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Expectation formula in Dirac notation.

  1. Oct 31, 2009 #1
    Expectation value of operator A is given by following formula in Dirac notation.

    <A> = <x|A|x>

    where
    A : Operator
    <A> : Expectation value of A
    |x> : State

    Somehow I am unable to convince myself that this formula is true.
    Would someone please explain it to me?

    Thanks
     
  2. jcsd
  3. Oct 31, 2009 #2
    [tex]\langle \hat{A} \rangle = \langle x|\hat{A}|x \rangle = \int \psi_{x}^{"*"} \hat{A} \psi_{x} \,d \tau[/tex]

    This is the expectation value postulate of quantum mechanics.

    If the wavefunction [tex]\psi_{x}[/tex] is not an eigenfunction of the operator [tex]\hat{A}[/tex] then the measured value of the observable [tex]A[/tex] is variable. When [tex]\psi_{x}[/tex] is normalized such that
    [tex]\int \psi_{x}^{"*"}\psi_{x} \,d \tau=1[/tex]
    the expectation value above yeilds the average measurement of all possible measurements of [tex]A[/tex]. (Note that [tex]d \tau[/tex] indicates an integration over all space.) This works because, since [tex]\hat{A}[/tex] is an Hermitian operator, its eigenvectors form an orthonormal basis. It follows that any function can be written as an infinite linear combination of eigenfunctions of [tex]\hat{A}[/tex].
     
    Last edited: Oct 31, 2009
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