WKB review

Tonight I reviewed Griffiths’ and Liboff’s quantum textbooks on WKB theory.  Being exposed to the Integrating Factor from ODEs really helped me not stumble over the main points.

Griffiths’ motivates the WKB in a non-standard way, noting that the Schrodinger equation can be broken up into two equations on the real and complex parts.  The approximation in this derivation becomes that the second differential of the amplitude of the wavefunction over the amplitude is small.  He leaves the more standard WKB expansion in powers of \hbar as an exercise.

Liboff examines the correspondence principal, stating that the change in DeBroglie wavelength over a wavelength must be small in the classical regime, and then derives this in terms of the particle momentum and arrives at the criterion

\left | \frac{\delta \lambda }{\lambda }\right | = \left | \frac{m h }{p^3} \frac{dV}{dx}\right |

Then when doing the WKB expansion, uses this criterion to validate truncating the series after the second term.

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