WKB approximation: Difference between revisions

Revision as of 20:02, 10 November 2024

Revision as of 04:08, 1 February 2025

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<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>

where <math>Q(x) \neq 0</math>. Substituting

<math display="block">y(x) = \exp \left[\frac{1}{\delta} \sum_{n=0}^\infty \delta^n S_nS_{n}(x)\right]</math>

results in the equation

<math display="block">\epsilon^2\left[\frac{1}{\delta^2} \left(\sum_{n=0}^\infty \delta^nS_n'nS_{n}^{\prime}\right)^2 + \frac{1}{\delta} \sum_{n=0}^{\infty}\delta^n S_n''S_{n}^{\prime\prime}\right] = Q(x).</math>

To [[leading-order|leading order]] in ''ϵ'' (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as

<math display="block">\frac{\epsilon^2}{\delta^2} S_0'S_{0}^{\prime}^2 + \frac{2\epsilon^2}{\delta} S_0'S_{0}^{\prime} S_1'S_{1}^{\prime} + \frac{\epsilon^2}{\delta} S_0''S_{0}^{\prime\prime} = Q(x).</math>

In the limit {{math|''δ'' → 0}}, the [[Method of dominant balance|dominant balance]] is given by

<math display="block">\frac{\epsilon^2}{\delta^2} S_0'S_{0}^{\prime}^2 \sim Q(x).</math>

So {{mvar|δ}} is proportional to ''ϵ''. Setting them equal and comparing powers yields

<math display="block">\epsilon^0: \quad S_0'S_{0}^{\prime}^2 = Q(x),</math>

which can be recognized as the [[eikonal equation]], with solution

<math display="block">S_0S_{0}(x) = \pm \int_{x_0}^x \sqrt{Q(x')}\,dx'.</math>

Considering first-order powers of {{mvar|ϵ}} fixes

<math display="block">\epsilon^1: \quad 2 S_0'S_{0}^{\prime} S_1'S_{1}^{\prime} + S_0''S_{0}^{\prime\prime} = 0.</math>

This has the solution

<math display="block">S_1S_{1}(x) = -\frac{1}{4} \ln Q(x) + k_1,</math>

where {{math|''k''₁}} is an arbitrary constant.

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Higher-order terms can be obtained by looking at equations for higher powers of {{mvar|δ}}. Explicitly,

<math display="block"> 2S_0'2S_{0}^{\prime} S_n'S_{n}^{\prime} + S''^{\prime\prime}_{n-1} + \sum_{j=1}^{n-1}S'_j^{\prime}_{j} S'^{\prime}_{n-j} = 0</math>

for {{math|''n'' ≥ 2}}.