1. The problem statement, all variables and given/known data three distiguishable spin 1/2 particles interact via [tex] H = \lamda ( S_1 \cdot S_2 + S_2 \cdot S_3 + S_3 \cdot S_1 ) [/tex] a) What is the demension of the hilbert space? b) Express H in terms of [tex] J^2 [/tex] where [tex] J = S_1 + S_2 + S_3 [/tex] c) I then need to find the energy and eigenstates, but i think i can due this once i know the hamilitonian. 2. Relevant equations 3. The attempt at a solution a) would this be 6D, 2 from each particle? or (2*3/2 + 1) = 4 or am i all wrong? b) [tex] J^2 = S_1^2 + S_2^2 + S_3^2 + 2S_1S_2 + 2S_3S_2 + 2S_1S_3 [/tex] [tex] H = \lamda (1/2 J^2 - _1^2 - S_2^2 - S_3^2) [/tex] but i still have S's in my H. Is this ok? I feel like its not. should i use.. [tex] J^2 + \hbar J_z + J_+J_- [/tex]? why cant i just write everything as a 2x2 matrix, for the energies, then solve it that way with out using J^2?