Phy5646/AddAngularMomentumProb: Difference between revisions

Revision as of 17:31, 25 April 2010

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

Express $\ S^{2}$ as a matrix for two spin-1/2 particles in the direct product basis.

1.) First express $\ S^{2}$ in terms of $\ S_{1}^{2}$ , $\ S_{2}^{2}$ , $\ S_{1z}$ , $\ S_{2z}$ , $\ S_{1\pm }$ and $\ S_{2\pm }:S^{2}=({\vec {S_{1}}}\cdot {\vec {S_{2}}})^{2}=S_{1}^{2}+S_{2}^{2}+2{\vec {S_{1}}}\cdot {\vec {S_{2}}}=S_{1}^{2}+S_{2}^{2}+2(S_{1x}S_{2x}+S_{1y}S_{2y}+S_{1z}S_{2z})=S_{1}^{2}+S_{2}^{2}+$ . Now use the idendity

${\hbar }^{2}{\begin{pmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&2\\\end{pmatrix}}$

@@ Line 1: / Line 1: @@
 Based on exercise 15.1.1. from Principles of Quantum Mechanics, 2nd ed. by Shankar:
-Express <math> \frac{}{}S^2 </math> 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.
-<math> \frac{}{}S^2 = (\vec{S_1} \cdot \vec{S_2})^2 = S_1^2 + S_2^2 +2\vec{S_1} \cdot \vec{S_2}</math>
+.) 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>.
+Now use the idendity

Phy5646/AddAngularMomentumProb: Difference between revisions

Revision as of 17:31, 25 April 2010

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