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Hermitian Operators Eigenvalues

  1. Oct 26, 2014 #1
    1. The problem statement, all variables and given/known data
    I have a hermitian Operator A and a quantum state |Psi>=a|1>+b|2> (so we're an in a two-dim. Hilbert space)
    In generally, {|1>,|2>} is not the eigenbasis of the operator A.

    I shall now show that the Eigenvaluse of A are the maximal (minimal) expection values <Psi|A|Psi>.


    3. The attempt at a solution
    I tried to calculate the expectation, which yields to:

    |a|^2 A_11 + |b|^2 A_22 +a b* A_12 +b a* A_21

    , where A_kl are the matrix elements of the operator A in the given basis of the Hilbert space.
    Now I could try to maximize this w.r.t to a and b, under the constraint that a^2 + b^2 =1. Didn't work very well...

    Does anyone have an idea?
     
  2. jcsd
  3. Oct 26, 2014 #2

    ShayanJ

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    Gold Member

    Show us what you did and what went wrong!
     
  4. Oct 26, 2014 #3
    Hey Shyan,

    thanks for your answer. I tried to find the maximum by using the Lagrange function, so:

    L=|a|^2 A_11 + |b|^2 A_22 +a b* A_12 +b a* A_21+lambda(a^2 + b^2 - 1)

    Now I got the problem that I don't know how derive L w.r.t c* (conjugated c). I also think that there is a better way to solve this exercise or?

    Best regards
     
  5. Oct 26, 2014 #4

    ShayanJ

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    Gold Member

    Hi
    The usual way of dealing with such a situation is to take c and c* as two independent variables(which is reasonable since c* is not a differentiable function of c). So you have a function of four independent variables to maximise with the constraint equation aa*+bb*=1.
     
  6. Oct 30, 2014 #5
    Thank you very much! :-)
     
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