please answer the question below with complete steps and solution A particle has
ID: 3896727 • Letter: P
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please answer the question below with complete steps and solution
A particle has orbital angular momentum given by the quantum number 1 = 3 and spin angular momentum given by the quantum number s = 1. How many distinct states are there with different values for the z components of the orbital and spin angular momenta? What are the possible values for the quantum number j that describes the total angular momentum of the particle? How many distinct states are there with different values for the magnitude and z component of total angular momentum? (Note that the rules for the addition of angular momenta given by Eq. (8.6) are such that, when angular momenta with quantum numbers l and s are combined to give total angular momenta with quantum numbers j=l + s,l + s-l, ... |/ - j|, then the number of distinct states with different values for m/ and ms is equal to the number of distinct states with different values for j and mj.)Explanation / Answer
(a) ORBITAL : ml = -3 to ml = +3
l = 3 so ml = -3,-2,-1,0,1,2,3 so there are 7 states
Spin : 2s+1 = 3 states
(b) j
J = | l+s | , l+s-1 , ... |l-s|
J = 2,3,4 hence 4 states
(c) J = 2 , mj = -2,-1,0,1,2 hence 5 states
J = 3 , mj = -3,-2,-1,0,1,2,3 hence 7 states
J = 4 , mj = -4,-3,-2,-1,0,1,2,3,4 hence 9 states
So in total there are 21 states
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