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Homework Help: Eigenvalues of operator in dirac not* (measurement outcomes)

  1. Apr 8, 2015 #1
    1. The problem statement, all variables and given/known data
    A measurement is described by the operator:

    |0⟩⟨1| + |1⟩⟨0|

    where, |0⟩ and |1⟩ represent orthonormal states.

    What are the possible measurement outcomes?

    2. Relevant equations

    Eigenvalue Equation: A|Ψ> = a|Ψ>

    3. The attempt at a solution

    Apologies for the basic question but just very unsure of myself when it comes to this stuff. I have had a go and come up with a solution but I'm not sure if its right so any help would be much appreciated.

    We're told that:
    A = |0⟩⟨1| + |1⟩⟨0|

    Can I then assume something like: Ψ = α|1> + β|0>?

    using this I've then solved the eigenvalue equation, AΨ=aΨ, and found:

    α|0> + β|1> = aα|1> + aβ|0>


    α=aβ & β = aα



    a = (+-) 1

    hence, my eigenvalues are -1 and 1.

    and these are the possible outcomes?
  2. jcsd
  3. Apr 8, 2015 #2
    You don't assume that [itex]|\psi>=\alpha |1>+ \beta |2>[/itex], there is a reason for that. The completeness relation gives,

    [itex]I=|1><1|+|2><2|[/itex] [look up completeness relation if you don't know about it.]

    which means, [itex]|\psi>=I|\psi>=|1><1|\psi>+|2><2|\psi>[/itex]

    or, [itex] |\psi>=\alpha |1>+ \beta |2> [/itex],

    where [itex]\alpha=<1|\psi>[/itex]

    and [itex]\beta=<2|\psi>[/itex] are c-number.

    Except that there are no more problem with your work.
  4. Apr 8, 2015 #3


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    Yes, that looks just fine to me. The possible measurements of an experiment are the eigenvalues of the operator.
    Last edited: Apr 9, 2015
  5. Apr 9, 2015 #4
    thank you. will look up the completeness relation
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