We consider a system of two electrons scattering off each other. The mass of the electron is noted m. We choose to describe the system in the centre of mass frame so only the relative vector r = r1 - 12, where r1,2 are the position vectors of the electrons, is relevant (see Fig. a). We note r and e the polar coordinates of r. path 1 path 4 path 2 path 3 Detector Detector (a) (b) (c) Q1 (20 marks) Show that the Lagrangian of the system can be expressed as L = (12 + +202) _ kez T Q2 (20 marks) Show that the canonical momentum pe = " is conserved. Give its expression. What is the name of this quantity? Consider the two electrons in a head-on collision, scattering at 90 degrees. A detector is placed on one side and can detect only one of the electrons. The electrons which are detected may have followed path 1 and 2 simultaneously (see Fig. b). Both paths have the same action So. Q3 (20 marks) Show that paths 1 and 2 interfere constructively. We now slightly change the position of the detector. The detected electrons may have taken path 3 with action So + 65 and path 4 with action So - 65 simultaneously (see Fig. c). Q4 (20 marks) Show that the number of detected electrons is proportional to cos (6S/h).The detector is now positioned where the first destructive interference is observed. Q5 (20 marks) Give the corresponding expression for 6S and calculate its value in J.s. Useful formula: L = K - V Vcoul. = kq192 F v = rep + ree d BL A = N eis.in h ~ 1.055 x 10- J.s