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Eigenvalues and Probabilities

  1. Nov 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state:

    |psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>.

    Provide the possible measured values of a and corresponding probabilities.


    2. Relevant equations

    (A)(psi) = sum[anCnPsin]

    3. The attempt at a solution
    It would seem that A1 = (3)^(-1/2), A2 = 2((3)^(-1/2), and A3 = (5/3)^(1/2), but the state is not normalized so the probabilities don't add up to one... so I am confused about how to handle this.
     
  2. jcsd
  3. Nov 13, 2009 #2

    MathematicalPhysicist

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    Gold Member

    So normalise |psi>.
    BTW, the probabilities of recieving the eigenvalue a_k is the square of the coefficient of |A_k> in |psi>.
     
  4. Nov 13, 2009 #3

    MathematicalPhysicist

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    Gold Member

    In general, if [tex]|\psi>=\sum c|n>[/tex] the the probability to find |psi> in the state |n> (and measuring its eigenvalue) is |c|^2=cc* where c* is the complex conjugate of c.
     
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