Chapter 11 Symmetries and Their Consequences

11.1 Overview

11.2 Translational Invariance in Quantum Theory

Exercise 11.2.1 Verify Eq. (11.2.11b).

Exercise 11.2.2 Using $T^{\dagger}(\varepsilon)T(\varepsilon)=I$ to order $\varepsilon$, deduce that $G^{\dagger}=G$.

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 $\overline{x}$ and $\overline{y}$ to $x$ and $y$ by the infinite string of Poisson brackets, you get the same result, $\overline{x}=x \cos \theta-y \sin \theta$, etc. (Recall the series for $\sin \theta$, etc.)

11.3 Time Translational Invariance

11.4 Parity Invariance

Exercise 11.4.1 Prove that if $[\Pi, H]=0$, 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 $x \rightarrow x+m a$, where $m$ 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 $x \rightarrow-x$ 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 $x-y$ plane), only one ($z$) coordinate gets reversed, whereas the parity transformation reverses all three coordinates.

Verify that reflection on a mirror in the $x-y$ plane is the same as parity followed by $180^{\circ}$ rotation about the $z$ axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.

11.5 Time-Reversal Symmetry