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Expectation Value Question with Unknown Operator

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider an observable A associated to an operator A with eigenvalues an.
    Using the formula <A> = ∫ψ*Aψ compute the expectation value of A for the following wave function:

    [itex]\Psi[/itex]=[itex]\frac{1}{\sqrt{3}}[/itex][itex]\phi_{1}[/itex]+[itex]\frac{1}{\sqrt{6}}[/itex][itex]\phi_{2}[/itex]+[itex]\frac{1}{\sqrt{2}}[/itex][itex]\phi_{3}[/itex]

    where [itex]\phi_{1,2,3}[/itex] are normalized and orthogonal.

    2. Relevant equations
    The only other equation I can think of is the eigenvalue equation A[itex]\phi_{n}[/itex]=[itex]a_{n}[/itex][itex]\phi_{n}[/itex] but it really just puts the first part of the question into an equation, which doesn't help. I really can't think of any other relevant equations.


    3. The attempt at a solution
    I understand that I need to determine the operator A in order to compute the integral, I'm just having trouble determining A from the given wave equation. I get the feeling this question should be straight forward, yet I'm stuck right off the bat...

    Also, I'm sorry if this is in the wrong part of the forum. I tried reading about what constitutes intro vs. upper level physics, but I'm not familiar with the divisions for junior and senior years in U.S. institutions. Anyways, this question is from my third year Elements of QM course, if that changes anything.
    Thanks in advance for any help you can offer!

    -Mike
     
  2. jcsd
  3. Mar 13, 2012 #2
    Since nothing else is mentioned in the question, I think it is safe to assume the eigenvalue equation you have written, i.e. [itex]\hat{A} \phi_n = a_n \phi_n [/itex]. In this case, what is [itex]\hat{A}\psi[/itex]?

    Then explicitly find the integral. Remember that [itex]\phi_{1,2,3}[/itex] are all normalized. (What does that mean?)
     
  4. Mar 13, 2012 #3
    So the [itex]\phi_{n}[/itex] functions being normalized means that when I integrate [itex]\Psi[/itex], they will each become 1, correct?

    I'm still confused as to what happens with the operator. Based on the given information,

    [itex]A\Psi=a_{n}[\frac{1}{\sqrt{3}}\phi_{1}+\frac{1}{\sqrt{6}}\phi_{2}+\frac{1}{\sqrt{2}}\phi_{3}][/itex]

    does it not?

    Do I not need to know what A represents in order to calculate [itex]a_{n}[/itex], to explicitly calculate the integral?

    Sorry, this class is the first time I've been exposed to concepts of QM. They're still pretty confusing for me.
     
  5. Mar 13, 2012 #4
    Hi,

    It's ok if these are confusing you. Let me explain.

    When the question says [itex] A\phi_n = a_n \phi_n [/itex] it means

    [tex]A\phi_1 = a_1 \phi_1,~ A\phi_2 = a_2 \phi_2,~A\phi_3 = a_3 \phi_3 [/tex]

    The question is asking you to find the expectation of [itex]A[/itex] in terms of [itex]a_1,a_2,a_3[/itex]. You do NOT need to calculate what [itex]a_1,a_2,a_3[/itex] are.
     
  6. Mar 13, 2012 #5
    Oh okay, so the values of [itex]a_1, a_2,[/itex] and [itex]a_3[/itex] are given, and I just need to compute [itex]<A>[/itex] as the sum of three separate integrals?
     
  7. Mar 13, 2012 #6
    yes.
     
  8. Mar 13, 2012 #7
    Thank you so much! I understand the general idea much better now.
     
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