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    Ferromagnet problem

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    Please see the attached file in proper format.

    I have not followed the Hint in question 3 as I think it is too cumbersome, and instead I offer a solution I like better for its simplicity.

    Here is the plain TEX source

    centerline{bf Spin $1over 2$ Ferromagnet }

    bf 1. rm

    The dimension of the Hilbert space at each site is $2cdot{1over 2} +1 = 2$, therefore the dimension of the Hilbert space for the entire system is $2^N$.

    bf 2. rm

    The Hamiltonian contains only the pair interactions
    vecsigma_acdotvecsigma_b = 4 vec S_acdotvec S_b
    = 2l[ (vec S_a + vec S_b)^2 - vec S_a^2 - vec S_b^2 r]
    = 2l[ vec S_{ab}^2 - vec S_a^2 - vec S_b^2 r] =
    = 2l[ vec S_{ab}^2 - S_a(S_a+1) - S_b(S_b+1) r]
    = 2l[ vec S_{ab}^2 - {3over 4} - {3over 4} r]
    = 2vec S_{ab}^2 - 3,
    where $vec S_{ab} = vec S_a + vec S_b$.
    For every such pair $a,b$, the state $|+_a,+_bra$ belongs to the $S_{ab}=1$ triplet, so that equation (2.1) gives
    vecsigma_acdotvecsigma_b = 2S_{ab}(S_{ab}+1) - 3
    = 1.
    Taking the full Hamiltonian and the full state $|+_1,+_2,...,+_Nra$, we see that the full state is an eigenstate of every pair term with eigenvalue $-J$, and therefore an eigenstate of the whole Hamiltonian with eigenvalue
    E_g = -Jsum_{j=1}^N 1 = ...

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