# Why are general wave functions expressed in terms of energy eigenfunctions?

I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? Could I express the hydrogen electron in terms of momentum eigenfunctions?

• That' s more of physicist speak for "I don't care that the mathematicians have spent a century on figuring out functional analysis for me and that it's actually much more complicated than I ever want to know, I just assume it's slap happy simple and move on.". The problem lies in "any function", which is a mighty large space and which is definitely not correct. If you replace it with "most functions I will ever have to concern myself with in physics", then you are getting somewhere, though. – CuriousOne Jul 11 '16 at 3:04
• You use spherical harmonics when expressing the hydrogen atom solution, right? Spherical harmonics are eigenfunctions of $L_z$ and $L^2$. – valerio Jul 11 '16 at 6:23
• Wavefunction in the coordinate representation can be on the physical level of rigorousness is the decomposition in terms of the eigenfunctions of the coordinate operator. Analogously its Fourier transform - the wavefunction in the momentum represenetation, is a decomposition in terms of the eigenfunctions of the momentum operator. Physical level of rigorousness because those "eigenfunctions" don't belong to the Hilbert space. – OON Jul 11 '16 at 7:51