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 Mathematical 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.

0. Basic Concepts of Photometric Quantities

In everyday language, we often say things such as “this lamp is very bright,” “the desk is very bright,” or “the screen is very bright.”

However, in physics and lighting engineering, the word “bright” cannot be represented by just one quantity, because it may refer to different things:

• how strongly a light source emits in a particular direction;
• how much light a source emits in total;
• how much light falls on a particular surface;
• how much light a surface itself emits outward;
• how bright a surface appears to the human eye from a particular direction.