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Eigenfunctions and dirac notation for a quantum mechanical system.

  1. Dec 2, 2013 #1
    QUESTION
    A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
    |n> with associate eigenvalues, En. The operator [itex]\widehat{A}[/itex] corresponds to an observable such that
    Aˆ|1> = |2>
    Aˆ|2> = |1>
    Aˆ|n> = |0>, n ≥ 3
    where |0> is the null ket. Find a complete orthonormal set of eigenfunctions for
    [itex]\widehat{A}[/itex]. The observable is measured and found to have the value +1. The system is unperturbed and then after a time t is remeasured . Calculate the probability
    that +1 is measured again.


    I would really appreciate any guidance on this. I cannot find this in my book at all. I know my teacher has spoken about it bra-ket notation and such. but it never really made sense. where can I find examples of problems like this but not this.
     
  2. jcsd
  3. Dec 2, 2013 #2

    Dick

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    You could start out by telling me what ##{\widehat A}(|1>+|2>)## and ##{\widehat A}(|1>-|2>)## are.
     
  4. Dec 3, 2013 #3
    Aˆ(|1>+|2>) = |2> + |1> ?
    Aˆ(|1>−|2>) = |2> - |1> ? I really don't know
     
  5. Dec 3, 2013 #4
    I found this I feel like it should help me but I don't know...

    For every observable A, there is an operator [itex]\hat{A}[/itex] which acts upon the
    wavefunction so that, if a system is in a state described by |ψ>, the
    expectation value of A is
    <A>= <ψ|[itex]\hat{A}[/itex]|ψ>= ∫ dx ψ*(x) [itex]\hat{A}[/itex] ψ(x)

    that integral is from -∞ to ∞
     
  6. Dec 3, 2013 #5
    ok so I am reading a little more and.

    Aˆ(|1>+|2>) = Aˆ|1>+Aˆ|2> ==>α=1, β=1
    Aˆ(|1>−|2>) = Aˆ|1>-Aˆ|2> ==>α=1,β=-1

    <1|2> = <-2|1>


    idk this does not really make sense to me
     
  7. Dec 3, 2013 #6

    Dick

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    Yes, A^(|1>+|2>)=|1>+|2> and A^(|1>-|2>)=|2>-|1>=(-1)*(|1>-|2>). What does that tell about eigenvalues and eigenvectors of A^?
     
  8. Dec 3, 2013 #7
    that they are linear and Hermitian

    or I guess that's for A^.....
     
  9. Dec 3, 2013 #8

    Dick

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    Be more specific. Looking at those two equations I see two eigenvectors of A^ and their corresponding eigenvalues.
     
  10. Dec 3, 2013 #9
    maybe that En is 1 and 2 ?
     
  11. Dec 3, 2013 #10

    Dick

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    Noo. x is an eigenvector of A^ if A^(x)=cx for some constant c. What does A^(|1>+|2>)=|1>+|2> tell you?
     
  12. Dec 3, 2013 #11
    is the probability that |1> will give 2 at t=0 |En|2 and that equals 1 ?
     
  13. Dec 3, 2013 #12
    I guess that means that α and β are 1?
     
  14. Dec 3, 2013 #13

    Dick

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    That's still not saying anything about eigenvalues or eigenvectors. Look, A^(|1>+|2>)=|1>+|2> tells me that |1>+|2> is an eigenvector of A^. Why do I say that and what's the corresponding eigenvalue?
     
  15. Dec 3, 2013 #14
    I don't know I give up. thanks for trying man.
     
  16. Dec 3, 2013 #15

    Dick

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    Yeah, something isn't clicking here. But in case you decide to take another crack at it, I'm trying to get you to see that |1>+|2> is an eigenvector of A^ with an eigenvalue of +1 and |1>-|2> is an eigenvector of A^ with an eigenvalue of -1. Review the definition of eigenvector, there's nothing very hard about this part. You are just getting confused by other stuff.
     
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