Chapter 12 Rotational Invariance and Angular Momentum

12.1 Translations in Two Dimensions

Exercise 12.1.1 Verify that $\hat{a} \cdot \mathbf{P}$ is the generator of infinitesimal translations along a by considering the relation

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).

Chapter 10 Systems with $N$ Degrees of Freedom

10.1 $N$ Particles in One Dimension

Exercise 10.1.1 Show the following:

(1)

(2)

(3)If

then

and similarly with $1\to 2$.

(4)

Chapter 9 The Heisenberg Uncertainty Relations

9.1 Introduction

9.2 Derivation of the Uncertainty Relations

9.3 The Minimum Uncertainty Packet

9.4 Applications of the Uncertainty Principle

Exercise 9.4.1 Consider the oscillator in the state $|n=1\rangle$ and verify that

Chapter 8 The Path Integral Formulation of Quantum Theory

8.1 The Path Integral Recipe

8.2 Analysis of the Recipe

8.3 An Approximation to $U(t)$ for the Free Particle

8.4 Path Integral Evaluation of the Free-Particle Propagator

8.5 Equivalence to the Schrödinger Equation

8.6 Potentials of the Form $V=a+bx+cx^{2}+d\dot{x}+ex\dot{x}$

Exercise 8.6.1 Verify that

agrees with the exact result, Eq. (5.4.31), for $V(x)=-fx$. Hint: Start with $x_{\text{cl}}(t^{\prime\prime})=x_{0}+v_{0}t^{\prime\prime}+\dfrac{1}{2}(f/m)t^{\prime\prime 2}$ and find the constants $x_{0}$ and $v_{0}$ from the requirement that $x_{\text{cl}}(0)=x^{\prime}$ and $x_{\text{cl}}(t)=x$.