I Why do Hydrogen bound states span the Hilbert space?

[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.

 

[blue_leaf77]

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.

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.

 

[A. Neumaier]

why does the set of hydrogen bound states form an orthonormal basis?

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.

 

[A. Neumaier]

See also https://www.physicsforums.com/posts/3950445/ and the surrounding discussion. SO(4,2) produces the full spectral basis, including the scattering part. The thread https://www.physicsforums.com/threa...oup-symmetry-but-not-general-symmetry.552990/ also contains related information.