For the typical bound state problem, WKB is good everywhere except the classical turning points. Solve the schrodinger equation near the classical turning points and match to the WKB solution in the other regions. There are 3 regions divided by the 2 classical turning points.

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Solution to schrodinger equation near turning points:

Then .

So the schrodinger equation is:

define

Then .

Take asymptotic forms for Ai(y) and Bi(y) as y goes to and match to get , . In general, use WKB connection formulas to get:

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match the 2 classically allowed solutions.

. This means

Matching to dying exponential means Then This is similar to the Bohr-Sommerfeld quantization.

source psfile jl@crush.caltech.edu index