Are Finite Well Bound States Enough for a Complete Basis in Quantum Mechanics?

[lttung]

Dear Physics Forum,

I have a question about quantum mechanics.
I know that the solutions of a Hamiltonian will form a complete basis.

However, for a case of a finite well, in the region of bound states (E<0), the number of eigenfunctions is finite, I wonder that they are enough to form a complete basis or not?

For the case of infinite square well or Hydrogen potential, the basis is complete.
But for the case of an attractive Delta well, or a finite well, do the bound states form a complete one, or we need to include the wave function for E>0?

Thank you very much.

Bests,

Tung LE

 

[TMFKAN64]

A finite set of functions cannot be complete. So yes, you need to include E>0 eigenfunctions as well.

 

[Entropia]

Dear Tung LE,

Thank you for your question regarding the completeness of a basis in quantum mechanics. The completeness of a basis is an important concept in quantum mechanics, as it ensures that any state in a given system can be expressed as a linear combination of the basis states. In the case of a finite well, the number of bound states is indeed finite, and it is natural to wonder if they are enough to form a complete basis.

The answer to your question is yes, the bound states in a finite well do form a complete basis. This is because the bound states are the eigenfunctions of the Hamiltonian, which means they span the space of all possible states in the system. This is true even though the number of bound states is finite. The completeness of the basis is not dependent on the number of states, but rather on the fact that they are eigenfunctions of the Hamiltonian.

In the case of an attractive Delta well or a finite well, the bound states are the only states that contribute to the Hamiltonian, and therefore they form a complete basis. Including the wave functions for E>0 would not be necessary for completeness.

I hope this answers your question and clarifies any confusion. If you have any further questions, please do not hesitate to ask.

Best regards,