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QM: Probabilities for an operator

  1. Mar 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    I have the following state at t=0 in a 3D Hilbert-space (it is in the eigenspace of the 3x3 Hamiltonian):

    [tex]
    \left| \psi \right\rangle = \frac{1}{{\sqrt 2 }}\left| \psi \right\rangle _1 + \frac{1}{2}\left| \psi \right\rangle _2 + \frac{1}{2}\left| \psi \right\rangle _3.
    [/tex]

    Now I have an operator representing an observal given by:

    [tex]
    \hat A = \left( {\begin{array}{*{20}c}
    1 & 0 & 0 \\
    0 & 0 & 1 \\
    0 & 1 & 0 \\
    \end{array}} \right)
    [/tex]

    I have to find the possible eigenvalues of A and the corresponding probabilities.

    3. The attempt at a solution

    The possible eigenvalues of A are easy. I am more concerned about the probabilities. I reasoned that they are the same, because the above state at t=0 is independent of the Hilbert space in is written in. So it will look the same if I write it in the eigenspace of A, but the unit-vectors (i.e. the possible states) are now different.

    So my attempt: The probabilities are the same, i.e. 1/2, 1/4 and 1/4. Can you confirm this?

    Thanks in advance.

    Best regards,
    Niles.
     
  2. jcsd
  3. Mar 17, 2009 #2
    Ok, I think I got it now.

    My above suggestion is not correct. I think I have to find the coefficients by writing the state |psi> in the basis of A by the following method:

    [tex]
    a_{cofficient\,\,1} = \left\langle {{\psi _{a\,\,1} }}
    \mathrel{\left | {\vphantom {{\psi _{a\,\,1} } \psi }}
    \right. \kern-\nulldelimiterspace}
    {\psi } \right\rangle = \sum\limits_i {\left\langle {{\psi _{a\,\,1} }}
    \mathrel{\left | {\vphantom {{\psi _{a\,\,1} } {\psi _i }}}
    \right. \kern-\nulldelimiterspace}
    {{\psi _i }} \right\rangle \left\langle {{\psi _i }}
    \mathrel{\left | {\vphantom {{\psi _i } \psi }}
    \right. \kern-\nulldelimiterspace}
    {\psi } \right\rangle },
    [/tex]

    where psi_i are the eigenfunctions of the Hamiltonian and psi_{a 1} are the eigenfunctions of A. The above method gives me the first coefficient for the representation of |psi> in the basis of A.

    Is this correct?
     
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