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    A Two State System Spanned by Two Orthonormal Vectors

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    Consider a two state system spanned by two orthonormal vectors, |1> and |2>. The action of an operator  is defined via:

    Â|1> = 2|1> + i|2>
    Â|2> = -i|1> + 3|2>

    Find Â|ѱ> where

    |ѱ> = |1> + |2>

    Can you now verify your answer for Â|ѱ> by doing the calculation in matrix representation?

    And then compute the four "matrix elements":

    A_ij ≡ <i|Â|j>

    Where i, j = 1, 2.

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    Solution Summary

    The solution shows, step by step, how to get the matrix representation of an operator under the given basis and show that the result is equivalent to direct calculation.