This repository serves as documentation for my 3-years long self-study program designed to build the foundation for studying quantum field theory.
Due to my seek for rigorous foundation, I self-studied Tom M. Apostol's Calculus (Vol. 1) to master derivation from first principles. This mathematical foundation enabled me to progress through MIT OpenCourseWare syllabi for Multivariable Calculus, Linear Algebra, and Quantum Mechanics.
Objective: To build the mathematical foundation required to understand Quantum Field Theory and engineer quantum simulations.
The study was conducted in two distinct phases: Mathematics (The Toolset) and Quantum Mechanics (The Application).
| Phase | Subject | Source Material | Key Output |
|---|---|---|---|
| I | Rigorous Calculus | Calculus Vol. 1 (Apostol) | Derivation of Limits & Integrals |
| I | Multi-Variable Calc | MIT 18.02 (Denis Auroux) | Vector Fields, Flux, Stokes' Theorem |
| I | Linear Algebra | MIT 18.06 (Gilbert Strang) | Vector Spaces, Eigenvalues |
| I | Differential Eq. | MIT 18.03 (Arthur Mattuck) | Second-order ODEs, Fourier Series |
| II | Quantum Physics I | MIT 8.04 (Allan Adams) | Schrödinger Eq, 1-3D Potentials, Spin Theory |
| II | Quantum Physics II | MIT 8.05 (Barton Zwiebach) | Spin-1/2 Systems, Quantum Dynamics |
- Focus: Vector Analysis and Field Theory.
- Content:
- Vector_Calculus_Notes.pdf: Notes covering Gradient, Divergence, and Curl. Includes derivations of Green's Theorem and Stokes' Theorem (essential prerequisites for Electromagnetism).
- Focus: Matrix Factorization and Spectral Theory.
- Content:
- Linear_Algebra_Notes.pdf: Detailed notes on Singular Value Decomposition (SVD) and the Spectral Theorem for Symmetric Matrices. Includes exploration of Positive Definite Matrices and their role in optimization.
- Focus: Dynamic Systems and Transform Methods.
- Content:
- Differential_Equations_Notes.pdf: Analysis of Non-Homogeneous Linear ODEs using Variation of Parameters. Covers Laplace Transforms, Convolution Integrals, and Delta Functions for impulse response.
- Focus: Wave mechanics, Operator formalism, and Entanglement.
- Content:
- Quantum_Physics_1_Notes.pdf: Synthesized notes covering the Schrödinger Equation, 1D Potentials (Infinite Well), and the Harmonic Oscillator (Algebraic & Analytic methods). Includes analysis of Scattering and Tunneling phenomena. Derivation of Spin Theory from its first principle.
- Quantum_Physics_2_Notes.pdf: Advanced notes on General Formalism (Dirac Notation), Spin-1/2 Systems (Pauli Matrices), and Angular Momentum. Detailed derivation of Bell's Inequalities and Quantum Entanglement dynamics on multi qubit systems.
- Selected_Hard_Problems.pdf: Solutions to high-difficulty problem sets (e.g., Stern-Gerlach experiments).
These notes are open-sourced under the MIT License to encourage every interested person regardless of age to pursue rigorous theoretical study.