Final Lab Frame Moller Electron 4-momentum components in XZ Plane

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Final Lab Frame Moller Electron 4-momentum components in XZ Plane

Figure 3: Definition of Moller electron variables in the Lab Frame in the x-z plane. Using [math]\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)[/math]

[math]\Longrightarrow {p^'_{2(z)}=p^'_{2}\cos(\theta '_2)}[/math]

Since,

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

[math]\Longrightarrow p^'_{2(z)}[/math] should always be positive with respect to the total momentum. This is effectively a ratio of momentum in the z direction versus total momentum.

Checking on the sign resulting from the cosine function, we are limited to:

[math]0^\circ \le \theta '_2 \le 90^\circ \equiv 0 \le \theta '_2 \le \pi \ Radians[/math]

At [math]\theta=0^{\circ} \quad \cos{0}=1 \Rightarrow p=p_z[/math] and all motion is in the z direction.

At [math]\theta=90^{\circ} \quad \cos{90}=0 \Rightarrow p=\sqrt{p_x^2+p_y^2}[/math] The motion is in the x-y plane.