parts c and d

The quantum mechanical spin operators are the 2 x 2 matrices: where are the three 2 x 2 Pauli matrices. (a) Use the properties of the Pauli matrices to prove the important identity: (-7)(6-) = (-7)1+i (x 7) - where and are vectors. (b) Use this to show that the spin rotation operator R(a, ) = exp - S, for a rotation through angle a with axis defined by the unit vector = (sin ( cos o, sin sin o, cos 6), can be written as R(a,0,0) = cos 6 (a) (10) E la / cos sind e-io) ( 1) - i sin ) ( sin 0 cio"- cose ) (c) Compute the eigenvalues and eigenstates of this 2 x 2 matrix R. (You are welcome to use Math- ematica, or similar, or do it by hand). Comment on these results. (d) Suppose the Hamiltonian H involves spin only through a term coupling the spin to a uniform magnetic field aligned along the z direction. Describe the symmetry of this system under spin rotations. The quantum mechanical spin operators are the 2 x 2 matrices: where are the three 2 x 2 Pauli matrices. (a) Use the properties of the Pauli matrices to prove the important identity: (-7)(6-) = (-7)1+i (x 7) - where and are vectors. (b) Use this to show that the spin rotation operator R(a, ) = exp - S, for a rotation through angle a with axis defined by the unit vector = (sin ( cos o, sin sin o, cos 6), can be written as R(a,0,0) = cos 6 (a) (10) E la / cos sind e-io) ( 1) - i sin ) ( sin 0 cio"- cose ) (c) Compute the eigenvalues and eigenstates of this 2 x 2 matrix R. (You are welcome to use Math- ematica, or similar, or do it by hand). Comment on these results. (d) Suppose the Hamiltonian H involves spin only through a term coupling the spin to a uniform magnetic field aligned along the z direction. Describe the symmetry of this system under spin rotations