Proving Orthogonality of Eigenfunctions for Hermitian Operators

[leila]

Hi there,

Was wondering if anyone could point me in the right direction for this one?

Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal?

Thanks

 

[LeonhardEuler]

Alright, say you have an hermitian operator, O with eigenfunctions |a₁> and |a₂>, with eigenvalues of a₁ and a₂ respectively. Then:
O|a₁>=a₁|a₁> (1)
<a₁|O=a₁*<a₁| (2)
O|a₂>=a₂|a₂> (3)
and
<a₂|O=a₂*<a₂| (4)
Now right multiply |a₁> in equation (4) and left multiply by <a₂| in equation (1) to get two expressions for <a₂|O|a₁>. Subtract the two equations and observe.
-edit: keep in mind that the eigenvalues of hermitian operators are real. You can prove this by letting a₁=a₂