WKB approximation: Difference between revisions

Revision as of 03:55, 16 October 2024

Revision as of 21:21, 30 October 2024

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We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at <math>\pm \infty </math>, we conclude:

<math>BA=-D=N </math>,

<math>AB=C=0 </math> and <math>\alpha = \frac \pi 4 </math>.

Thus, letting some normalization constant be <math>N </math>, the wavefunction is given for increasing potential (with x) as:<ref name=":1" />

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We cannot use Bairy function since it gives growing exponential behaviour for positive x. When compared to WKB solutions and matching their behaviours at <math>\pm \infty </math>, we conclude:

<math>2A2B=C=N' </math>,

<math>D=BA=0 </math> and <math>\alpha = \frac \pi 4 </math>.

Thus, letting some normalization constant be <math>N' </math>, the wavefunction is given for increasing potential (with x) as:<ref name=":1" />