Atomic and Solid State Physics Atomic and Solid State Physics Help Forum Jan 4th 2015, 06:40 AM #1 Junior Member   Join Date: Oct 2014 Posts: 18 Finding angle between L and S angular momentum. 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.   Jan 13th 2015, 11:37 AM   #2

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 Originally Posted by pranimaboity2050 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 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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Last edited by topsquark; Jan 13th 2015 at 11:43 AM.  Tags angle, angular, finding, momentum ,
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angle between the total and orbital angular momentum vector

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