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Homework Help: Question concerning Hamiltonian and eigenstates

  1. Sep 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Two spin-1/2 particles are placed in a system described by Hamiltonian H=S(x1)S(x2), (S(x) being the spin operator in the x-direction). States are written like |[tex]\uparrow\downarrow[/tex]>, (and can be represented by 2 x 2 matrix) so that there are 4 possible states. (|[tex]\uparrow\uparrow>, |\uparrow\downarrow>, |\downarrow\downarrow>, |\downarrow\uparrow>[/tex]

    Given: |[tex]\phi> = |\uparrow\downarrow> - |\downarrow\uparrow>[/tex]

    Find the normalized eigenstate of |[tex]\phi[/tex]>
    2. Relevant equations

    S(x1) and S(x2) are both matrices represented by [tex]\hbar[/tex]/2 *
    0 1 (I don't know how to write matrices), where x1 operates on the first particle, and x2 on the second.
    1 0

    3. The attempt at a solution

    S(x) inverts the spin, and multiplies the state by [tex]\hbar/2[/tex]; there are 2 spin operators working on the different states so that the whole state in the end will be multiplied by the square of that: [tex]\hbar\stackrel{2}{}/4[/tex].
    The difficulty is that I don't understand what is meant by normalized eigenstate. I do end up with a eigenstate (or energy) of -[tex]\hbar\stackrel{2}{}/4[/tex] by calculating <[tex]\phi|\phi>[/tex], but I'm quite sure that's not the definite answer, because something still needs to be normalized.
  2. jcsd
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