A standard problem in quantum mechanics is the solution of the
harmonic oscillator problem with the Hamiltonian

The eigenvalue problem is

The notation is simplified if we change variables to

so that the Hamiltonian is now

with

If energy is measured in units of , then the eigenvalue problem reads

The eigenenergies are for and the eigenfunctions are

The are Hermite polynomials.

Often the harmonic oscillator form is just the first term in a
Taylor's series approximation to a more complicated potential. In
some cases it is important to keep higher terms. Thus we consider the
symmetric anharmonic oscillator potential, obtained by adding a
quartic term:

where may be small or large.

This problem cannot be solved in closed form, but can be solved
numerically in the basis of the eigenstates of . In that case
the eigenvalue problem becomes a matrix eigenvalue problem. It reads

where is a column vector containing the expansion coefficients of the anharmonic oscillator wave function in terms of harmonic oscillator wave functions

and the hamiltonian matrix is

The integral can be done exactly with the result that

(1) |

(2) |