Quantum Mechanics for Engineers 

© Leon van Dommelen 

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 .