2025-05-30
Solutions to Principles of Quantum Mechanics
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目录

Chapter 11 Symmetries and Their Consequences
11.1 Overview
11.2 Translational Invariance in Quantum Theory
11.3 Time Translational Invariance
11.4 Parity Invariance
11.5 Time-Reversal Symmetry

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(ε)T(ε)=IT^{\dagger}(\varepsilon)T(\varepsilon)=I to order ε\varepsilon, deduce that G=GG^{\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 x\overline{x} and y\overline{y} to xx and yy by the infinite string of Poisson brackets, you get the same result, x=xcosθysinθ\overline{x}=x \cos \theta-y \sin \theta, etc. (Recall the series for sinθ\sin \theta, etc.)

11.3 Time Translational Invariance

11.4 Parity Invariance

Exercise 11.4.1 Prove that if [Π,H]=0[\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

V(x)=V0sin(2πx/a)V(x)=V_0 \sin (2 \pi x / a)

which is invariant under the translations xx+max \rightarrow x+m a, where mm 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 xxx \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 xyx-y plane), only one (zz) coordinate gets reversed, whereas the parity transformation reverses all three coordinates.

Verify that reflection on a mirror in the xyx-y plane is the same as parity followed by 180180^{\circ} rotation about the zz axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.

11.5 Time-Reversal Symmetry

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