Difference between eigenvalue and an expectation value

[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?

 

[Ilmrak]

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?

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