Scattering Amplitude - Revision history

Vanwdani at 21:12, 29 December 2018

2018-12-29T21:12:41Z

Vanwdani at 21:11, 29 December 2018

2018-12-29T21:11:19Z

Vanwdani at 21:09, 29 December 2018

2018-12-29T21:09:19Z

Vanwdani at 21:04, 29 December 2018

2018-12-29T21:04:03Z

Vanwdani at 21:02, 29 December 2018

2018-12-29T21:02:01Z

Vanwdani at 20:57, 29 December 2018

2018-12-29T20:57:24Z

Vanwdani: /* Scattering Amplitude */

2017-06-27T15:30:44Z

Scattering Amplitude

Vanwdani: /* Scattering Amplitude */

2017-06-27T15:26:44Z

Scattering Amplitude

Vanwdani: /* Scattering Amplitude */

2017-06-27T15:25:29Z

Scattering Amplitude

Vanwdani at 15:22, 26 June 2017

2017-06-26T15:22:54Z

← Older revision		Revision as of 21:12, 29 December 2018
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−	<center><math>-i \mathfrak{M}_{1}=ie\left(\mathbf p_{1}+\mathbf p_{1}^{'}\right)^{\mu} \left (\frac{-ig_{\mu \nu}}{q^{2}} \right) ie \left( \mathbf p_{2}+\mathbf p_{2}^{'}\right)^{\nu} \qquad \qquad -i \mathfrak{M}_{2}=ie\left(\mathbf p_{1}+\mathbf p_{2}^{'})^{\mu} ~~\left~~ \left(\frac{-ig_{\mu \nu}}{q^2} \right) ie \left( \mathbf p_{2}+\mathbf p_{1}^{'}\right)^{\nu}</math></center>	+	<center><math>-i \mathfrak{M}_{1}=ie\left(\mathbf p_{1}+\mathbf p_{1}^{'}\right)^{\mu} \left(\frac{-ig_{\mu \nu}}{q^{2}} \right) ie \left( \mathbf p_{2}+\mathbf p_{2}^{'}\right)^{\nu} \qquad \qquad -i \mathfrak{M}_{2}=ie\left(\mathbf p_{1}+\mathbf p_{2}^{'}\right)^{\mu} \left(\frac{-ig_{\mu \nu}}{q^2} \right) ie \left( \mathbf p_{2}+\mathbf p_{1}^{'}\right)^{\nu}</math></center>

← Older revision		Revision as of 21:11, 29 December 2018
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−	<center><math>-i \mathfrak{M}_{1}=ie(\mathbf p_{1}+\mathbf p_{1}^{'})^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf ~~p_2~~^')^{\nu} \qquad \qquad -i \mathfrak{M}_{2}=ie(\mathbf p_{1}+\mathbf p_{2}^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf ~~p_1~~^')^{\nu}</math></center>	+	<center><math>-i \mathfrak{M}_{1}=ie\left(\mathbf p_{1}+\mathbf p_{1}^{'}\right)^{\mu} \left (\frac{-ig_{\mu \nu}}{q^{2}} \right) ie \left( \mathbf p_{2}+\mathbf p_{2}^{'}\right)^{\nu} \qquad \qquad -i \mathfrak{M}_{2}=ie\left(\mathbf p_{1}+\mathbf p_{2}^{'})^{\mu} \left \left(\frac{-ig_{\mu \nu}}{q^2} \right) ie \left( \mathbf p_{2}+\mathbf p_{1}^{'}\right)^{\nu}</math></center>

← Older revision		Revision as of 21:09, 29 December 2018
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−	<center><math>-i \mathfrak{M}_{1}=ie^{2}\left (\frac{(\mathbf p_{1}+\mathbf p_{1}^{'})_{\mu} (\mathbf p_{2}+\mathbf p_{2}^{'})^{\mu}}{(\mathbf p_{2}^{'}-\mathbf p_{2})^{2}} \right ) \qquad \qquad -i \mathfrak{M}_{2}=ie^{2}\left (\frac{(\mathbf p_{1}+\mathbf p_{2}^')_{\mu} (\mathbf p_{2}+\mathbf p_{1}^{'})^{\mu}}{(\mathbf p_{1}^{'}-\mathbf p_{2})^{2}} \right ) </math></center>	+	<center><math>-i \mathfrak{M}_{1}=ie^{2}\left (\frac{\left(\mathbf p_{1}+\mathbf p_{1}^{'}\right)_{\mu} \left(\mathbf p_{2}+\mathbf p_{2}^{'}\right)^{\mu}}{\left(\mathbf p_{2}^{'}-\mathbf p_{2}\right)^{2}} \right ) \qquad \qquad -i \mathfrak{M}_{2}=ie^{2}\left (\frac{\left(\mathbf p_{1}+\mathbf p_{2}^{'}\right)_{\mu} \left(\mathbf p_{2}+\mathbf p_{1}^{'}\right)^{\mu}}{\left(\mathbf p_{1}^{'}-\mathbf p_{2}\right)^{2}} \right ) </math></center>

← Older revision		Revision as of 21:04, 29 December 2018
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−	<center><math>-i \mathfrak{M}_{1}=ie^2\left (\frac{(\mathbf p_{1}+\mathbf p_{1}^{'})_{\mu} (\mathbf p_{2}+\mathbf p_{2}^{'})^{\mu}}{(\mathbf p_{2}^{'}-\mathbf p_{2})^2} \right ) \qquad \qquad -i \mathfrak{M}_{2}=ie^2\left (\frac{(\mathbf p_{1}+\mathbf p_{2}^')_{\mu} (\mathbf p_{2}+\mathbf p_{1}^')^{\mu}}{(\mathbf p_{1}^{'}-\mathbf p_{2})^2} \right ) </math></center>	+	<center><math>-i \mathfrak{M}_{1}=ie^{2}\left (\frac{(\mathbf p_{1}+\mathbf p_{1}^{'})_{\mu} (\mathbf p_{2}+\mathbf p_{2}^{'})^{\mu}}{(\mathbf p_{2}^{'}-\mathbf p_{2})^{2}} \right ) \qquad \qquad -i \mathfrak{M}_{2}=ie^{2}\left (\frac{(\mathbf p_{1}+\mathbf p_{2}^')_{\mu} (\mathbf p_{2}+\mathbf p_{1}^{'})^{\mu}}{(\mathbf p_{1}^{'}-\mathbf p_{2})^{2}} \right ) </math></center>

@@ Line 10: / Line 10: @@
 =Scattering Amplitude=
-In the Møller scattering <math>(\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')</math> we have identical particles in the initial and final states.  This that the amplitude to be symmetric under interchange of particles <math>(\mathbf P_1^' \leftrightarrow \mathbf P_2^' </math> or <math> \mathbf P_1  \leftrightarrow \mathbf P_2)</math>. Due to this symmetry we can determine two 1st level Feynman diagrams to describe this scattering.
+In the Møller scattering <math>(\mathbf P_{1} + \mathbf P_{2} \rightarrow \mathbf P_{1}^{'} + \mathbf P_{2}^{'})</math> we have identical particles in the initial and final states.  This that the amplitude to be symmetric under interchange of particles <math>(\mathbf P_{1}^{'} \leftrightarrow \mathbf P_{2}^{'} </math> or <math> \mathbf P_{1}  \leftrightarrow \mathbf P_{2})</math>. Due to this symmetry we can determine two 1st level Feynman diagrams to describe this scattering.
 <center>[[File:Feynman1stLevel.png | 600 px]]</center>
 The amplitudes of the individual Feynman diagrams add linearly to form the total amplitude
-<center><math>\mathfrak{M}=\mathfrak{M}_1+\mathfrak{M}_2</math></center>
+<center><math>\mathfrak{M}=\mathfrak{M}_{1}+\mathfrak{M}_{2}</math></center>
@@ Line 30: / Line 30: @@
-<center><math>-i \mathfrak{M}_1=ie(\mathbf p_1+\mathbf p_1^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_2^')^{\nu} \qquad \qquad -i \mathfrak{M}_2=ie(\mathbf p_1+\mathbf p_2^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_1^')^{\nu}</math></center>
+<center><math>-i \mathfrak{M}_{1}=ie(\mathbf p_{1}+\mathbf p_{1}^{'})^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_2^')^{\nu} \qquad \qquad -i \mathfrak{M}_{2}=ie(\mathbf p_{1}+\mathbf p_{2}^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_1^')^{\nu}</math></center>
-<center><math>-i \mathfrak{M}_1=ie(\mathbf p_1+\mathbf p_1^')^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_2^'-\mathbf p_2)^2} \right ) ie( \mathbf p_{2}+\mathbf p_2^')^{\nu} \qquad  \qquad -i \mathfrak{M}_2=ie(\mathbf p_1+\mathbf p_2^')^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_1^'-\mathbf p_2)^2} \right ) ie( \mathbf p_{2}+\mathbf p_1^')^{\nu}</math></center>
+<center><math>-i \mathfrak{M}_1=ie(\mathbf p_{1}+\mathbf p_{1}^{'})^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_{2}^{'}-\mathbf p_{2})^{2}} \right ) ie( \mathbf p_{2}+\mathbf p_{2}^{'})^{\nu} \qquad  \qquad -i \mathfrak{M}_{2}=ie(\mathbf p_{1}+\mathbf p_{2}^{'})^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_{1}^{'}-\mathbf p_{2})^{2}} \right ) ie( \mathbf p_{2}+\mathbf p_{1}^{'})^{\nu}</math></center>
-<center><math>-i \mathfrak{M}_1=ie^2\left (\frac{(\mathbf p_1+\mathbf p_1^')_{\mu} (\mathbf p_{2}+\mathbf p_2^')^{\mu}}{(\mathbf p_2^'-\mathbf p_2)^2} \right ) \qquad \qquad -i \mathfrak{M}_2=ie^2\left (\frac{(\mathbf p_1+\mathbf p_2^')_{\mu} (\mathbf p_{2}+\mathbf p_1^')^{\mu}}{(\mathbf p_1^'-\mathbf p_2)^2} \right ) </math></center>
+<center><math>-i \mathfrak{M}_{1}=ie^2\left (\frac{(\mathbf p_{1}+\mathbf p_{1}^{'})_{\mu} (\mathbf p_{2}+\mathbf p_{2}^{'})^{\mu}}{(\mathbf p_{2}^{'}-\mathbf p_{2})^2} \right ) \qquad \qquad -i \mathfrak{M}_{2}=ie^2\left (\frac{(\mathbf p_{1}+\mathbf p_{2}^')_{\mu} (\mathbf p_{2}+\mathbf p_{1}^')^{\mu}}{(\mathbf p_{1}^{'}-\mathbf p_{2})^2} \right ) </math></center>

← Older revision		Revision as of 20:57, 29 December 2018
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		+	<center><math>\underline{\textbf{Navigation}}</math></center>
		+
		+	<center>
		+	[[Lorentz_Transformation_to_Lab_Frame\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame\|<math>\triangle </math>]]
		+	[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance\|<math>\vartriangleright </math>]]
		+	</center>
		+
		+
	=Scattering Amplitude=		=Scattering Amplitude=

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	<center><math>\mathfrak{M}_{e^-e^-}=e^2 \left (\frac{u-s}{t}+\frac{t-s}{u} \right )</math></center>		<center><math>\mathfrak{M}_{e^-e^-}=e^2 \left (\frac{u-s}{t}+\frac{t-s}{u} \right )</math></center>
		+
		+
		+	----
		+
		+	<center><math>\underline{\textbf{Navigation}}</math></center>
		+
		+	<center>
		+	[[Lorentz_Transformation_to_Lab_Frame\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame\|<math>\triangle </math>]]
		+	[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance\|<math>\vartriangleright </math>]]
		+	</center>

← Older revision		Revision as of 15:26, 27 June 2017
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	<center><math>q \equiv \mathbf p_{final}-\mathbf p_{initial}</math></center>		<center><math>q \equiv \mathbf p_{final}-\mathbf p_{initial}</math></center>

−	and <math>g_{\mu \nu}</math> is the Mandelstam metric which allows the transformation from the contravariant to covariant form needed for tensor multiplication.	+	and <math>g_{\mu \nu}</math> is the Mandelstam metric which allows the transformation from the contravariant to covariant form needed for tensor multiplication. Examining both Feynman diagrams seperately, we find for their individual amplitudes

← Older revision		Revision as of 15:25, 27 June 2017
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	where q is the momentum of the photon		where q is the momentum of the photon
	<center><math>q \equiv \mathbf p_{final}-\mathbf p_{initial}</math></center>		<center><math>q \equiv \mathbf p_{final}-\mathbf p_{initial}</math></center>
		+
		+	and <math>g_{\mu \nu}</math> is the Mandelstam metric which allows the transformation from the contravariant to covariant form needed for tensor multiplication.

@@ Line 1: / Line 1: @@
 =Scattering Amplitude=
-In the Møller scattering <math>(\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')</math> we have deal with identical particles in the initial and final states, which means that the amplitude has to be symmetric under interchange of particles <math>(\mathbf P_1^' \leftrightarrow \mathbf P_2^' </math>or<math> \mathbf P_1  \leftrightarrow \mathbf P_2)</math>. As a consequence there will be two Feynman diagrams to describe this scattering.
+In the Møller scattering <math>(\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')</math> we have deal with identical particles in the initial and final states, which means that the amplitude has to be symmetric under interchange of particles <math>(\mathbf P_1^' \leftrightarrow \mathbf P_2^' </math> or <math> \mathbf P_1  \leftrightarrow \mathbf P_2)</math>. As a consequence there will be two Feynman diagrams to describe this scattering.
 <center>[[File:Feynman1stLevel.png | 600 px]]</center>