# I Why do Hydrogen bound states span the Hilbert space?

1. Aug 29, 2016

### HomogenousCow

As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.

2. Aug 29, 2016

### blue_leaf77

The eigenvectors of a Hermitian operator is complete, the prove of this for infinite dimensional space is not an easy task (unfortunately this area of math is not my specialty, so I can only refer you to another example like in here). It's important to know that the functions that span the space are not only the bound states, the scattering states which are also solutions of the time-independent Schroedinger equation should also be included in the basis functions.

3. Aug 29, 2016

### A. Neumaier

They don't. The bound state vectors form an orthonormal set, they form a basis iff the spectrum is purely discrete. This is not the case for the hydrogen atom.

Last edited: Aug 29, 2016
4. Aug 29, 2016