Relativistic Quantum Mechanics: Spin & Dirac Equation

Mathematical Intuition behind Relativistic Quantum Mechanics

What you'll learn
Learn the properties of (intrinsic) Angular Momentum
Learn how the concept of spin naturally arises in Quantum Mechanics
Learn how spin helps derive Relativistic Quantum theory
Learn how to derive the Dirac equation
Learn how to solve the Dirac equation
Learn how to derive the relativistic spectrum of the hydrogen atom
Learn how the Dirac equation contains in the Schrodinger equation
Requirements
operators in quantum mechanics
commutators
wave functions
Schrodinger equation
Description
This is a course on Relativistic Quantum Mechanics. Why did I create this new course even if there is already a course on Quantum Mechanics and Quantum Field Theory? The answer is simple: the main reason is that I am passionate about these topics, but another reason is the fact that my previous course on QM and QFT already contained roughly 40 hours of content, so it would have been too "chaotic" if I added another 10 hours of content.Besides, this course is developed on its own (even if we do not start from scratch). In fact, the topics covered here are not covered in the other course. Here we start from some commutation relations regarding angular momentum, from which we derive the concept of spin. It is therefore recommended to have a prerequisite knowledge of operators and commutators, and how the latter are related to the possibility of measuring two physical quantities simultaneously.After a first part on angular momentum (in particular, intrinsic angular momentum), we use the concepts therein developed to construct the Dirac equation. We will see that the concept of spin is naturally incorporated into the relativistic theory.Once we have the Dirac equation, we will start solving it in the case of a free particle, and we also derive conserved quantities from it (the Hamiltonian, current, etc.).From other commutation relations that we derive, we finally find the spectrum of the hydrogen atom in the relativistic case, and compare it with the non-relativistic solution.

Overview

Section 1: How spin naturally arises in Quantum Mechanics

Lecture 1 Introduction to Angular Momentum Operators

Lecture 2 Properties of Angular Momentum Operators Part 1

Lecture 3 Properties of Angular Momentum Operators part 2

Lecture 4 Matrix Representation of Angular Momentum

Lecture 5 Spin 1/2 particles

Section 2: Derivation of the Dirac Equation from Pauli matrices

Lecture 6 Hamiltonian For an Electron in an Electromagnetic Field

Lecture 7 Dirac Equation Derived from Pauli Matrices

Section 3: Conserved Current

Lecture 8 Conserved Quantity from the Dirac Equation

Section 4: Non Relativistic Limit of the Dirac Equation

Lecture 9 Non Relativistic Limit of the Dirac Equation part 1

Lecture 10 Non Relativistic Limit of the Dirac Equation part 2

Section 5: Solution of the Dirac Equation

Lecture 11 Solution of the Dirac Equation for a Free Particle part 1

Lecture 12 Solution of the Dirac Equation for a Free Particle part 2 (Particles at Rest)

Lecture 13 Solution of the Dirac Equation for a Free Particle in the General Case

Lecture 14 Problem with the Negative Energy Components

Lecture 15 Interpretation of Negative Energy Solutions

Section 6: Hamiltonian and Other Important Operators

Lecture 16 Hamiltonian + commutator between Hamiltonian and Total Angular Momentum

Lecture 17 Hamiltonian and Spin Component along the Total Angular Momentum

Lecture 18 Commutator between H and K

Section 7: Relativistic Spectrum of the Hydrogen Atom from the Dirac Equation

Lecture 19 Solving the Dirac Equation for Hydrogen Part 1

Lecture 20 Solving the Dirac Equation for Hydrogen Part 2

Lecture 21 Solving the Dirac Equation for Hydrogen Part 3

Lecture 22 Solving the Dirac Equation for Hydrogen Part 4

Lecture 23 Solving the Dirac Equation for Hydrogen Part 5

Lecture 24 Solving the Dirac Equation for Hydrogen Part 6

Lecture 25 Solving the Dirac Equation for Hydrogen Part 7

Lecture 26 Final Considerations on the Relativistic Hydrogen Spectrum

Section 8: Appendix: Non-Relativistic Spectrum of the Hydrogen Atom

Lecture 27 Intro to This Appendix

Lecture 28 Hydrogen-Like Atoms

Lecture 29 Hamiltonian of a Hydrogen-Like Atom

Lecture 30 More on Potential Energy and How to Find the Spectrum

Lecture 31 Separation of Variables in the Schrodinger Equation

Lecture 32 Time Independent Schrodinger Equation in Spherical Coordinates

Lecture 33 Separating the Variables in the Time-Independent Schrodinger Equation

Lecture 34 Radial Schrodinger Equation

Lecture 35 Working on the Radial Schrodinger Equation

Lecture 36 Solution to the Radial Schrodinger Equation

Lecture 37 Derivation of the Discrete Energy Spectrum

physics students,physics enthusiast,students who want to understand the mathematics behind relativistic quantum mechanics