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

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The discussion centers on the completeness of basis states in quantum mechanics, specifically regarding finite well potentials. It is established that while bound states (E<0) in a finite well yield a finite number of eigenfunctions, they are insufficient to form a complete basis. To achieve completeness, it is necessary to include eigenfunctions corresponding to positive energy states (E>0), as evidenced by the cases of infinite square wells and Hydrogen potentials, which do provide complete bases.

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

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