### D.46 Derivation of the WKB approximation

The purpose in this note is to derive an approximate solution to the Hamiltonian eigenvalue problem

where the classical momentum is a known function for given energy. The approximation is to be valid when the values of are large. In quantum terms, you can think of that as due to an energy that is macroscopically large. But to do the mathematics, it is easier to take a macroscopic point of view; in macroscopic terms, is large because Planck’s constant is so small.

Since either way is a large quantity, for the left hand side of the Hamiltonian eigenvalue problem above to balance the right hand side, the wave function must vary rapidly with position. Something that varies rapidly and nontrivially with position tends to be hard to analyze, so it turns out to be a good idea to write the wave function as an exponential,

and then approximate the argument of that exponential.

To do so, first the equation for will be needed. Taking derivatives of using the chain rule gives in terms of

Then plugging and its second derivative above into the Hamiltonian eigenvalue problem and cleaning up gives:
 (D.30)

For a given energy, will depend on both what is and what is. Now, since is small, mathematically it simplifies things if you expand in a power series with respect to :

You can think of this as writing as a Taylor series in . The coefficients will depend on . Since is small, the contribution of and further terms to is small and can be ignored; only and will need to be figured out.

Plugging the power series into the equation for produces

where primes denote -​derivatives and the dots stand for powers of greater than that will not be needed. Now for two power series to be equal, the coefficients of each individual power must be equal. In particular, the coefficients of 1 must be equal, , so there are two possible solutions

For the coefficients of 1 to be equal, , or plugging in the solution for ,

It follows that the -​derivative of is given by

and integrating gives as

where is an integration constant. Finally, now gives the two terms in the WKB solution, one for each possible sign, with equal to the constant or .