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Homework Help: Quantum hw problem

  1. Dec 19, 2006 #1
    1. The problem statement, all variables and given/known data

    I can't figure this one out: A two-state quantum system is psi = c1|1> + c2|2> and the states are orthonormal and c1 and c2 are complex and normalized condition is c1c1* + c2c2* = 1

    where * denotes complex conjugate

    The probability of measuring the quantum system to be in state |2> is given by the expectation value of a certain operator P_2

    What is the operator P_2


    2. Relevant equations

    standard expectation value and operator equations

    3. The attempt at a solution

    I think i figured that the probability to measure the system in state |2> is c2c2*

    then c2c2* = <P_2> = <psi|P_2|psi> but going through that I can't figure out how to determine P_2 .... i think i am missing something simple and any help would be very much appreciated! :)
     
  2. jcsd
  3. Dec 19, 2006 #2

    OlderDan

    User Avatar
    Science Advisor
    Homework Helper

    Have you done anything with projection operators?
     
  4. Dec 19, 2006 #3
    Let's try thinking about this just a bit.

    What's the expectation value of an operator for a state [tex] | a\rangle[/tex]? It's [tex] \langle a | \hat{\mathcal{O}} | a \rangle [/tex]. Well, we can write this as [tex] \sum_i \langle a | o_i \rangle \langle o_i \rangle \hat{\mathcal{O}} | a \rangle [/tex] where [tex]\mathcal{O}[/tex] is the observable in question, and the [tex]|o_i \rangle [/tex] are the eigenstates of the observable. The the above sum becomes

    [tex]
    \langle \hat{\mathcal{O}} \rangle = \sum_i | \langle o_i | a \rangle |^2 o_i
    [/tex]

    If we interpret this probabilistically, this is the sum over the probability of measuring the value [tex]o_i[/tex] times this value, which is pretty much the definition of an average.

    So what's the [tex]\hat{\mathcal{O}}[/tex] that you want?
     
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