Eigenfunctions corresponding to a particular energy value

[hokhani]

Suppose we want to get eigenfunctions of a One-Particle Hamiltonian corresponding to one of its eigenvalues, say E, in bases of free particle eigenfunctions (plane waves). Can we use the plane waves corresponding to energies near E to get a reasonable solution?

 

[WannabeNewton]

No it depends entirely on the Hamiltonian. This isn't like the variational principle where just about any trial wave-function gives you a good estimate of the ground state energy. You're trying to go backwards and it can certainly fail to give a good approximation in general. For example, the plane wave is an absolutely terrible approximation of the ground state wave-function of the harmonic oscillator. You can't approximate a Gaussian with a sinusoid.