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?
The concept of bound states is important because it helps us understand how electrons are confined around the nucleus of a Hydrogen atom. This is crucial in understanding the electronic structure and properties of atoms, which has significant implications in fields such as chemistry, physics, and materials science.
The Hilbert space is a mathematical concept that represents the set of all possible states of a quantum system, including the bound states of a Hydrogen atom. It is a fundamental tool in quantum mechanics and allows us to describe the behavior of particles at the atomic and subatomic level.
Hydrogen bound states span the Hilbert space because they represent all possible energy levels and configurations of the electron in a Hydrogen atom. This means that any state of the electron, whether it is bound to the nucleus or not, can be described as a combination of these bound states.
The fact that Hydrogen bound states span the Hilbert space is significant because it allows us to fully describe the behavior of the electron in a Hydrogen atom. This is important in understanding chemical bonding, spectral lines, and other phenomena related to the electronic structure of atoms.
Yes, there are exceptions to this rule in certain cases, such as when external forces are applied to the Hydrogen atom or when considering relativistic effects. However, in most scenarios, the bound states of Hydrogen do indeed span the Hilbert space and provide a comprehensive understanding of the atom's behavior.