Fundamentals of physics

Alessio Mangoni

• Editore: Alessio Mangoni
• Anno: 2020
• ISBN:
  9791220233460
• Argomento: Naturwissenschaften
• Categoria: Quantentheorie

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...