Phy5646/character: Difference between revisions

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<math>=\sum_{\mathit{l=\left | j_{1}-j_{2} \right |}}^{\mathit{j_{1}+j_{2}}}\; \; \frac{\epsilon ^{\mathit{l+1}}-\epsilon^{\mathit{-l}}}{\epsilon -1}</math>
<math>=\sum_{\mathit{l=\left | j_{1}-j_{2} \right |}}^{\mathit{j_{1}+j_{2}}}\; \; \frac{\epsilon ^{\mathit{l+1}}-\epsilon^{\mathit{-l}}}{\epsilon -1}</math>


<math>= \chi _{\mathit{j_{1}+j_{2}}}(\phi)\; +...\: +\:\chi _{\left |\mathit{j_{1}-j_{2}}\right |}(\phi)</math>
<math>=\chi _{\mathit{j_{1}+j_{2}}}(\phi)\; +...\: +\:\chi _{\left |\mathit{j_{1}-j_{2}}\right |}(\phi)</math>


This shows that the product representation is reducible to a sum of the known irreducible representations:
This shows that the product representation is reducible to a sum of the known irreducible representations:

Revision as of 20:23, 25 April 2010

Angular Momentum Addition by Characters

Rotation matrices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{{\mathit{i}}{\vec{\omega }.\vec{J}}}} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2\mathit{j}+1)\times (2\mathit{j}+1)} matrix functions of rotating angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\omega }} in some representation of spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{j}} . To indicate more explicitly the representation we are in we write them as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{\mathit{j}}(\vec{\omega })} .Let us define the character by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi _{\mathit{j}}(\vec{\omega })=\mathit{tr}\; D_{\mathit{j}}(\vec{\omega})} For a rotation about the z-axis, the rotation matrix is diagonal

and the character is easy to compute

But any rotation may be brought to diagonal form by a similarity transform, so this is the most general character. It depends on the rotation angle, not the direction.

If we tensor together the states and , they transform under the tensor product representation .These matrices have characters which are just products of the elementary characters.

.

This expression can then be manipulated into a sum of the irreducible representation characters:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\chi _{\mathit{j_{1}+j_{2}}}(\phi)\; +...\: +\:\chi _{\left |\mathit{j_{1}-j_{2}}\right |}(\phi)}

This shows that the product representation is reducible to a sum of the known irreducible representations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{\mathit{j_{1}}}\times D_{_{\mathit{j_{2}}}}\; =\; D_{\mathit{j_{1}+j_{2}}}+\; ...\; +D_{\mathit{j_{1}-j_{2}}}.}

This is another way of aproach to the essential content of the angular momentum addition theorem.