"Mott scattering" "Verify equation (1)" f = 1 / (r exp(epsilon r)) exp(i / hbar q r cos(theta)) r^2 sin(theta) -- integrate over r F = integral(f,r) EXP = exp(i / hbar q r cos(theta) - epsilon r) F = eval(F,EXP,0) - eval(F,r,0) F = eval(F,epsilon,0) -- integrate over theta F = defint(F, theta, 0, pi) F = eval(F, q, 2 p sin(theta/2)) -- integrate over phi F = 2 pi F check(F == pi hbar^2 / (p^2 sin(theta/2)^2)) "ok" E = sqrt(p^2 + m^2) p1 = (E, 0, 0, p) p2 = (E, p sin(theta) cos(phi), p sin(theta) sin(phi), p cos(theta)) u11 = (E + m, 0, p1[4], p1[2] + i p1[3]) / sqrt(E + m) u12 = (0, E + m, p1[2] - i p1[3], -p1[4]) / sqrt(E + m) u21 = (E + m, 0, p2[4], p2[2] + i p2[3]) / sqrt(E + m) u22 = (0, E + m, p2[2] - i p2[3], -p2[4]) / sqrt(E + m) check(dot(conj(u11),u11) == 2 E) check(dot(conj(u12),u12) == 2 E) check(dot(conj(u21),u21) == 2 E) check(dot(conj(u22),u22) == 2 E) M1 = dot(conj(u21),u11) M2 = dot(conj(u21),u12) M3 = dot(conj(u22),u11) M4 = dot(conj(u22),u12) "Verify equation (2)" check(1/2 (conj(M1) M1 + conj(M2) M2 + conj(M3) M3 + conj(M4) M4) == 4 m^2 + 4 p^2 cos(theta/2)^2) "ok" M = (Z alpha pi hbar^2 / (p^2 sin(theta/2)^2))^2 * (4 m^2 + 4 p^2 cos(theta/2)^2) M = M / (2 E V)^2 dsigma = (E V / (2 pi hbar^2))^2 M "Verify equation (3)" check(dsigma == (Z alpha / (4 p^2 sin(theta/2)^2))^2 (4 m^2 + 4 p^2 cos(theta/2)^2)) "ok" "Verify equation (4)" check(dsigma == (Z alpha m / (2 p^2 sin(theta/2)^2))^2 (1 + (p/m)^2 cos(theta/2)^2)) "ok" Run