Problem 1 - angular momentum Angular momentum in classical mechanics measures the 'amount of rotation'. It is anal- ogous to linear momentum, which measures the 'amount of motion'. In classical mechanics, angular momentum is conserved. The definition of angular momentum for a point particle is given by L= Ix2 (1) namely by the cross product of the position vector with the (linear) momentum vector. a) From this definition, show that, in quantum mechanics, the z-component of the angular momentum operator is given by L= = -in Jay (2) b) Let us assume that our particle motion is restricted to the x-y plane. In this case, we can transform to plane polar coordinates. The transformations and reverse transformations are I = rose y = r sine 0 = arctan(y/x) (3) Show that, in plane polar coordinates, L= = -ih- (4) Hint: you must use the chain rule Or a 20 a and ay dy Or dy 20 (5) and you will need the derivative of arctan: O arctan(I) (6)