# Ferromagnet problem

See attached file for full problem description.

Ferromagnet

© BrainMass Inc. brainmass.com June 3, 2020, 8:06 pm ad1c9bdddfhttps://brainmass.com/physics/energy/ferromagnet-problem-123589

#### Solution Preview

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,

eqno(2.1)

$$

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.

eqno(2.2)

$$

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 = ...

#### Solution Summary

With full explanations and calculations, the problems are solved.