The WKB approximation is most commonly used to approximate
large *n* eigenenergies
in bound state problems. Here is the summary of the usual WKB results:

For this bouncing ball problem, WKB produces:

where in our dimensionless variables *k ^{2}*=

is the start of a well known formula for the zeros of the Airy function
(e.g., Abramowitz & Stegun, §10.4.94&105 p. 450). Abramowitz & Stegun
report (for large *n*) the fractional error in this result is
of order: .005/*n ^{2}*.

The Rayleigh-Ritz is most commonly used to give an upper bound for
the groundstate
eigenenergy. Consider, for example, the following guess at the
groundstate wavefunction (a "trial" wavefunction): *f(z')=z'(a ^{2}-z'^{2})* for
0<

This expectation value is guaranteed to be above the groundstate
energy for every *a*. To get the lowest upper-bound, we want
to find the value of *a* that produces the smallest expectation
value; that value of *a* should produce the wavefunction and
expectation value closest to the actually groundstate wavefunction and
groundstate energy. In this particular example, we can find the
minimum value of the expectation value by taking the derivative
w.r.t. *a* and setting the result equal to zero:

Better guesses for our approximate wavefunction *f(z')*,
produce better (closer) results. For example, the guess:

*f(z')*=*z'* exp(*-az'*),

produces a upper bound for the groundstate energy of 2.476 just 6%
above the correct result. In the below plot the "best *a*"
trial wavefunctions are plotted (blue: the polynomial; red: the exponential)
and compared to the exact Airy groundstate wavefunction (black)

Note that a fairly approximate trial wavefunction (e.g., red), produced moderately close energy. This is the "quadratic accuracy" that makes Rayleigh-Ritz so useful for eigenenergies (but not nearly so useful for wavefunctions).

We consider here the simplest version of perturbation theory: time independent, first-order, and non-degenerate. To start the perturbation theory game we need a solution to a Schrödinger's equation. Perturbation theory then approximates the effect of small additional forces (beyond those in the Schrödinger's equation already solved) on eigenenergies and wavefunctions. For example, the already-solved-Schrödinger's-equation might be for the hydrogen atom, the small additional force might be an external magnetic field, and our aim to calculate the small Zeeman shift of energy levels in the (slightly) changed situation of strong nuclear attraction plus small magnetic bending.

The result of perturbation theory is that the small shift in energy is given by the following formula:

That is that the first order shift in the energy of the
*n ^{th}* energy eigenfunction
(

Consider then the (fairly bogus) example of a small additional constant force in our bouncing-ball problem. It's "fairly bogus" because we can exactly solve this problem, but thats the point: we can then compare the perturbation theory approximation to the exact answer.

We consider then a potential:

*V(z)=mgz*(1+)

where represents the small additional
perturbing force (i.e, *V'(z)=mgz*)
Since the problem remains that of a linear potential, the solution is
exactly as above (Airy function, etc.) but with modified scales
*l* and *e*. Thus the eigenenergies are:

Hence to verify perturbation theory we must show the following:

or simply:

Proving that the Airy functions satisfy the above for every *n*
looks to be a chore, but the Virial theorem (Saxon p.115&297,
Gasiorowicz p.208, Schiff p.180)
comes to our rescue and proves the above.