Homework Help: Addition of spin and degeneracy

1. May 2, 2017

Nico045

1. The problem statement, all variables and given/known data

S=1/2
I1=I2=1/2 (nuclear spin)

I=I1+I2
J=S+I

2. Relevant equations

H= a S . I1 + a S.I2

3. The attempt at a solution

A) Find the values of I, the eigenvalues of I2, their degree of degeneracy and show that [H,I2]=0

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.In

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 problem

C) Find the eigenvalues of H= a ( S . I1 + S.I2) and their degeneracy

I 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.

2. May 4, 2017

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 ?