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How nature works at the smallest scales. Build physical intuition before touching a single qubit. Wave-particle duality, superposition, entanglement, and measurement.
Probability, normalization, and the Schrödinger equation
This module introduces the wave function $\psi(x,t)$, the Schrödinger equation, Born's statistical interpretation, normalization, and the Heisenberg uncertainty principle. It establishes the mathematical and conceptual foundation for all of quantum mechanics.
Separation of variables and the time-independent Schrodinger equation
For time-independent potentials, the Schrodinger equation can be solved by separation of variables. This module develops the theory of stationary states: their properties, the separation procedure, and why they form the foundation for solving quantum mechanics problems.
Boundary conditions, quantization, and expansion in eigenstates
The infinite square well is the simplest quantum system that exhibits energy quantization. This module solves the time-independent Schrodinger equation inside the well, applies boundary conditions, finds the energy spectrum and eigenfunctions, and shows how arbitrary initial states evolve by expansion in the energy eigenbasis.
Algebraic and analytic methods
The quantum harmonic oscillator is one of the most important systems in physics. This module develops both the algebraic method (ladder operators) and the analytic method (Hermite polynomials), derives the energy spectrum, and introduces coherent states — the quantum states closest to classical behavior.
Plane waves, wave packets, and the delta-function well
This module treats two foundational problems: the free particle, which introduces the concepts of plane waves, wave packets, and group velocity; and the delta-function potential, the simplest model that exhibits both bound and scattering states.
Bound states, scattering, and quantum tunneling
The finite square well is a more realistic model than the infinite well or delta potential. It exhibits both bound states and scattering states, and demonstrates the phenomenon of quantum tunneling — a particle passing through a region where classically it would be forbidden.
Vector spaces, Dirac notation, and changing bases
Quantum mechanics lives in Hilbert space. This module develops the mathematical framework: vector spaces, inner products, operators, complete orthonormal bases, Dirac notation, and the transformation between different representations.
Hermitian operators, determinate states, and spectra
Every measurable physical quantity corresponds to a Hermitian operator. This module develops the theory of observables: Hermitian operators, determinate states, discrete and continuous spectra, and the generalized statistical interpretation that connects operators to measurement outcomes.
Proof, minimum-uncertainty states, and energy-time uncertainty
The uncertainty principle is a fundamental theorem about the mathematical structure of quantum mechanics. This module presents the full proof of the generalized uncertainty principle, constructs minimum-uncertainty wave packets, and explores the energy-time uncertainty relation.
The Schrodinger equation in 3D, separation of variables, and central potentials
The real world is three-dimensional. This module extends quantum mechanics to 3D, derives the three-dimensional Schrodinger equation, shows how to separate variables in Cartesian and spherical coordinates, and introduces angular momentum as the conserved quantity for central potentials.
Radial equation, energy spectrum, and hydrogenic wave functions
The hydrogen atom is the quantum mechanical system that explains the structure of matter. This module solves the radial equation for the Coulomb potential, derives the quantized energy levels and Bohr radius, and presents the full set of hydrogenic wave functions with their quantum numbers.
Intrinsic angular momentum, spin-$1/2$, and Clebsch-Gordan coefficients
Spin is intrinsic angular momentum with no classical analog. This module introduces the spin operators, constructs the spin-$1/2$ and spin-1 representations, and develops the theory of adding angular momenta using Clebsch-Gordan coefficients.
Bosons, fermions, exchange symmetry, and the helium atom
Quantum mechanics imposes strict constraints on systems of identical particles. This module derives the exchange symmetry requirement, explains the spin-statistics theorem, develops the formalism for many-particle wave functions, and applies it to the helium atom.
Exchange energy, perturbation theory, and the variational principle
This final module covers advanced techniques and applications of quantum mechanics: perturbation theory for weakly perturbed systems, the variational principle for finding upper bounds on ground state energies, and an overview of how quantum mechanics explains atoms, molecules, and solids.