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Urgent: Confusing matrix operator equation

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm new to matrix mechanics in quantum mechanics and I admit that linear algebra is the weakest part of my maths toolkit but I have an operator to convert to matrix form. I'm about 85% sure I've done it right, however I can't solve for the eigenvectors. I'm sure I'm missing something obvious and was wondering if someone could help me find it.

    The operator is a Hamiltonian of the form.

    [tex] H=c \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] [/tex]

    If I solve for the eigenvalues I get [itex] \lambda = \pm c [/itex]

    Which seems right. However, if I try to solves for the eigenvectors using [itex] H'=(H-\lambda I) [/itex] and [itex]H'x = 0 [/itex] for the eigenvalue +c, I get a set of equations that don't make sense to me.

    [tex]H'=c \left[ \begin{array}{cc} 1-1 & 1 \\ 1 & -1-1 \end{array} \right]= \left[ \begin{array}{cc} 0 & 1 \\ 1 & -2 \end{array} \right] [/tex]

    Using x=(x,y), and if I'm reading this right, this would give a set of equations where y=0 but also y=x. This would imply the answer is a null vector which makes no sense.

    Can anyone see where I've gone wrong/what I'm missing?
     
  2. jcsd
  3. Sep 28, 2011 #2
    Nevermind , apparently I had forgotten how to take a determinant.

    Thanks anyway.
     
  4. Sep 28, 2011 #3
    Stuck again because now I need to solve this equations:

    [tex] (1-\sqrt{2})x+y = 0 ; x+(-1-\sqrt{2})y = 0 [/tex]


    I've been pounding my head against it buit can't seem to find the way, standard Gaussian techniques seem to be leading to a mess.
     
  5. Sep 28, 2011 #4
    Just set x or y to one, then you get the value of the other, (1, something) or (something, 1)
     
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