Chapter 5 Simple Problems in One Dimension

5.1 The Free Particle

Exercise 5.1.1 Show that Eq. (5.1.9) may be rewritten as an integral over $E$ and a sum over the $\pm$ index as

Chapter 4 The Postulates——a General Discussion

4.1 The Postulates

4.2 Discussion of Postulates I-III

Exercise 4.2.1 Consider the following operators on a Hilbert space $\mathbb{V}^{3}(C)$:

(1) What are the possible values one can obtain if $L_{z}$ is measured?

(2) Take the state in which $L_{z}=1$. In this state what are $\langle L_{x}\rangle$, $\langle L_{x}^{2}\rangle$ and $\Delta L_{x}$?

(3) Find the normalized eigenstates and the eigenvalues of $L_{x}$ in the $L_{z}$ basis.

(4) If the particle is in the state with $L_{z}=-1$, and $L_{x}$ is measured, what are the possible outcomes and their probabilities? (5) Consider the state

in the $L_{z}$ basis. If $L_{z}^{2}$ is measured in this state and a result $+1$ is obtained, what is the state after the measurement? How probable was this result? If $L_{z}$ is measured, what are the outcomes and respective probabilities?

(6) A particle is in a state for which the probabilities are $P(L_{z}=1)=1/4$, $P(L_{z}=0)=1/2$, and $P(L_{z}=-1)=1/4$. Convince yourself that the most general, normalized state with this property is

It was stated earlier on that if $|\psi\rangle$ is a normalized state then the state $e^{i \theta}|\psi\rangle$ is a physically equivalent normalized state. Does this mean that the factors $e^{i \delta_i}$ multiplying the $L_z$ eigenstates are irrelevant? [Calculate for example $P\left(L_x=0\right)$.]

Chapter 3 All Is Not Well with Classical Mechanics

3.1 Particles and Waves in Classical Physics

Chapter 2 Review of Classical Mechanics

2.1 The Principle of Least Action and Lagrangian Mechanics

Exercise 2.1.1 Consider the following system, called a harmonic oscillator. The block has a mass $m$ and lies on a frictionless surface. The spring has a force constant $k$. Write the Lagrangian and get the equation of motion.

Chapter 1 Mathemtical Introduction

1.1 Linear Vector Spaces: Basics

Exercise 1.1.1 Verify these claims. For the first consider $|0\rangle+|0^{\prime}\rangle$ and use the advertised properties of the two null vectors in turn. For the second start with $|0\rangle=(0+1)|V\rangle+|-V\rangle$. For the third, begin with $|V\rangle+(-|V\rangle)=0|V\rangle=|0\rangle$. For the last, let $|W\rangle$ also satisfy $|V\rangle+|W\rangle=|0\rangle$. Since $|0\rangle$ is unique, this means $|V\rangle+|W\rangle=|V\rangle+|-V\rangle$. Take it from here.