Question

The Hamiltanion due to interaction of particle of spin S with a magnetic field B is 

given by H=-S.B . Where S is the spin.Calculate the commutator [S,H]?

Answer 1

Hamiltonian can be written as

$H=-S_zB_z$

When magnetic field is in z direction.

Now the commutator [S,H] is given by

By components form x y z

$[S_x, -S_zB_z]=-B_z[S_x, S_z]$

=$i\hbar B_zS_y$

$[S_y, -S_zB_z]=-B_z[S_y, S_z]$

=$-i\hbar B_zS_x$

$[S_z, -S_zB_z]=-B_z[S_z,S_z]=0$

Therefore adding of all above three commutator gives the final answer as below

the commutator [S,H] =

$[S_x, -S_zB_z]+[S_y, -S_zB_z]$

$+[S_z, -S_zB_z]$

=$i\hbar B_z(S_y-Sx)$