Phy5646/AddAngularMomentumProb: Difference between revisions

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Express <math>\ S^2 </math> as a matrix for two spin-1/2 particles in the direct product basis.
Express <math>\ S^2 </math> as a matrix for two spin-1/2 particles in the direct product basis.


1.) First express <math>\ S^2</math> in terms of <math>\ S_1^2</math>, <math>\ S_2^2</math>, <math>\ S_{1z}</math>, <math>\ S_{2z}</math>, <math>\ S_{1\plusmn}</math> and <math>\ S_{2\plusmn}: S^2 = (\vec{S_1} \cdot \vec{S_2})^2 = S_1^2 + S_2^2 +2\vec{S_1} \cdot \vec{S_2} = S_1^2 + S_2^2 + 2(S_{1x}S_{2x} + S_{1y}S_{2y} + S_{1z}S_{2z}) = S_1^2 + S_2^2 + </math>.
1.) First express <math>\ S^2</math> in terms of <math>\ S_1^2</math>, <math>\ S_2^2</math>, <math>\ S_{1z}</math>, <math>\ S_{2z}</math>, <math>\ S_{1\plusmn}</math> and <math>\ S_{2\plusmn}</math>:  
Now use the idendity
<math>\ S^2 = (\vec{S_1} + \vec{S_2})^2 = S_1^2 + S_2^2 +2\vec{S_1} \cdot \vec{S_2} = S_1^2 + S_2^2 + 2(S_{1x}S_{2x} + S_{1y}S_{2y} + S_{1z}S_{2z}) = S_1^2 + S_2^2 + 2S_{1z}S_{2z}+S_{1+}S_{2-}+S_{1-}S_{2+}</math>.
 
2.) Act with this on a 2-electron state with arbitrary <math>\ S_z </math> quantum numbers:
 




Revision as of 18:51, 25 April 2010

Based on exercise 15.1.1. from Principles of Quantum Mechanics, 2nd ed. by Shankar:

Express as a matrix for two spin-1/2 particles in the direct product basis.

1.) First express in terms of , , , , and : .

2.) Act with this on a 2-electron state with arbitrary quantum numbers:





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