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Expectation Value of an Operator

  1. Aug 6, 2008 #1
    Problem
    Consider an operator [tex]\hat{A}[/tex] whose commutator with the Hamiltonian [tex]\hat{H}[/tex] is the constant [tex]c[/tex]... ie [tex] [\hat{H}, \hat{A}] = c[/tex]. Find [tex]\langle A \rangle[/tex] at [tex]t > 0,[/tex] given that the system is in a normalized eigenstate of [tex]\hat{A}[/tex] at [tex]t=0,[tex] corresponding to the eigenvalue [tex]a[/tex].

    Attempt Solution
    We know that

    [tex]\frac{\partial \langle A \rangle}{dt} = \langle \frac{i}{\hbar} [\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t} \rangle = \langle \frac{i c}{\hbar} + 0 \rangle = \frac{i c}{\hbar}[/tex].

    Is this correct? (I'm just confirming that [tex]d\hat{A}/dt = 0[/tex] since we're in an eigenstate of [tex]\hat{A}[/tex]). But this means that the expected value of [tex]A[/tex] is complex... clearly, [tex]\hat{A}[/tex] is not Hermitian then, right?
     
  2. jcsd
  3. Aug 6, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    None of this is very clear. What's the exact problem? ic/hbar may be real if c is complex. It's certainly true that the expectation values of a hermitian operator are real. That I'll give you, clearly.
     
  4. Aug 7, 2008 #3
    Hmm... I couldn't edit my previous post, so here's the new problem... (slight LaTeX error in previous post):

    Problem
    Consider an operator [tex]\hat{A}[/tex] whose commutator with the Hamiltonian [tex]\hat{H}[/tex] is the constant [tex]c[/tex]... ie [tex] [\hat{H}, \hat{A}] = c[/tex]. Find [tex]\langle A \rangle[/tex] at [tex]t > 0,[/tex] given that the system is in a normalized eigenstate of [tex]\hat{A}[/tex] at [tex]t=0,[/tex] corresponding to the eigenvalue [tex]a[/tex].

    This is the correct problem. Notation-wise, we have that [tex]\langle A \rangle[/tex] denotes the expected value of the operator [tex]\hat{A}[/tex] operating upon some wavefunction [tex]\psi[/tex]... here, we know that for our wavefunction, [tex] [\hat{H}, \hat{A}] = c [/tex].
     
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