Finding angle between L and S angular momentum.

Oct 2014
18
0
In a many electron atom the orbital spin and total angular momenta are denoted by L,S and J respectively.If L=2,
S=1, and J=2, find the angle between L,S
using vector atomic model.
Attempt:
J=L+S
J.J=|L|^2 +|S|^2 + 2|L|.|S| cosθ.
So cosθ= [|J|^2 -|L|^2-|S|^2]/ [2|L|.|S| ]
Now |L|=([L (L+1)]^1/2 ) hbar
|S|=([S(S+1)]^1/2 ) hbar ,|J|=([J(J+1)]^1/2 ) hbar
So cosθ=[J(J+1) - L (L+1)-S (S+1)]/[2([L (L+1)]^1/2 )×([S (S+1)]^1/2 ) ]
Now cosθ= 6-6-2/2×(6×2)^1/2 =-0.2886
θ=106.78º.Have I done this correctly? I couldn't find a suitable worked example to be sure by myself.
 

topsquark

Forum Staff
Apr 2008
2,978
631
On the dance floor, baby!
In a many electron atom the orbital spin and total angular momenta are denoted by L,S and J respectively.If L=2,
S=1, and J=2, find the angle between L,S
using vector atomic model.
Attempt:
J=L+S
J.J=|L|^2 +|S|^2 + 2|L|.|S| cosθ.
So cosθ= [|J|^2 -|L|^2-|S|^2]/ [2|L|.|S| ]
Now |L|=([L (L+1)]^1/2 ) hbar
|S|=([S(S+1)]^1/2 ) hbar ,|J|=([J(J+1)]^1/2 ) hbar
So cosθ=[J(J+1) - L (L+1)-S (S+1)]/[2([L (L+1)]^1/2 )×([S (S+1)]^1/2 ) ]
Now cosθ= 6-6-2/2×(6×2)^1/2 =-0.2886
θ=106.78º.Have I done this correctly? I couldn't find a suitable worked example to be sure by myself.
Your question is ill posed. There are a number of possible answers to your question.

Your overall method is okay but the value of l is not fixed when we give a value for |L|. In your example, |L| = 2 so we have three possible values for l: l = 0, 1, 2. These are the kinds of values you use in your cosine equation: we are not using J, L, and S...we are using j, l, and s. We can generate a list of possible values for cos(t) (I am using t = theta). The j(j + 1) equation is better to use as we might have l = 0 and/or s = 0.

For j = 1, l = 1, and s = 1 we have cos(t) = 1/2. For j = 2, l = 2, and s = 1 we have that cos(t) = -1/(2 sqrt(3)). As these two angles are not the same there is more than one answer to your question.

(You can also do this using mj = ml + ms. Given L = 2 implies l = 0, 1, 2 implies ml = -2, -1, 0, 1, and 2 and similar for J and S.)

-Dan

Edit: Apparently the codecogs website has changed something. You can barely make out the equation. It is the same as your J(J + 1) equation using the cosine rule, but with lowercase letters in place of the capitals.
 

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