Which boundary condition are you referring to?? If it's periodic then your calculation is right but if it's anti-periodic then your calculation is wrong.
I know physics, I graduated from physics. Currently doing masters in Theoretical Physics. But on the other hand my knowledge of biology is next to zero. How should I proceed? A suggested list of book with ordering will be helpful.
Yes we demand it, if it's not the case it's not much of a physics anymore. But even if law of physics changes(energy not conserved anymore) it changes slowly. The apple has not gone to heaven as far as civilization can remember :p
Did you understood the transfer matrix technique? the first n-1 spin sum just becomes the normal trace of the transfer matrix. But then you have \sigma_m, but the possible value of \sigma_m are 1 and -1, with introduction of pauli matrix(+1 and -1 remember?), it just becomes a normal trace once...
You don't assume that |\psi>=\alpha |1>+ \beta |2>, there is a reason for that. The completeness relation gives,
I=|1><1|+|2><2| [look up completeness relation if you don't know about it.]
which means, |\psi>=I|\psi>=|1><1|\psi>+|2><2|\psi>
or, |\psi>=\alpha |1>+ \beta |2> ,
where...