Classical Electrodynamics · Problem 11.23

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · relativistic-collision-kinematics-cm-frame

Problem

In a collision process a particle of mass m2m_2, at rest in the laboratory, is struck by a particle of mass m1m_1, momentum pLAB\mathbf{p}_{\rm LAB} and total energy ELABE_{\rm LAB}. In the collision the two initial particles are transformed into two others of mass m3m_3 and m4m_4. The configurations of the momentum vectors in the center of momentum (cm) frame (traditionally called the center-of-mass frame) and the laboratory frame are shown in the figure. (a) Use invariant scalar products to show that the total energy WW in the cm frame has its square given by W2=m12+m22+2m2ELABW^2 = m_1^2 + m_2^2 + 2m_2 E_{\rm LAB} and that the cms 3-momentum p′\mathbf{p}' is p′=m2pLABW\mathbf{p}' = \frac{m_2 \mathbf{p}_{\rm LAB}}{W} (b) Show that the Lorentz transformation parameters βcm\boldsymbol{\beta}_{\rm cm} and γcm\gamma_{\rm cm} describing the velocity of the cm frame in the laboratory are βcm=pLABm2+ELAB,γcm=m2+ELABW\boldsymbol{\beta}_{\rm cm} = \frac{\mathbf{p}_{\rm LAB}}{m_2 + E_{\rm LAB}}, \quad \gamma_{\rm cm} = \frac{m_2 + E_{\rm LAB}}{W} (c) Show that the results of parts a and b reduce in the nonrelativistic limit to the familiar expressions, W≈m1+m2+(m2m1+m2)pLAB22m1W \approx m_1 + m_2 + \left(\frac{m_2}{m_1 + m_2}\right)\frac{p^2_{\rm LAB}}{2m_1} p′≈(m2m1+m2)pLAB,βcm≈pLABm1+m2\mathbf{p}' \approx \left(\frac{m_2}{m_1 + m_2}\right)\mathbf{p}_{\rm LAB}, \quad \boldsymbol{\beta}_{\rm cm} \approx \frac{\mathbf{p}_{\rm LAB}}{m_1 + m_2}

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