Consider one-dimensional ferromagnet namely N spin-1/2 objects placed around a circle with the Hamil

hetriamhageh6k20

hetriamhageh6k20

Answered question

2022-05-17

Consider one-dimensional ferromagnet namely N spin-1/2 objects placed around a circle with the Hamiltonian
H = J n = 1 N S n S n + 1
where we assume the periodic boundary condition S N + 1 S 1 and J > 0
I'm trying to show that total spin ket is a good quantum number that is they commute with H and finding out the energy corresponding to
| ψ 0 = | 1 | 2 | N
By definition:
S 2 = ( n S n ) 2 = n S n 2 + i , j S i S j
The second term has our Hamiltonian but there are other terms also. I don't understand, How do I proceed from here?

Answer & Explanation

allstylekvsvi

allstylekvsvi

Beginner2022-05-18Added 16 answers

you should look at symmetries. The transformations that the total spin induces is a simultaneous rotations of all the spins. Such rotations leave the dot-product S i S j unchanged. Therefore this is a symmetry of the system, which means that S t o t = i S i are conserved, and represent good quantum numbers (of course one can only pick S t o t 2 and one of the vector component, say S t o t z )

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