Fundamentals of physics

By Alessio Mangoni

This book aims to provide solid bases for the study of physics for the university and it is divided into four parts, each dedicated to a fundamental branch of physics: quantum mechanics, theoretical physics, particle physics and condensed matter physics. In the first part we start with the concept of wave function, until the Heisenberg uncertainty principle. In the second part, after recalling the basic concepts of relativity, we treat the elementary particles and the hadrons, arriving to the notions of scattering and cross section. The third part is dedicated to the theoretical physics, where we analyze the field theory and the concepts of Lagrangian and Hamiltonian, introducing the quantum electrodynamics (QED), passing through the Klein-Gordon, Dirac and Maxwell fields. In the last part of the book we expose the basics of the condensed matter physics, including diffusion and Brownian motion, Drude and Sommerfeld models, the calculation of specific heat and the principal mechanical properties of solids, with references to lattice defects and semiconductors.

• Language: English
• Publisher: Alessio Mangoni
• Release date: Dec 6, 2020
• ISBN: 9791220233460

Book preview

2020

Contents

Contents

Introduction

Part I

Quantum Mechanics

Introduction

The wave function

The Schrödinger equation

Free particle equation

General equation

Continuity equation

Wave packets

Normalization

Fourier transform

Interval of length 2pi

Interval of length L

Infinite interval

Coordinate and momentum space

Expectation value

Operators

Position operator

Momentum operator

Energy operator

Angular momentum operator

Spherical coordinates

Commutation relations

Uncertainty principle

Eigenvalue equations

Position operator

Momentum operator

The third component of angular momentum operator

Part II

Particle Physics

Introduction

Natural units

Bases of relativity

Four-vectors

Lorentz transformations

Relativistic kinematics

Invariant mass

Particles

Elementary particles

Quarks

Leptons

Quark model

Fundamental interactions

Hadrons

Mesons

The Yukawa meson

Baryons

Nucleons

Cosmic rays

The pion

The muon

Particles with strangeness

Kaons

Hyperons

Energy loss

Ionization energy loss

Electron energy loss

Photon energy loss

Hadron energy loss

Quantum numbers and symmetries

The strangeness

The parity

Parity of the photon

Parity of a two-particle system

Charge conjugation

Charge conjugation of the photon

Charge conjugation of the pion

Time reversal

CPT theorem

Baryon number

Lepton number

Isospin

Hypercharge

The Gell-Mann-Nishijima formula

G-parity

Helicity

Chirality

Scattering and decays

Reference frames

The invariant quantity s

Mandelstam variables

Two‐body elastic scattering

Fermi's golden rule

Cross section

Beam intensity reduction

Luminosity

Two-body cross section

Decays

Part III

Theoretical Physics

Introduction

Lagrangian and Hamiltonian

Lagrangian field theory

Hamiltonian field theory

Symmetries and gauge invariance

Symmetries and conservation laws

Gauge invariance

Campo di Klein-Gordon

The Klein-Gordon field

Klein-Gordon Lagrangian

Klein-Gordon Hamiltonian

The electromagnetic field

Maxwell's equations

Gauge invariance

Maxwell Lagrangian

The Dirac field

Dirac equation

Properties of gamma matrices

Dirac Lagrangian

Dirac Hamiltonian

Free particle solutions

Quantum electrodynamics

Interaction Lagrangian

Interaction Hamiltonian

Field operators

The S matrix

Part IV

Condensed Matter Physics

Introduction

Brownian motion and diffusion

Introduction

Einstein relation

Fick's laws

Random walker

Langevin equation

Fokker-Planck equation

Boltzmann equation

Drude model

Introduction

Electric conductivity

Hall effect

Thermal conductivity

Seebeck effect

Sommerfeld model

Quantum treatment

Internal energy

Sommerfeld expansion

Mechanical properties of solids

Introduction

Young's modulus

Poisson's ratio

Lattice defects

Introduction

Point defects

Color centers

Dislocations

Semiconductors

Intrinsic semiconductor

Extrinsic semiconductor

Introduction

This book aims to provide solid bases for the study of physics for the university and it is divided into four parts, each dedicated to a fundamental branch of physics: quantum mechanics, theoretical physics, particle physics and condensed matter physics. In the first part we start with the concept of wave function, until the Heisenberg uncertainty principle. In the second part, after recalling the basic concepts of relativity, we treat the elementary particles and the hadrons, arriving to the notions of scattering and cross section. The third part is dedicated to the theoretical physics, where we analyze the field theory and the concepts of Lagrangian and Hamiltonian, introducing the quantum electrodynamics (QED), passing through the Klein-Gordon, Dirac and Maxwell fields. In the last part of the book we expose the basics of the condensed matter physics, including diffusion and Brownian motion, Drude and Sommerfeld models, the calculation of specific heat and the principal mechanical properties of solids, with references to lattice defects and semiconductors.

Part I

Quantum Mechanics

Introduction

In this first part we will provide a rigorous, but intuitive and therefore suitable for most, theoretical introduction of non-relativistic quantum mechanics. This theory describes systems of particles of atomic scale dimensions, but with small velocity compared to the speed of light in vacuum, for which the relativistic effects can be neglected. There are four fundamental forces in nature: the nuclear strong force, the electromagnetic force, the nuclear weak force and the gravitational one. The two theories that should be considered for a modern description of nature are the Einstein's special relativity and the quantum mechanics. Nowadays all the fundamentals interactions except the gravitation are described by quantum theories of fields (relativistic theories) such as the quantum electrodynamics (QED) and the quantum chromodynamics (QCD). For this reason the study of quantum mechanics represents a fundamental objective. In this part we will treat only the non-relativistic quantum mechanics which represents also the basis for its relativistic formulation (which is often formulated through field theories). In this part we will cover the following topics:

- the wave function;

- the Schrödinger equation (free particle, general equation and continuity equation);

- the wave packets;

- the normalization;

- complete systems and Fourier transform;

- coordinate and momentum space;

- the expectation value;

- the operators (position, momentum, energy, angular momentum);

- the operators in spherical coordinates;

- the commutation relations;

- the eigenvalue equations;

- the Heisenberg uncertainty principle.

The wave function

Let's start by saying that the description of a quantum system occurs through a function, called wave function, associated to the system. This is a function of time and space (x,y,z coordinates) and, in general, it is a complex number. It is usually denoted by the Greek letter



and must satisfy some properties which we will list shortly. First of all, the formulation of quantum mechanics is based on the so-called Copenhagen interpretation and asserts that everything that can be known about a system is contained in its wave function. In particular, the probability of finding the system in the volume element between (x,y,z) and (x + dx, y + dy, z + dz) at a certain moment t is given by



Note that it is a non-negative real number being the square modulus of a complex number. If we integrate the probability of finding a system on all the available volume we should obtain 1 (which corresponds to a percentage of 100%), that is, the certainty of finding it somewhere on the available volume. As we will see later, this cannot happen for a free particle since, also intuitively, the probability density of finding it somewhere is constant and if we integrate a constant on an infinite volume we will find infinite and not 1. The solution is to limit the available volume of the particle, in fact also in nature it can never be infinite. Such a normalization is called box normalization and will be discussed later. When the integral of the square modulus of the wave function, extended to the available volume, is 1 then it is said that the wave function is normalized to 1 and its square modulus gives the probability density of the particle presence. For normalized wave functions it therefore happens that



We now list the physical requirements that a wave function must satisfy in order to describe a quantum system:

the wave function must be everywhere continuous. Being connected with the probability of finding a particle in a volume in a certain time it cannot be discontinuous, otherwise there would be different probabilities depending on the way of calculating the volume.

the wave function must be limited everywhere. In fact, it makes no sense to speak of infinite probability of finding the system somewhere (the maximum probability is 1).

the wave function must be a single valued function, i.e. monodromic. In fact, you cannot have more probabilities for a given point and a given time.

To conclude this chapter on the wave function of a quantum system (or for a particle, in general) we illustrate the so-called superposition principle. Meanwhile, let's say that two wave functions that differ in the normalization constant or in a generic multiplicative complex constant describe the same system. In addition, given two wave functions that describe the same system then a linear combination of them will also describe that system. For practical purposes and for the concept of probability given to the square modulus of the wave function we will always choose a wave function normalized to 1 (when possible, for example for free particle we will adopt the so-called box normalization, as we will see later). We can multiply a normalized wave function by a phase factor of the type



with modulus 1. In general if



is the normalized wave function for a system then also



with



an arbitrary real constant, it will be a normalized wave function for the same system since



because



The Schrödinger equation

We now come to the equation on which all non-relativistic quantum mechanics is based. This is a partial differential equation called Schrödinger equation, from the name of the scientist who formulated it for the first time. The essential