Quantum Mechanics: The 360-Lesson Architect Course

A 3-year curriculum designed to take you from basic arithmetic to the frontiers of quantum physics.

Master Table of Contents

Year 1: Mathematical Foundations

Chapter 1: The Logic of Arithmetic & Algebra (Lessons 1-30)

  1. Order of Operations & The Logic of Grouping
  2. Signed Numbers: Addition & Subtraction on the Line
  3. Signed Numbers: Multiplication & The Negative Identity
  4. Fractions I: Equivalence & The Concept of Parts
  5. Fractions II: Addition & The Common Denominator
  6. Fractions III: Multiplication & Cross-Cancellation
  7. Fractions IV: Division as Reciprocal Multiplication
  8. Variables: Placeholders for the Unknown
  9. The Distributive Law: Expanding Logic
  10. Combining Like Terms: Mathematical Efficiency
  11. Translating Sentences into Equations
  12. The Golden Rule of Balance: Addition/Subtraction
  13. The Golden Rule of Balance: Multiplication/Division
  14. Multi-Step Equations: Unwrapping the Variable
  15. Variables on Both Sides: Consolidating Information
  16. The Ratio and Proportion: Scaling Reality
  17. Literal Equations: Rearranging Physical Formulas
  18. Introduction to Inequalities: The Concept of Range
  19. Compound Inequalities: Intersections & Unions
  20. Absolute Value Equations: Distance from Center
  21. Absolute Value Inequalities: Sandwich & Split Logic
  22. Laws of Exponents I: The Product Rule
  23. Laws of Exponents II: The Quotient Rule
  24. Zero and Negative Exponents: Inverse Logic
  25. Scientific Notation: Calculating the Cosmic & Atomic
  26. Multiplying Polynomials: The Geometry of FOIL
  27. Factoring I: The Greatest Common Factor
  28. Factoring II: Simple Trinomials
  29. Factoring III: The Difference of Squares
  30. Factoring IV: Completing the Square

Chapter 2: Advanced Algebra & Geometry (Lessons 31-60)

  1. The Quadratic Formula I: Derivation
  2. The Quadratic Formula II: Application & Roots
  3. Radical Expressions: Simplifying Roots
  4. Rationalizing Denominators: Mathematical Convention
  5. Radical Equations: Undoing the Root
  6. Fractional Exponents: The Bridge to Calculus
  7. The Cartesian Plane: Geometry Meets Algebra
  8. Slope: The Measure of Change
  9. Slope-Intercept Form: The Line as a Function
  10. Point-Slope Form: Building Lines from Data
  11. Parallel and Perpendicular Lines
  12. Systems of Equations I: Substitution
  13. Systems of Equations II: Elimination
  14. 3-Variable Systems: A Prelude to Matrices
  15. Function Notation: The Machine Analogy
  16. Domain and Range Deep-Dive
  17. Composite Functions: Nesting Logic
  18. Inverse Functions: Reversing the Process
  19. Exponential Growth and Decay
  20. Introduction to Logarithms
  21. Properties of Logarithms: Expansion & Compression
  22. The Natural Logarithm and the Constant 'e'
  23. Solving Logarithmic Equations
  24. Geometric Shapes and Perimeter
  25. Area Formulas and Integration Prep
  26. The Pythagorean Theorem: The Distance Core
  27. Special Right Triangles (45-45-90 and 30-60-90)
  28. Introduction to Trigonometric Ratios
  29. The Unit Circle I: Angle as Rotation
  30. The Unit Circle II: Sine and Cosine as Coordinates

Chapter 3: Trigonometry & Periodic Functions (Lessons 61-90)

  1. Tangent and Cotangent: Slope Ratios
  2. Secant and Cosecant: Reciprocal Identities
  3. Radians: The Natural Measure of Rotation
  4. Arc Length and Sector Area
  5. Graphing Sine and Cosine: The Waveform
  6. Amplitude and Period: Scaling the Wave
  7. Phase Shift: Translating Waves in Time
  8. Vertical Shift: The Equilibrium Point
  9. Fundamental Trig Identities: Pythagorean Theory
  10. Sum and Difference Formulas I: Sine
  11. Sum and Difference Formulas II: Cosine
  12. Double Angle Formulas: Frequency Doubling
  13. Half Angle Formulas: Frequency Halving
  14. Product-to-Sum Identities
  15. Sum-to-Product Identities
  16. Solving Trigonometric Equations I
  17. Solving Trigonometric Equations II: Multiple Angles
  18. Law of Sines: Non-Right Triangles
  19. Law of Cosines: The Generalized Pythagorean Theorem
  20. Ambiguous Case of the Law of Sines
  21. Polar Coordinates I: From Grid to Circle
  22. Polar Coordinates II: Conversion Equations
  23. Graphing Polar Equations: Roses and Limacons
  24. Harmonic Motion I: The Simple Pendulum
  25. Harmonic Motion II: Springs and Restoring Forces
  26. Damped Oscillations: The Concept of Decay
  27. Forced Oscillations and Resonance
  28. Superposition I: Adding Waves
  29. Superposition II: Constructive vs Destructive Interference
  30. Beats and Phase Velocity

Chapter 4: Complex Numbers & Infinite Series (Lessons 91-120)

  1. The Imaginary Unit 'i': Beyond the Real Line
  2. Arithmetic of Complex Numbers
  3. The Complex Plane: Argand Diagrams
  4. Modulus and Argument: Distance and Angle
  5. Polar Form of Complex Numbers
  6. De Moivre's Theorem: Powers of Complex Numbers
  7. Roots of Unity: Splitting the Circle
  8. Euler's Identity: The Most Beautiful Equation
  9. The Complex Exponential: e^(iθ)
  10. Complex Conjugation and Normalization
  11. Sequences: Patterns of Infinity
  12. Arithmetic Series: Linear Summation
  13. Geometric Series: Exponential Summation
  14. Infinite Geometric Series and Convergence
  15. Sigma Notation: The Language of Sums
  16. The Binomial Theorem: Expanding Powers
  17. Power Series: Functions as Polynomials
  18. Taylor Series I: Approximation at a Point
  19. Taylor Series II: Maclaurin Series of e^x
  20. Taylor Series III: Sin(x) and Cos(x)
  21. Euler's Formula via Power Series
  22. Hyperbolic Functions: Sinh, Cosh, Tanh
  23. Limits I: Approaching the Infinite
  24. Limits II: One-Sided Limits and Continuity
  25. Limits III: The Squeeze Theorem
  26. Infinity in Math: Horizontal Asymptotes
  27. L'Hôpital's Rule: Resolving Indeterminate Forms
  28. Introduction to Vectors in 2D
  29. Vector Addition and Scalar Multiplication
  30. Year 1 Capstone: The Mathematical Framework of Waves

Year 2: Calculus & Classical Physics

Chapter 5: Differential Calculus: Rates of Change (Lessons 121-150)

  1. The Derivative: Instantaneous Rate of Change
  2. Definition of the Derivative via Limits
  3. The Power Rule: Efficient Differentiation
  4. The Constant Multiple and Sum Rules
  5. Derivatives of e^x and ln(x)
  6. Derivatives of Sine and Cosine
  7. The Product Rule: Multiplying Rates
  8. The Quotient Rule: Dividing Rates
  9. The Chain Rule: Composition of Change
  10. Implicit Differentiation
  11. Higher-Order Derivatives: Acceleration
  12. Mean Value Theorem: Average vs Instantaneous
  13. Increasing and Decreasing Functions
  14. The First Derivative Test: Finding Extrema
  15. Concavity and the Second Derivative Test
  16. Inflection Points: Where Change Changes
  17. Optimization I: Maximizing Physical Efficiency
  18. Optimization II: The Principle of Least Time
  19. Related Rates I: Geometry in Motion
  20. Related Rates II: Expansion of Space
  21. Differentials and Linear Approximation
  22. Partial Derivatives I: Multivariable Change
  23. Partial Derivatives II: Chain Rule for Fields
  24. The Gradient Vector: Direction of Maximum Increase
  25. Directional Derivatives
  26. Divergence and Curl: Vector Calculus Intro
  27. Laplace's Equation and Harmonic Functions
  28. Velocity and Acceleration as Vector Derivatives
  29. Tangent Lines and Normal Lines to Curves
  30. The Geometric Meaning of the Differential

Chapter 6: Integral Calculus: Summation & Area (Lessons 151-180)

  1. The Antiderivative: Reversing the Clock
  2. Indefinite Integrals and the Constant C
  3. Riemann Sums: Area by Infinite Rectangles
  4. The Definite Integral: Total Accumulation
  5. The Fundamental Theorem of Calculus Part I
  6. The Fundamental Theorem of Calculus Part II
  7. U-Substitution: The Chain Rule in Reverse
  8. Integration by Parts: The Product Rule in Reverse
  9. Trigonometric Integrals: Powers of Sin and Cos
  10. Trigonometric Substitution: Area of a Circle
  11. Partial Fraction Decomposition in Integration
  12. Improper Integrals I: Infinite Limits
  13. Improper Integrals II: Discontinuous Integrands
  14. Numerical Integration: Simpson's Rule
  15. Area Between Curves
  16. Volumes of Revolution I: Disc Method
  17. Volumes of Revolution II: Shell Method
  18. Arc Length and Surface Area of Solids
  19. Work as an Integral: Force over Distance
  20. Center of Mass and Moments
  21. Double Integrals over Rectangles
  22. Double Integrals over General Regions
  23. Triple Integrals in Cartesian Space
  24. Integration in Polar Coordinates
  25. Integration in Cylindrical and Spherical Coordinates
  26. Line Integrals: Integration Along a Path
  27. Green's Theorem: Linking Area and Path
  28. Surface Integrals and Flux
  29. Stokes' Theorem: The Boundary of Curl
  30. The Divergence Theorem: Sources and Sinks

Chapter 7: Differential Equations: The Language of Nature (Lessons 181-210)

  1. Introduction to Differential Equations: Modeling Growth
  2. Classification: Order, Linearity, and Type
  3. Separable Differential Equations
  4. First-Order Linear Equations: Integrating Factors
  5. Exact Equations and Potential Functions
  6. Population Models and Logistic Growth
  7. Euler's Method: Numerical Solutions
  8. Second-Order Homogeneous Linear Equations
  9. The Characteristic Equation and Real Roots
  10. Complex Roots and Oscillatory Motion
  11. Repeated Roots and Critical Damping
  12. Non-Homogeneous Equations: Undetermined Coefficients
  13. Variation of Parameters: A General Method
  14. Mechanical Vibrations: The Spring-Mass Model
  15. RLC Circuits: Electrical Analogs of Motion
  16. The Laplace Transform I: Definition
  17. The Laplace Transform II: Solving Initial Value Problems
  18. Step Functions and Dirac Delta Impulse
  19. Systems of Linear Differential Equations
  20. Phase Portraits and Stability Analysis
  21. Power Series Solutions Near Ordinary Points
  22. Legendre Polynomials: Symmetry in Spheres
  23. Bessel Functions: Waves in Cylinders
  24. Introduction to Partial Differential Equations (PDEs)
  25. Separation of Variables for the Heat Equation
  26. The Wave Equation I: Strings and Sound
  27. Fourier Series I: Periodic Function Decomposition
  28. Fourier Series II: Orthogonality and Coefficients
  29. The Fourier Transform: From Time to Frequency
  30. Uncertainty in Fourier Analysis: Bandwidth vs Duration

Chapter 8: Analytical Mechanics: Hamiltonian & Lagrangian (Lessons 211-240)

  1. Newton's Laws: The Differential Form
  2. Work-Energy Theorem Revisited
  3. Conservative Forces and Potential Energy Wells
  4. Central Forces and Planetary Motion
  5. Generalized Coordinates: Breaking Free from X-Y
  6. The Principle of Least Action
  7. The Calculus of Variations: Euler-Lagrange Equation
  8. The Lagrangian (L = T - V)
  9. Solving the Pendulum via Lagrangian Mechanics
  10. Symmetry and Conservation: Noether's Theorem
  11. Cyclic Coordinates and Conserved Momentum
  12. Generalized Momentum and Conjugate Variables
  13. The Hamiltonian (H = T + V)
  14. Hamilton's Equations of Motion
  15. Phase Space: The State of a System
  16. Poisson Brackets: A Prelude to Commutators
  17. Canonical Transformations
  18. The Hamilton-Jacobi Equation
  19. Rigid Body Rotation and Inertia Tensors
  20. Normal Modes: Coupled Oscillators
  21. The Failure of Classical Mechanics I: Blackbody Radiation
  22. The Failure of Classical Mechanics II: Photoelectric Effect
  23. The Failure of Classical Mechanics III: Atomic Spectra
  24. Einstein and the Quantization of Light
  25. The Bohr Model of the Atom
  26. De Broglie's Hypothesis: Matter Waves
  27. Wave-Particle Duality: The Double Slit Experiment
  28. The Davisson-Germer Experiment: Electron Diffraction
  29. Phase Velocity vs Group Velocity in Matter Waves
  30. Year 2 Capstone: The End of Determinism

Year 3: Quantum Mechanics

Chapter 9: Linear Algebra I: Vectors & Spaces (Lessons 241-264)

  1. Vector Spaces: The Rules of the Game
  2. Linear Independence and Span
  3. Basis Sets: Defining a Coordinate System
  4. Change of Basis: Viewing Reality from a New Angle
  5. Inner Product Spaces: Defining Length and Angle
  6. Orthogonality and Orthonormal Bases
  7. Gram-Schmidt Orthogonalization
  8. Completeness and Hilbert Spaces
  9. Introduction to Matrices as Transformations
  10. Matrix Multiplication and Composition
  11. The Determinant: Scaling Factor of Space
  12. Matrix Inversion and the Identity
  13. Transpose and Adjoint: Reversing the Map
  14. Unitary Matrices: Preserving the Probabilities
  15. Hermitian Matrices: Real Observables
  16. Trace and Rank: Invariants of Space
  17. Dirac Notation I: Bras and Kets
  18. Dirac Notation II: Inner and Outer Products
  19. Operators as Matrices in Hilbert Space
  20. Projectors and Identity Resolution
  21. Tensors and Product Spaces
  22. Continuous Bases: From Sums to Integrals
  23. Delta Function Normalization
  24. Coordinate vs Momentum Representations

Chapter 10: Linear Algebra II: Operators & Eigenvalues (Lessons 265-288)

  1. The Eigenvalue Problem: Finding Invariant Directions
  2. The Characteristic Equation Revisited
  3. Diagonalization of Matrices
  4. Spectral Theorem for Hermitian Operators
  5. Functions of Operators
  6. Commutators: When Order Matters
  7. Simultaneous Eigenstates and Compatibility
  8. The Position Operator and its Eigenstates
  9. The Momentum Operator in Position Space
  10. Fundamental Commutation Relation [x, p]
  11. The Heisenberg Uncertainty Principle Derivation
  12. Minimum Uncertainty Wavepackets
  13. Expectation Values: The Average of Reality
  14. Variance and Standard Deviation in Quantum States
  15. Time Evolution and the Hamiltonian Operator
  16. Unitary Evolution Operators
  17. Ehrenfest's Theorem: Quantum to Classical Bridge
  18. The Virial Theorem in Quantum Mechanics
  19. Symmetries and Conservation Laws (Quantum Noether)
  20. Translation and Rotation Operators
  21. Parity and Reflection Symmetry
  22. Time Reversal Symmetry
  23. Degeneracy: Multiple States, Same Energy
  24. The Density Matrix: Mixed States

Chapter 11: The Schrödinger Equation (Lessons 289-312)

  1. The Wavefunction: Postulates of Quantum Mechanics
  2. Born's Rule: Probability Interpretation
  3. Normalization of the Wavefunction
  4. Time-Dependent Schrödinger Equation (TDSE)
  5. Separation of Variables: Time and Space
  6. Time-Independent Schrödinger Equation (TISE)
  7. Stationary States and Energy Eigenvalues
  8. General Solution as a Superposition
  9. Probability Current and Continuity Equation
  10. Boundary Conditions for the Wavefunction
  11. The Free Particle: Wave Packets and Dispersion
  12. Phase Velocity vs Group Velocity Revisited
  13. The Gaussian Wave Packet
  14. Spreading of the Wave Packet over Time
  15. The Infinite Square Well: Particle in a Box
  16. Energy Quantization and Nodes
  17. Orthogonality of the Square Well States
  18. Expansion in Infinite Well Eigenstates
  19. The Finite Square Well: Bound States
  20. Transcendental Equations for Energy
  21. Scattering States and Transmission
  22. Quantum Tunneling I: Potential Barriers
  23. Quantum Tunneling II: The Transmission Coefficient
  24. Applications: Scanning Tunneling Microscope

Chapter 12: One-Dimensional Systems (Lessons 313-330)

  1. The Delta Function Potential Well
  2. Bound States of the Delta Potential
  3. Scattering from a Delta Barrier
  4. The Step Potential: Reflection and Transmission
  5. Quantum Paradox: Total Reflection
  6. Periodic Potentials and Bloch's Theorem
  7. The Kronig-Penney Model: Energy Bands
  8. Conductors, Insulators, and Semiconductors
  9. The Harmonic Oscillator I: The Potential Well
  10. Solving the Oscillator via Power Series
  11. Hermite Polynomials
  12. Ground State and Zero-Point Energy
  13. The Algebraic Method: Ladder Operators
  14. Raising and Lowering the Energy
  15. Coherent States: The Most Classical Quantum States
  16. Numerical Solutions: The Shooting Method
  17. The WKB Approximation I: Semiclassical Limit
  18. The WKB Approximation II: Tunneling Rates

Chapter 13: The Harmonic Oscillator & Symmetry (Lessons 331-342)

  1. 3D Schrödinger Equation in Cartesian Coordinates
  2. Separation of Variables in 3D
  3. Central Potentials and Spherical Symmetry
  4. The Angular Equation: Spherical Harmonics
  5. Orbital Angular Momentum Operators L^2 and Lz
  6. Eigenvalues of Angular Momentum
  7. The Radial Equation and Effective Potential
  8. The Hydrogen Atom I: The Coulomb Potential
  9. Solving the Radial Equation for Hydrogen
  10. Laguerre Polynomials and Energy Levels
  11. Quantum Numbers: n, l, m
  12. Probability Clouds and Orbitals

Chapter 14: Hydrogen Atom & Angular Momentum (Lessons 343-354)

  1. Spin I: The Stern-Gerlach Experiment
  2. Spin 1/2 and Pauli Matrices
  3. Spinors and Rotation in Spin Space
  4. Addition of Angular Momentum
  5. Clebsch-Gordan Coefficients
  6. Fine Structure of Hydrogen
  7. Spin-Orbit Coupling
  8. The Zeeman Effect: Atoms in Magnetic Fields
  9. Identical Particles: Bosons and Fermions
  10. The Pauli Exclusion Principle
  11. The Helium Atom and Exchange Energy
  12. The Periodic Table from First Principles

Chapter 15: Entanglement, Computing & Interpretation (Lessons 355-360)

  1. EPR Paradox and Hidden Variables
  2. Bell's Theorem and the Death of Local Realism
  3. Quantum Entanglement and Teleportation
  4. Quantum Computing I: Qubits and Gates
  5. Quantum Computing II: Shor's and Grover's Algorithms
  6. Final Capstone: Interpretations of Quantum Mechanics