# Hermitian Operators

1. Dec 31, 2005

### 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

2. Dec 31, 2005

### 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

Last edited: Dec 31, 2005