Phy5646/character

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

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}}(\phi\hat{z})=\; diag(e^{\mathit{ij}\phi}e^{\mathit{i(j-1)}\phi}...e^{-\mathit{ij}\phi})}

and the character is easy to compute

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}}(\phi)=\sum_{\mathit{m=-j}}^{\mathit{j}}\; e^{\mathit{im\phi}}}

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 =\frac{\epsilon^{\mathit{j+1}}-\epsilon^{-\mathit{j}}}{\epsilon-1}\; \; \; where\; \; \epsilon=e^{\mathit{i\phi}}}

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 =\frac{sin(\mathit{j+\frac{1}{2})\phi}}{sin(\frac{\phi}{2})}}

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 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_{1}m_{1}}\rangle } and 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_{2}m_{2}}\rangle} , they transform under the tensor product representation 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}}}} .These matrices have characters which are just products of the elementary 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 D_{\mathit{j_{1}}}\times D_{\mathit{j_{2}}} :\; \; \; \chi_{}\mathit{j_{1}\times\mathit{j_{2}}}(\phi)=\chi _{\mathit{j_{1}}}(\phi)\chi _{\mathit{j_{2}}}(\phi)} .

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_{j_{1}}(\phi)\chi_{j_{2}}(\phi)=(\sum_{\mathit{m_{2}=-j_{2}}}^{\mathit{j_{2}}}\; \; \epsilon^{\mathit{m_{2}}})\; \frac{\epsilon^{\mathit{j_{1}+1}}-\epsilon^{-\mathit{j_{1}}}}{\epsilon-1} }

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

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.