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Addition of spin and degeneracy

  1. May 2, 2017 #1
    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


    Using 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.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. jcsd
  3. May 4, 2017 #2
    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 ?
     
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