Homework #11 (due 5/6/08)

1.       Transformation between LS coupling and JJ coupling.

Add four angular momenta with  =0, =1, s1=s2=1/2 to form total J.

(a)    Write down the quantum numbers in the LS coupling scheme, and in the JJ coupling scheme separately.  Note that the states are labelled by |LSJMJ> and |j1j2JMJ>. List all of them and identify the dimension of the Hilbert space.

(b)   For MJ=0, write down the total wavefunction  in terms of the angular momentum functions. Use Ylm for orbital angular momentum functions and  and β for spin 1/2 .  Note that for J=0 and J=2, the two functions in the LS and the JJ coupling schemes are identical. There are two J=1 states in each coupling scheme and they are not identical.

(c)    Show that the two J=1 MJ=0 states in the jj coupling can be written as the linear combination of the two J=1 MJ=0 states in the LS coupling. Find the coefficients.

Note that you need to get the C-G coefficients from the internet. The C-G coefficient can always be written as plus or minus of  , where p and q are integers. Express all the coefficients in this form in your answer.

2.       Let the magnetic dipole moment  = g1 j1+ g2 j2. Find the expectation value of the dipole moment with respect to the eigenstates of total angular momentum J and MJ.