Vanwdani at 22:17, 15 June 2017

2017-06-15T22:17:19Z

Vanwdani: Created page with "=Final Lab Frame Scattered Electron 4-momentum components= We can perform a Lorentz transformation from the Center of Mass frame, with zero total momentum, to the Lab frame. <…"

2017-05-31T19:43:20Z

Created page with "=Final Lab Frame Scattered Electron 4-momentum components= We can perform a Lorentz transformation from the Center of Mass frame, with zero total momentum, to the Lab frame. <…"

New page

=Final Lab Frame Scattered Electron 4-momentum components=

We can perform a Lorentz transformation from the Center of Mass frame, with zero total momentum, to the Lab frame.

<center><math>\left( \begin{matrix}E'_{1}+E'_{2}\\ p'_{1(x)}+p'_{2(x)} \\p'_{1(y)}+ p'_{2(y)} \\ p'_{1(z)}+p'_{2(z)}\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & \beta^* \gamma^*\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ \beta^* \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E^*_{1}+E^*_{2}\\ p^*_{1(x)}+p^*_{2(x)} \\ p^*_{(1(y)}+p^*_{2(y)} \\ p^*_{1(z)}+p^*_{2(z)}\end{matrix} \right)</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>p^*_{1(x)}= p'_{1(x)}</math>
|}

{| class="wikitable" align="center"
| style="background: gray" | <math>p^*_{1(y)}= p'_{1(y)}</math>
|}

<center><math>\Longrightarrow\begin{cases}
E'=\gamma E^*+\beta \gamma p^*_{z} \\
p'_{z}=\beta \gamma E^*+ \gamma p^*_{z}
\end{cases}</math></center>

<center><math>\Longrightarrow\begin{cases}
\gamma = \frac{E'}{E^*} \\
\beta =\frac{p'_{z}}{E'}
\end{cases}</math></center>

Without knowing the values for gamma or beta, we can use the fact that lengths of the two 4-momenta are invariant

<center><math>s={\mathbf P^*} ^2=(E^*_{1}+E^*_{2})^2-(\vec p\ ^*_{1}+\vec p\ ^*_{2})^2</math></center>

<center><math>s={\mathbf P}^2=(E'_{1}+E'_{2})^2-(\vec p\ '_{1}+\vec p\ '_{2})^2</math></center>

Setting these equal to each other, we can use this for the collision of two particles of mass m<sub>1</sub> and m<sub>2</sub>. Since the total momentum is zero in the Center of Mass frame, we can express total energy in the center of mass frame as

<center><math>(E^*_{1}+E^*_{2})^2-(\vec p\ ^*_{1}+\vec p\ ^*_{2})^2=s=(E'_{1}+E'_{2})^2-(\vec p\ '_{1}+\vec p\ '_{2})^2</math></center>

<center><math>(E^*)^2-(\vec p\ ^*)^2=(E'_{1}+E'_{2})^2-(\vec p\ ')^2</math></center>

<center><math>(E^*)^2=(E')^2-(\vec p\ ')^2</math></center>

<center><math>(E^*)^2+(\vec{p'})^2=(E')^2</math></center>

<center><math>E'=\sqrt{(E^*)^2+(\vec p\ ')^2}</math></center>

Since momentum is conserved, the initial momentum from the incident electron and stationary electron is still the same in the Lab frame, therefore <math>p\ '_{Lab}=11000 MeV</math>

<center><math>E'=\sqrt{(106.031 MeV)^2+(11000 MeV)^2}\approx 11000.511 MeV</math></center>

This is the same as the initial total energy in the Lab frame, which should be expected since scattering is considered to be an elastic collision.

Since,

<center><math>E'\equiv E'_{1}+E'_{2}</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>\Longrightarrow E'_{1}=E'-E'_{2}</math>
|}

Using the relation

<center><math>E'^2\equiv p^2+m^2</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>\Longrightarrow p_{1}^{'\ 2}=E_{1}^{'\ 2}-m_{1}^2=E_{1}^{'\ 2}-(.511 MeV)^2</math>
|}

Using

<center><math>p^2 \equiv p_x^2+p_y^2+p_z^2</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>\Longrightarrow p_{1(z)}'=\sqrt {p_{1}^{'\ 2}-p_{1(x)}^{'\ 2}-p_{1(y)}^{'\ 2}}</math>
|}

← Older revision		Revision as of 22:17, 15 June 2017
Line 1:		Line 1:
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Limits_based_on_Mandelstam_Variables\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Final_Lab_Frame_Moller_Electron_4-momentum_components\|<math>\triangle </math>]]
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
	=Final Lab Frame Scattered Electron 4-momentum components=		=Final Lab Frame Scattered Electron 4-momentum components=

Final Lab Frame Scattered Electron 4-momentum components - Revision history

Vanwdani at 22:17, 15 June 2017

Vanwdani: Created page with "=Final Lab Frame Scattered Electron 4-momentum components= We can perform a Lorentz transformation from the Center of Mass frame, with zero total momentum, to the Lab frame. <…"