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Expectation values and trace math

  1. Oct 27, 2007 #1
    1. The problem statement, all variables and given/known data


    How do I get the expectation value of operator [itex]\sigma[/itex] using density matrix [itex]\rho[/itex] in a trace: [itex]Tr\left(\sigma\rho\right)[/itex]

    I have [itex]\sigma[/itex] and [itex]\rho[/itex] in matrix form but how do I get a number out of the trace?
     
  2. jcsd
  3. Oct 27, 2007 #2

    nrqed

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    I am not sure I follow your question. Do you know what it means to take the trace of a matrix?
     
  4. Oct 27, 2007 #3
    I do if it involves just a bra and ket

    i.e. [tex]Tr\left(|a><b|\right) = <b|a>[/tex]

    I've been shown [itex]<\Lambda> = Tr\left(\Lambda\rho\right)[/itex]

    But I have rho and lambda in matrix form and not as a product of bra's and ket's
     
  5. Oct 27, 2007 #4

    Hurkyl

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    The trace is additive, so

    [tex]
    \mathrm{Tr}\left( \sum_i | a_i \rangle \langle b_i| \right) = \sum_i \mathrm{Tr}\left( | a_i \rangle \langle b_i| \right)
    [/tex]

    If you have a matrix, this greatly simplifies. It's just the sum of the diagonal entries.
     
    Last edited: Oct 27, 2007
  6. Oct 27, 2007 #5
    Ahh I remember that now.

    So I just take the matrix product [itex]\Lambda\rho[/itex] and then sum the diagonal entries to compute the trace?
     
  7. Oct 27, 2007 #6
    I got the correct answer. Thanks guys.
     
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