What Are the Eigenvalues and Degeneracies in Spin and Degeneracy Calculations?

[Nico045]

Homework Statement

S=1/2
I₁=I₂=1/2 (nuclear spin)

I=I₁+I₂
J=S+I

Homework Equations

H= a S . I₁ + a S.I₂

The Attempt at a Solution

A) Find the values of I, the eigenvalues of I², their degree of degeneracy and show that [H,I²]=0Using momentum addition I get
I = 0 or 1
I² has for eigenvalues : ħ²I(I+1)

degree of degeneracy :
for I=1 ==> d=3
for I=0 ==> d=1[H,I²]= [H, I₁²] +[H,I₂²] + 2 [H,I₁I₂] = [H, I₁²] +[H,I₂²]+ 2 I₁ [H,I₂] + 2 I₂ [H,I₁]

H= a S . I₁ + a S.I₂

I don't see how to calculate this since we have S.I_n


B) Find the values of J, the eigenvalues of J², their degree of degeneracy and show that [H,J²]=0

J=S+I , it gives 2 values :
J= 1/2 { 1-1/2 // 0+1/2 }
J= 3/2 { 1+1/2 }
J² has for eigenvalues : ħ²J(J+1)

degree of degeneracy :
J=1/2 ==> d=6
J=3/2 ==> d=2

[H,J²]= [H, I²] + [H,S²] + 2 [H,I.S]
same problemC) Find the eigenvalues of H= a ( S . I₁ + S.I₂) and their degeneracyI guess I have to use J² :

S.I = S.I₁ + S.I₂ = 1/2 ( J^2 -S^2 -I^2 )

and the eigenvalues of H are : a * ħ²/2 ( J(J+1) - S(S+1) - I(I+1) )I am not sure to know how degeneracy is calculated for example : in (A) d(I) = 4 so d( I²) = 16 ?D) is {J², J₂z} forming a CSCO ? What about {I²,J², J₂z} ?

This concept isn't very clear to me.

 

[Nico045]

I managed to solve a few things but the remaining problem is about degeneracy :

for I₁ = I₂ = 1/2
I=I₁+I₂

what are the degeneracies for I² ?
for I we have (2I₁+1)*(2I₂+1) = 2*2 = 4
but for I² is it just d(I) * d(I) = 16 ?