# Difference between eigenvalue and an expectation value

1. Aug 16, 2012

### solas99

difference between eigenvalue and an expectation value of an observable. in what circumstances may they be the same?

from what i understand, an expectation value is the average value of a repeated value, it might be the same as eigen value, when the system is a pure eigenstate..

am i right?

2. Aug 16, 2012

### Ilmrak

Yes, you are right

Given an operator $A: H \rightarrow H$, $H$ an Hilbert space, $|\psi\rangle \in H$, then

i) $a \equiv \frac{\langle \psi | A| \psi \rangle}{\langle \psi | \psi \rangle}$ is the expectation value of $A$ over the state $|\psi \rangle$;
ii) if there exists $\alpha \in \mathbb{C}$ such that $A|\psi \rangle = \alpha |\psi \rangle$, then $\alpha$ is the eigenvalue of $A$ associated with the eigenstate $|\psi\rangle$.

So if $|\psi\rangle$ is an eigenstate of $A$ with eigenvalue $\alpha$ and $\langle \psi|\psi\rangle=1$, then $a=\alpha$.

Ilm