Solving Spin 1/2 Interactions with Hilbert Space Dimensions and J$^2$

[marlow6623]

Homework Statement

three distiguishable spin 1/2 particles interact via

$$H = \lamda ( S_1 \cdot S_2 + S_2 \cdot S_3 + S_3 \cdot S_1 )$$

a) What is the demension of the hilbert space?

b) Express H in terms of $$J^2$$ where $$J = S_1 + S_2 + S_3$$

c) I then need to find the energy and eigenstates, but i think i can due this once i know the hamilitonian.

Homework Equations

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) $$J^2 = S_1^2 + S_2^2 + S_3^2 + 2S_1S_2 + 2S_3S_2 + 2S_1S_3$$

$$H = \lamda (1/2 J^2 - _1^2 - S_2^2 - S_3^2)$$

but i still have S's in my H. Is this ok? I feel like its not.

should i use.. $$J^2 + \hbar J_z + J_+J_-$$?

why can't i just write everything as a 2x2 matrix, for the energies, then solve it that way without using J^2?

 

[marlow6623]

so i know now that a) is 8D

and i think that H = J^2 - S^2 - S^2 - S^2