Final CM Frame Moller Electron 4-momentum components

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Final CM Frame Moller Electron 4-momentum components

Relativistically, the x and y components remain the same in the conversion from the Lab frame to the Center of Mass frame, since the direction of motion is only in the z direction.

[math]p^*_{2(x)}\Leftrightarrow p_{2(x)}'[/math]

[math]p^*_{2(y)}\Leftrightarrow p_{2(y)}'[/math]

[math]p^*_{2(z)}=-\sqrt {(p^*_2)^2-(p^*_{2(x)})^2-(p^*_{2(y)})^2}[/math]

We choose negative, since the incoming electron in the lab frame is traveling in the positive direction, and the Moller electron is initially at rest, which translates to negative motion in the CM frame.

Redefining the components in simpler terms, we use the fact that

[math]E^*\equiv E^*_1+E^*_2[/math]

[math]s \equiv (P_1^*+P_2^*)^2=(E^*_1+E^*_2)^2=(2E^*_2)^2=(P_1+P_2)^2=2m^2+2E_1E_2-2 \vec p_1 \cdot \vec p_2[/math]

Initially [math]\vec p_2=0 \quad \Rightarrow E^2=p^2+m^2=m^2[/math]

[math]\sqrt{2}E^*_2=\sqrt{2m(m+E_1)}[/math]

[math]E^*_2=\sqrt{\frac{m(m+E_1)}{2}}[/math]

[math] p^*_{2}=\sqrt{E_{2}^{*2}-m^2}[/math]

Initially, before the collision in the CM frame, p₂ was in the negative z direction. After the collision, the direction should reverse to the positive z direction. This same switching of the momentum direction alters p₁ as well.

Using

[math]\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)[/math]

[math]\Longrightarrow \theta ^*_2=\arccos \left(\frac{p^*_{2(z)}}{p^*_{2}}\right)[/math]

For the case where [math]\theta = 180^{\circ}[/math]

This implies

[math]\cos{\frac{p_z}{p}} =-1 [/math]

This implies that the z component of the momentum is equal in magnitude but opposite in direction to the total momentum. From this situation there can by no x or y components. Since there is only the z component, the total can be equal to it's oppposite, so no negative z components are allowed. Even if there where x and y components included with a negative z component summing to a positive total momentum, this would violate the conservation of total momentum / energy in the CM frame. The largest [math]\theta[/math] then is [math]90^{\circ}[/math] where all momentum in the CM frame has been transfered to the xy plane from an initial z only momentum.

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