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Projection operator

  1. Dec 1, 2013 #1
    Hello,

    Suppose P is a projection operator.
    How can I show that I+P is inertible and find (I+P)^-1?
    And is there a phisical meaning to a projection operator?

    (Please be patient I have just started with QM).

    Thanks.
    Y.
     
  2. jcsd
  3. Dec 1, 2013 #2

    bhobba

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    Science Advisor
    Gold Member

    Well I + |u><u| is diagonalizeable, hence invertable. If A is diagonalizeable it is of the form P(-1)DP where D is a diagonal matrix and hence easily and obviously invertable (eigenvalues non zero). Simply take the inverse to get P(-1)D(-1)P.

    |u><u| is a projection operator and represents an operator that as an observable gives the expected value of observing a state to determine if its in state |u> - with 1 if it is in that state and zero otherwise ie the expected number of times it gives a true result.

    This follows directly from the Born rule, which is the expected value of observing a system in state P with observable O is trace(OP).

    Actually a projection operator in general form is Ʃ |ui><ui|, but I will leave you to do the grunt work of generalizing it - it's a good exercise.

    Thanks
    Bill
     
    Last edited: Dec 1, 2013
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