Difference between revisions of "Forest AngMomRecoupling"

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The recoupling of two subsystems <math>\psi</math> with angular momenta <math>j_1</math> and <math>j_2</math> to a new system<math> \Psi</math> with total angular momentum <math>J</math> is written as
 
The recoupling of two subsystems <math>\psi</math> with angular momenta <math>j_1</math> and <math>j_2</math> to a new system<math> \Psi</math> with total angular momentum <math>J</math> is written as
  
−
<math>\Psi^{J}_{M} = \sum_{m_1,m_2} C^{j_1,j_2,J}_{m_1,m_2,M} \psi^{j_1}_{m_1} \psi^{j_2}_{m_2}</math>
+
<math>\Psi^{J}_{M} = \sum_{m_1,m_2} C^{j_1,j_2,J}_{m_1,m_2,M} \psi^{j_1}_{m_1} \psi^{j_2}_{m_2}</math> = expansion of the systems total angular momentum in terms of the uncoupled original basis states of each individual constituent
  
  
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<math>C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}</math>
 
<math>C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}</math>
  
 +
<math>C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{0,\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}</math>
 +
 +
<math>C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}</math>
 +
 +
<math>C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{0,\frac{1}{2},\frac{1}{2}}= -\frac{1}{\sqrt{3}}</math>
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 +
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<math>\Gamma_1 = \int_0^{x_o} g_1(x,Q^2) dx</math>
 +
 +
<math>g_1 = \frac{F_1}{1+\gamma^2} (A_1+\gamma A_2)</math>
  
 
<math>\sigma \propto |M_{if}|^2</math>
 
<math>\sigma \propto |M_{if}|^2</math>
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<math>M_{fi} = <\Psi_f | H_{int} | \Psi_i></math>
 
<math>M_{fi} = <\Psi_f | H_{int} | \Psi_i></math>
  
−
<math>A = \frac{\sigma_{\frac{1}{2}} - \sigma_{\frac{3}{2}}}{\sigma_{\frac{1}{2}} + \sigma_{\frac{3}{2}}}</math>
+
<math>A = \frac{\sigma_{\frac{1}{2}} - \sigma_{\frac{3}{2}}}{\sigma_{\frac{1}{2}} + \sigma_{\frac{3}{2}}} = \frac{\frac{1}{3} - 1}{\frac{1}{3} + 1} = -1/2</math>
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[http://wiki.iac.isu.edu/index.php/Forest_Classes] [[ Forest_Classes]]

Latest revision as of 21:03, 11 January 2010

The recoupling of two subsystems [math]\psi[/math] with angular momenta [math]j_1[/math] and [math]j_2[/math] to a new system[math] \Psi[/math] with total angular momentum [math]J[/math] is written as

[math]\Psi^{J}_{M} = \sum_{m_1,m_2} C^{j_1,j_2,J}_{m_1,m_2,M} \psi^{j_1}_{m_1} \psi^{j_2}_{m_2}[/math] = expansion of the systems total angular momentum in terms of the uncoupled original basis states of each individual constituent


[math]C^{j_1,j_2,J}_{m_1,m_2,M}[/math] : Clebsch-Gordon Coefficient

[math]C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{1}{2},\frac{3}{2}}=1[/math]

[math]C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}[/math]

[math]C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{0,\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}[/math]

[math]C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{2}{\sqrt{3}}[/math]

[math]C^{1,\;\;\frac{1}{2},\frac{1}{2}}_{0,\frac{1}{2},\frac{1}{2}}= -\frac{1}{\sqrt{3}}[/math]


[math]\Gamma_1 = \int_0^{x_o} g_1(x,Q^2) dx[/math]

[math]g_1 = \frac{F_1}{1+\gamma^2} (A_1+\gamma A_2)[/math]

[math]\sigma \propto |M_{if}|^2[/math]

[math]M_{fi} = \lt \Psi_f | H_{int} | \Psi_i\gt [/math]

[math]A = \frac{\sigma_{\frac{1}{2}} - \sigma_{\frac{3}{2}}}{\sigma_{\frac{1}{2}} + \sigma_{\frac{3}{2}}} = \frac{\frac{1}{3} - 1}{\frac{1}{3} + 1} = -1/2[/math]


[1] Forest_Classes

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