Mathematical Intuition behind Relativistic Quantum Mechanics
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.
Introduction to Angular Momentum Operators
FREE PREVIEWProperties of Angular Momentum Operators Part 1
FREE PREVIEWProperties of Angular Momentum Operators part 2
Matrix Representation of Angular Momentum
Spin 1/2 particles
Hamiltonian For an Electron in an Electromagnetic Field
FREE PREVIEWDirac Equation Derived from Pauli Matrices
Conserved Quantity from the Dirac Equation
Non Relativistic Limit of the Dirac Equation part 1
Non Relativistic Limit of the Dirac Equation part 2
Solution of the Dirac Equation for a Free Particle part 1
Solution of the Dirac Equation for a Free Particle part 2 (Particles at Rest)
Solution of the Dirac Equation for a Free Particle in the General Case
Problem with the Negative Energy Components
Interpretation of Negative Energy Solutions
Hamiltonian + commutator between Hamiltonian and Total Angular Momentum
Hamiltonian and Spin Component along the Total Angular Momentum
Commutator between H and K
Solving the Dirac Equation for Hydrogen Part 1
FREE PREVIEWSolving the Dirac Equation for Hydrogen Part 2
FREE PREVIEWSolving the Dirac Equation for Hydrogen Part 3
Solving the Dirac Equation for Hydrogen Part 4
Solving the Dirac Equation for Hydrogen Part 5
Solving the Dirac Equation for Hydrogen Part 6
Solving the Dirac Equation for Hydrogen Part 7
Final Considerations on the Relativistic Hydrogen Spectrum
Intro to This Appendix
Hydrogen-Like Atoms
Hamiltonian of a Hydrogen-Like Atom
More on the Potential Energy and How to Find the Spectrum
Separation of Variables in the Schrodinger Equation
Time Independent Schrodinger Equation in Spherical Coordinates
Separating the Variables in the Time-Independent Schrodinger Eq
Radial Schrodinger Equation
Working on the Radial Schrodinger Equation
9 Solution to the radial Schrodinger equation
10 derivation of the discrete energy spectrum corrected
1 Orbital angular momentum and its square in QM
FREE PREVIEW2 Laplacian in spherical coordinates
3 Legendre differential equation
4 Solution to the Legendre differential equation
5 Spherical harmonics
6 Eigenfunctions and eigenvalues of Lz and L squared
7 properties of Legendre Polynomials part 1
8 Properties of legendre polynomials part 2: Rodrigues formula
9 Normalization of Legendre polynomials
10 Relation between Beta and Gamma function
11 Completeness relation for the Legendre polynomials
12 Properties of the generalized Legendre polynomials part 1
13 Properties of the generalized Legendre polynomials part 2 : Orthogonality
14 Full formula for the spherical harmonics
15 Symmetry property of spherical harmonics
16 Completeness of spherical harmonics
17 theorem of Addition of Spherical Harmonics
18 Final considerations on the Addition Theorem for Spherical Harmonics