The total angular momentum vector J > of an electron is the sum of its spin (int

Question

The total angular momentum vector J> of an electron is the sum of its spin (intrinsic angular momentum) vector and its orbital angular momentum vector, J> = L> + S> However, since J> is itself an angular momentum vector, it has the same restrictions. Namely only one component (the z component) and its magnitude can be measured with certainty. The process of finding the possible z components and magnitudes of J vector from L vector and S vector is known as addition of angular momentum.

For this problem, suppose the electron is in the p-shell (l = 1).

(a) From the above equation we can conclude that Jz = Lz + Sz. From this result, list the possible values of the z components of Jvector.

(b) If the z components of J vector are given by hbarmj , where mj is the secondary total angular momentum quantum number for Jvector, what are the possible values of mj ?

(d) From the possible values of j in part (c), what are the possible values of the magnitude J of the vector J~?

Explanation / Answer

a)   J = L+S     and its magnitude

         J = sqrt(j(j + 1)) *

=>      For,   j = 3/2,1/2

=>     z components of Jvector   ------->      J = (sqrt(15)/2) *             ,    J = (sqrt(3)/2) *

b)      possible values of mj    =   3/2 , 1/2   , -3/2   , -1/2   .

c)     possible values that j could take   = 3/2 , 1/2   .

d)     possible values of the magnitude J    =>    J = (sqrt(15)/2) *     

        J = (sqrt(3)/2) *

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