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 &gt; 0I'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?

Question

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    &amp;gt;  0I&#039;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&#039;t understand, How do I proceed from here?

allstylekvsvi · Accepted Answer

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  )

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

Answered question

Answer & Explanation

New Questions in Electromagnetism