# Homework Help: Question concerning Hamiltonian and eigenstates

1. Sep 11, 2010

### Gordijnman

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 |$$\uparrow\downarrow$$>, (and can be represented by 2 x 2 matrix) so that there are 4 possible states. (|$$\uparrow\uparrow>, |\uparrow\downarrow>, |\downarrow\downarrow>, |\downarrow\uparrow>$$

Given: |$$\phi> = |\uparrow\downarrow> - |\downarrow\uparrow>$$

Find the normalized eigenstate of |$$\phi$$>
2. Relevant equations

S(x1) and S(x2) are both matrices represented by $$\hbar$$/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 $$\hbar/2$$; 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: $$\hbar\stackrel{2}{}/4$$.
The difficulty is that I don't understand what is meant by normalized eigenstate. I do end up with a eigenstate (or energy) of -$$\hbar\stackrel{2}{}/4$$ by calculating <$$\phi|\phi>$$, but I'm quite sure that's not the definite answer, because something still needs to be normalized.