Probability density function problem

[Chronos000]

Homework Statement

Is the PDF of something between two different bases or wavefunctions always 0?

For instance, if you have the lowering operator \hat{}a -

<n|\hat{}a|n>

that changes to <n|\sqrt{}n|n-1> =0

I'm not sure I understand the physical scenario if this is true however.

 

[ideasrule]

Does the lowering operator produce a function orthogonal to the original? If it's guaranteed to produce an eigenfunction of the Hamiltonian with a different eigenvalue (it should, or else it's not a very useful lowering operator), then all functions produced by the operator have to be orthogonal. This is due to a more general theorem which says that all eigenfunctions of a Hermitian operator with different eigenvalues are mutually orthogonal.

 

[Chronos000]

ok, I think I get this thanks