Phase 1 of 3

Phase 1: Quantum Mechanics

How nature works at the smallest scales. Build physical intuition before touching a single qubit. Wave-particle duality, superposition, entanglement, and measurement.

Prerequisites

Introductory physics at the university level (classical mechanics, electromagnetism)
Multivariable calculus (partial derivatives, multiple integrals, vector calculus)
Fluency with complex numbers and Euler's formula
Linear algebra: vector spaces, eigenvalues, eigenvectors, matrix diagonalization
Ordinary differential equations (separation of variables, Fourier series)

QM-1

Beginner

The Wave Function

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.

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QM-2

Beginner

Stationary States

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.

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QM-3

Beginner

The Infinite Square Well

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.

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QM-4

Intermediate

The Harmonic Oscillator

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.

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QM-5

Intermediate

The Free Particle & Delta-Function Potential

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.

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QM-6

Intermediate

The Finite Square Well

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.

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QM-7

Intermediate

Hilbert Space

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.

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QM-8

Intermediate

Observables & Eigenfunctions

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.

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QM-9

Intermediate

The Uncertainty Principle

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.

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QM-10

Intermediate

Quantum Mechanics in Three Dimensions

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.

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QM-11

Advanced

The Hydrogen Atom

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.

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QM-12

Advanced

Spin & Angular Momentum Addition

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.

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QM-13

Advanced

Identical Particles

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.

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QM-14

Advanced

Quantum Symmetries & Applications

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.

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Next Phase

Phase 2: Quantum Computing

Phase 1: Quantum Mechanics

How nature works at the smallest scales. Build physical intuition before touching a single qubit. Wave-particle duality, superposition, entanglement, and measurement.

Prerequisites

Introductory physics at the university level (classical mechanics, electromagnetism)
Multivariable calculus (partial derivatives, multiple integrals, vector calculus)
Fluency with complex numbers and Euler's formula
Linear algebra: vector spaces, eigenvalues, eigenvectors, matrix diagonalization
Ordinary differential equations (separation of variables, Fourier series)