. Its zeros are therefore simple. Under the linearization of a Riccati equation, such a zero produces . Zeros separate the intervals on which the Riccati equation solution is finite.
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The WKB approximation, short for the Wentzel-Kramers-Brillouin approximation, is a mathematical technique used primarily in quantum mechanics to find approximate solutions to the Schrödinger equation in the semiclassical limit, where quantum effects can be approximated by classical trajectories. The WKB method arises when studying quantum systems with a potential that varies slowly compared to the wavelength of the particle.