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Finding Eigenvectors/values given matrix defined by bra-ket notation

  1. Feb 28, 2013 #1
    1. The problem statement, all variables and given/known data
    sorry about the lack of LaTex but I dont know how to do bra-ket notation in tex
    vectors |1> and |2> are a complete set of normalized basis vectors.
    the hamiltonian is defined as |1><1|-|2><2|+|1><2|+|2><1| find the eigenvalues and eigenvectors in ters of |1> and |2>


    2. Relevant equations
    for a normal matrix eigenvalues are

    Det(A-Iλ)=0 solve for λ
    Eigenvectors are then
    A*v=λv

    3. The attempt at a solution
    I tried just making up 2 generic 1X2 vectors and plugging it into mathematica, but its ugly and im assuming its not what they want, is there some stupid trick im missing here?
    If the first sign in the hamiltonian was + instead of minus I know the first set of terms would equal 1 but its minus so im just lost here
     
  2. jcsd
  3. Feb 28, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: Finding Eigenvectors/values given matrix defined by bra-ket notati

    I may be misunderstanding your hamiltonian but it looks to me like that corresponds to matrix
    [tex]\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}[/tex]

    If that is correct, then the characteristic equation is
    [tex]\left|\begin{array}{cc} 1-\lambda & 1 \\ 1 & -1-\lambda\end{array}\right|= (1-\lambda)(-1- \lambda)- 1= 0[/tex]
    [tex]= \lambda^2- 2= 0[/tex]

    so the eigenvalues are [itex]\sqrt{2}[/itex] and [itex]-\sqrt{2}[/itex].

    The corresponding eigenvectors are given by
    [tex]\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}\begin{bmatrix}x & y\end{bmatrix}= \sqrt{2}\begin{bmatrix} x \\ y \end{bmatrix}[/tex]
    so that [itex]x+ y= \sqrt{2}x[/itex] and [itex]x- y= \sqrt{2}y[/itex]

    and
    [tex]\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}\begin{bmatrix}x & y\end{bmatrix}= -\sqrt{2}\begin{bmatrix} x \\ y \end{bmatrix}[/tex]
    so that [itex]x+ y= -\sqrt{2}x[/itex] and [itex]x- y= -sqrt{2}y[/itex]
     
  4. Feb 28, 2013 #3
    Re: Finding Eigenvectors/values given matrix defined by bra-ket notati

    How did you know the hamiltonian corresponded to that matrix?
     
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