Eigenvalue of an hamiltonian with spin

[Nico045]

Homework Statement

Finding eigenvalues of an hamiltonian

Homework Equations

H = a S²z + b Sz
(hbar = 1)
what are the eigenvalues of H in |S,M> = |1,1>,|1,0>,|1,-1>

The Attempt at a Solution

H|1,1> = (a + b) |1,1>
H|1,0> = 0
H |1,-1> = (a-b) |1,-1>

which gives directly the energy :

a+b , 0 , a-b

is that right ? Actually I'm not sure about :

S²z |S,M> = M² |S,M>After, I need to calculate the average value of the magnetic moment so I need to be sure of the energy

 

[kuruman]

S²z |S,M> = M² |S,M>

That is correct. Think of it as two successive applications of S_z. You are good so far.