1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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>

    giving:

    α=aβ & β = aα

    thus,

    β=a2β

    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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Eigenvalues of operator in dirac not* (measurement outcomes)
Loading...