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Difference between eigenvalue and an expectation value

  1. Aug 16, 2012 #1
    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. jcsd
  3. Aug 16, 2012 #2
    Yes, you are right :smile:

    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].

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