Phy5646/character: Difference between revisions
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<math>D_{\mathit{j}}(\phi\hat{z})=\; diag(e^{\mathit{ij}\phi}e^{\mathit{i(j-1)}\phi}...e^{-\mathit{ij}\phi})</math> | <math>D_{\mathit{j}}(\phi\hat{z})=\; diag(e^{\mathit{ij}\phi}e^{\mathit{i(j-1)}\phi}...e^{-\mathit{ij}\phi})</math> | ||
and the character is easy to compute | and the character is easy to compute | ||
<math>\chi_{\mathit{j}}(\phi)=\sum_{\mathit{m=-j}}^{\mathit{j}}\; e^{\mathit{im\phi}}</math> | <math>\chi_{\mathit{j}}(\phi)=\sum_{\mathit{m=-j}}^{\mathit{j}}\; e^{\mathit{im\phi}}</math> | ||
<math>=\frac{\epsilon^{\mathit{j+1}}-\epsilon^{-\mathit{j}}}{\epsilon-1}\; \; \; where\; \; \epsilon=e^{\mathit{i\phi}}</math> | <math>=\frac{\epsilon^{\mathit{j+1}}-\epsilon^{-\mathit{j}}}{\epsilon-1}\; \; \; where\; \; \epsilon=e^{\mathit{i\phi}}</math> | ||
<math>=\frac{sin(\mathit{j+\frac{1}{2})\phi}}{sin(\frac{\phi}{2})}</math> | <math>=\frac{sin(\mathit{j+\frac{1}{2})\phi}}{sin(\frac{\phi}{2})}</math> | ||
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. | 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 <math> |\mathit{j_{1}m_{1}}\rangle </math> and <math>|\mathit{j_{2}m_{2}}\rangle</math>, they transform under the tensor product representation <math>D_{\mathit{j_{1}}}\times D_{\mathit{j_{2}}}</math>. | |||
If we tensor together the states <math> |\mathit{j_{1}m_{1}}\rangle </math> and <math>|\mathit{j_{2}m_{2}}\rangle</math>, they transform under the tensor product representation | |||
<math>D_{\mathit{j_{1}}}\times D_{\mathit{j_{2}}}</math>. | |||
Revision as of 19:32, 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
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}}}} .