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Homework Help: Conditions for a density matrix; constructing a density matrix

  1. Aug 19, 2011 #1
    1. The problem statement, all variables and given/known data
    What are the conditions so that the matrix [itex]\hat{p}[/itex] describes the density operator of a pure state?


    2. Relevant equations
    [PLAIN]http://img846.imageshack.us/img846/2835/densitymatrix.png [Broken]
    [itex]p=\sum p_{j}|\psi_{j}><\psi_{j}[/itex]

    3. The attempt at a solution
    I know that tr([itex]\rho[/itex])=1 for pure states.
    But I do not know how to construct the density matrix. Or is it [itex]\hat{p}[/itex] already?
    In my book, there is the example "hermitian, tr=1, [PLAIN]http://img89.imageshack.us/img89/4346/densitymatrixexample.png", [Broken] but I do not know how they constructed that.

    Maybe I can just say a+d+e=1; c,b arbitrary?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 19, 2011 #2

    vela

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  4. Aug 20, 2011 #3
    What do you think about that?

    [PLAIN]http://img155.imageshack.us/img155/7663/densitymatrixexamplep2.png [Broken]
    a²+bc=a (a[itex]\neq[/itex]0) => [itex]a+\frac{bc}{a}=1[/itex]
    ab+bd=b (b[itex]\neq[/itex]0) => a+d=1 <=> a= 1-d <=> d=1-a
    ca+dc=c (c[itex]\neq[/itex]0) => a+d = 1
    cb+d²=2 (d[itex]\neq[/itex]0) => [itex]d+\frac{cb}{d}=1[/itex]
    e=e²

    a=0 v a= 1 - d
    b € |R
    c € |R
    d=0 v d= 1 - a
    e=0 v 1
     
    Last edited by a moderator: May 5, 2017
  5. Aug 20, 2011 #4

    vela

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  6. Aug 20, 2011 #5
    Ok, thanks, I have read that.

    with the condition for the trace:
    a+d+e=1 => e=0: a+d=1;
    e=1: a=d=0

    and since the matrix is hermitian and p^T=p:
    b = c, c € |R

    On the page you linked to it says only one element on the diagonale can be different from zero when you chose the right basis, but I think I can not change the basis here. So I think I can not further restrict it from that.
    Right?
     
  7. Aug 20, 2011 #6

    vela

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    For e=1, you can say a+d=0, which isn't quite the same as saying a=d=0.
    Both b and c can be complex. At best you can say c=b*.
    Yes, I agree. I think the problem wants you to find a general form for the density matrix.
     
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