The discussion revolves around the nature of hydrogen bound states and their role in forming an orthonormal basis within the context of quantum mechanics. Participants explore the completeness of the eigenvectors associated with the hydrogen atom and the implications of different potential scenarios on the existence of bound states.
Participants express disagreement regarding the completeness and orthonormality of the hydrogen bound states as a basis. There is no consensus on whether these states can form a complete basis in the context discussed.
Participants highlight the complexity of proving completeness in infinite dimensional spaces and the need to consider both bound and scattering states, indicating potential limitations in the discussion.
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.HomogenousCow said: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.
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.HomogenousCow said:why does the set of hydrogen bound states form an orthonormal basis?