3.4. Quantum magnets Consider the Hamiltonian H = −1 2

R,R J(R−R

)

 S(R) ·  S(R

)

where J(R−R

) > 0 and  S(R) is the quantum operator of spin S.

a. Prove initially the following result: the largest (smallest) diagonal element that a Hermitian operator can have is equal to its largest (smallest) eigenvalue.

b. Use this result to prove that, for R = R,   S(R) ·  S(R ) ≤ S2.

c. Let | SR be the eigenvectors Sz(R) with the maximum eigenvalue Sz(R) | SR = S | SR.
Prove that the state | 0 =*i | SR is eigenvector of the Hamiltonian with eigenvalue E0 = −1 2 S2
R,R J(R−R ).
Hint. Express the Hamiltonian in terms of the ladder operators S±(R) = (Sx ±iSy)(R)
and use the condition S+(R) | SR = 0.

d. Using the result of

a, show that E0 is the smallest eigenvalue of the Hamiltonian.