Exercise 11.2.1 Verify Eq. (11.2.11b).
Exercise 11.2.2 Using to order , deduce that .
Exercise 11.2.3 Recall that we found the finite rotation transformation from the infinitesimal one, by solving differential equations (Section 2.8). Verify that if, instead, you relate the transformed coordinates and to and by the infinite string of Poisson brackets, you get the same result, , etc. (Recall the series for , etc.)
Exercise 11.4.1 Prove that if , a system that starts out in a state of even/odd parity maintains its parity. (Note that since parity is a discrete operation, it has no associated conservation law in classical mechanics.)
Exercise 11.4.2 A particle is in a potential
which is invariant under the translations , where is an integer. Is momentum conserved? Why not?
Exercise 11.4.3 You are told that in a certain reaction, the electron comes out with its spin always parallel to its momentum. Argue that parity is violated.
Exercise 11.4.4 We have treated parity as a mirror reflection. This is certainly true in one dimension, where may be viewed as the effect of reflecting through a (point) mirror at the origin. In higher dimensions when we use a plane mirror (say lying on the plane), only one () coordinate gets reversed, whereas the parity transformation reverses all three coordinates.
Verify that reflection on a mirror in the plane is the same as parity followed by rotation about the axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.
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