Quantum Mechanics Course

Lesson 1 • Differential Equations for Physicists

Reviews ordinary and partial differential equations essential for solving the Schrödinger equation. Emphasizes boundary conditions and separation of variables.

Lesson 2 • Operators, Matrices, and Eigenvalues

Introduces linear operators and their matrix representations, then derives eigenvalue equations. Eigenvalues become the measurable quantities in quantum theory.

Lesson 3 • Probability and Statistics Essentials

Establishes probability distributions, expectation values, and variance needed to interpret quantum measurement outcomes. Directly supports the Born rule introduced next.

Lesson 4 • Fourier Analysis and Transforms

Develops Fourier series and the Fourier transform as tools for switching between position and momentum representations. Lays groundwork for wave-packet analysis.

Lesson 1 • Dirac Notation and Hilbert Space

Formalizes bra-ket notation and the structure of Hilbert space as the arena for quantum states. Connects abstract notation to concrete matrix and function representations.

Lesson 2 • The Postulates of Quantum Mechanics

States the five core postulates governing state vectors, observables, measurement, and time evolution. Each postulate is linked to a concrete physical example.

Lesson 3 • Quantum Superposition and Interference

Analyzes superposition as a fundamental feature distinguishing quantum from classical systems. Demonstrates interference through the double-slit experiment and Mach-Zehnder interferometer.

Lesson 4 • Uncertainty Principles

Derives the Heisenberg uncertainty relation from the commutator algebra of operators. Extends to the generalized Robertson uncertainty relation for arbitrary observables.

Lesson 1 • Quantum Harmonic Oscillator

Solves the harmonic oscillator via both differential equation and algebraic ladder-operator methods. Establishes zero-point energy and Hermite polynomial wave functions.

Lesson 2 • Stationary States and the TISE

Separates variables to obtain the time-independent Schrödinger equation and defines stationary states. Shows how energy eigenstates form a complete basis for general solutions.

Lesson 3 • Particle in a Box

Solves the infinite square well exactly to illustrate quantization, normalization, and orthogonality. Extends to the finite square well to introduce tunneling boundary conditions.

Lesson 4 • Time-Dependent Schrödinger Equation

Derives the TDSE from the energy-operator correspondence and analyzes its general structure. Establishes probability current and continuity as conservation laws.

Lesson 1 • Spherical Harmonics

Constructs spherical harmonics as the angular eigenfunctions and analyzes their symmetry properties. Provides the angular part of all central-potential wave functions.

Lesson 2 • 3D Schrödinger Equation in Cartesian Coordinates

Generalizes the TISE to three dimensions and solves the 3D particle-in-a-box. Introduces degeneracy as a consequence of spatial symmetry.

Lesson 3 • Central Potential and Radial Equation

Separates the 3D TISE into radial and angular parts for any central potential. Solves the radial equation for the hydrogen atom and the 3D harmonic oscillator.

Lesson 4 • Angular Momentum Operators

Derives the orbital angular momentum operators and their commutation relations in spherical coordinates. Establishes the simultaneous eigenstates of L² and Lz.

Lesson 1 • Intrinsic Spin and the Stern-Gerlach Experiment

Motivates spin through the Stern-Gerlach experiment and defines spin-1/2 as a two-state system. Establishes spin operators and their commutation relations.

Lesson 2 • Addition of Angular Momenta

Combines two angular momenta using Clebsch-Gordan coefficients and the triangle rule. Applies the formalism to spin-orbit coupling and two-electron systems.

Lesson 3 • Magnetic Moments and Spin in Fields

Connects spin to magnetic moment and analyzes spin dynamics in external magnetic fields. Derives Larmor precession and the quantum description of magnetic resonance.

Lesson 4 • Pauli Matrices and Spin-1/2 Algebra

Introduces the Pauli matrices as the matrix representation of spin-1/2 operators. Derives spin measurement probabilities and expectation values for arbitrary spin states.

Lesson 1 • Indistinguishability and Exchange Symmetry

Establishes why quantum particles are fundamentally indistinguishable and introduces the exchange operator. Derives the requirement for symmetric or antisymmetric wave functions.

Lesson 2 • Applications to Condensed Matter

Applies quantum statistics to free electron gas, phonons, and Bose-Einstein condensation. Connects microscopic quantum statistics to macroscopic material properties.

Lesson 3 • Quantum Statistical Distributions

Derives Fermi-Dirac and Bose-Einstein distributions from symmetry constraints and compares them to Maxwell-Boltzmann statistics. Introduces the concept of chemical potential.

Lesson 4 • Bosons, Fermions, and the Spin-Statistics Theorem

Classifies particles as bosons or fermions based on integer or half-integer spin. States the spin-statistics theorem and its profound physical implications.

Lesson 1 • Time-Dependent Perturbation Theory

Develops Fermi's golden rule for transition rates under time-dependent perturbations. Applies the result to absorption, emission, and selection rules in atoms.

Lesson 2 • WKB Approximation

Derives the WKB wave functions in classically allowed and forbidden regions using the semiclassical limit. Applies connection formulas to tunneling and quantization conditions.

Lesson 3 • Variational Principle and Method

Proves the variational theorem and uses trial wave functions to estimate ground-state energies. Applies the method to helium and simple molecular systems.

Lesson 4 • Degenerate Perturbation Theory

Handles degeneracy by diagonalizing the perturbation within the degenerate subspace. Applies the method to the Stark effect and Zeeman effect in hydrogen.

Lesson 1 • Quantum Algorithms and Error Correction

Presents Grover's search and Shor's factoring algorithms as demonstrations of quantum advantage. Introduces stabilizer codes as the foundation of quantum error correction.

Lesson 2 • Bell Inequalities and Quantum Nonlocality

Derives Bell inequalities as tests of local hidden-variable theories and interprets experimental violations. Establishes quantum nonlocality as a verified physical phenomenon.

Lesson 3 • Quantum Entanglement

Defines entanglement through non-separability of composite system states and quantifies it with entropy measures. Analyzes Bell states and their role in quantum correlations.

Lesson 4 • Density Matrix Formalism

Introduces the density operator as the general description of pure and mixed quantum states. Derives the von Neumann equation as the density-matrix analog of the Schrödinger equation.