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 [itex]A: H \rightarrow H[/itex], [itex]H[/itex] an Hilbert space, [itex]|\psi\rangle \in H[/itex], then

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

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

Ilm