Phase 0 of 3

Mathematical Foundations of Quantum Mechanics

Modern quantum mechanics is written in Dirac notation. This track builds the linear-algebraic toolkit — vector spaces, operators, projectors, spectral decomposition, tensor products, and Hilbert spaces — with the rigor needed for the Phase 1 quantum mechanics curriculum.

Prerequisites

High-school algebra and trigonometry
Basic calculus (derivatives and integrals)
Willingness to work through proofs and derivations

MATH-1

Complex Numbers and Phases for Quantum Mechanics

The field $\mathbb{C}$ , polar form, and the interference of probability amplitudes

This module builds the complex-number toolkit used throughout quantum mechanics. We define the field $\mathbb{C}$ , study modulus and conjugation, explore polar form and Euler's formula, and learn why global phases are unobservable while relative phases control interference.

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MATH-2

Vector Spaces, Bases, and Linear Combinations

Abstract vector spaces over $\mathbb{C}$ , independence, span, and coordinates

This module introduces vector spaces over the complex numbers, the concepts of linear independence and span, the existence of bases and dimension, column-vector representations, change of basis, and the idea that the same vector can look different in different coordinate systems.

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MATH-3

Dirac Notation -- Bras and Kets

State vectors, dual space, inner products, and orthonormality

This module introduces Dirac notation, the compact language of quantum mechanics. We define kets as abstract state vectors, bras as dual vectors, the inner product as a bridge between the two, and the concepts of normalization, orthogonal states, and orthonormal bases.

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MATH-4

Outer Products, Projection Operators, and Completeness

From $|u\rangle\langle v|$ to the resolution of the identity

This module develops outer products, projection operators, the spectral family of a Hermitian operator, the completeness relation, and the technique of inserting the identity to change basis. These tools convert abstract Dirac notation into concrete calculations.

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MATH-5

Linear Operators and Matrix Elements

Linear operators, matrix elements $A_{ij}=\langle i|A|j\rangle$ , adjoints, and special classes

This module studies linear operators $A$ on Hilbert space, their action $A|\psi\rangle$ , matrix elements $A_{ij}=\langle i|A|j\rangle$ in Dirac notation, operator products, the adjoint, and the important classes of Hermitian, unitary, normal, inverse, and identity operators that appear in every quantum calculation.

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MATH-6

Eigenstructure and Spectral Decomposition

$A|\psi\rangle=\lambda|\psi\rangle$ , spectral decomposition, and the spectral theorem

This module covers the eigenvalue equation $A|\psi\rangle=\lambda|\psi\rangle$ , the characteristic polynomial, the reality and orthogonality of eigenvectors of Hermitian operators, spectral decomposition $A=\sum_i\lambda_i|e_i\rangle\langle e_i|$ , functions of operators, and a complete diagonalization of the Pauli-X matrix.

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MATH-7

Unitary Evolution, Observables, and Commutators

$|\psi(t)\rangle=U(t)|\psi(0)\rangle$ , observables, compatibility, and uncertainty

This module connects unitary operators with quantum time evolution $U(t)=e^{-iHt/\hbar}$ , Hermitian operators with observables and measurement, commutators with compatibility, and derives the generalized uncertainty relation in Dirac notation.

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MATH-8

Tensor Products and Composite Systems

Combining Hilbert spaces, entanglement, and multi-qubit states

This module defines the tensor product of Hilbert spaces, distinguishes product states from entangled states, describes operators on composite systems, introduces multi-qubit Dirac notation, and constructs the Bell states explicitly.

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MATH-9

Hilbert Spaces, Fourier Analysis, and Representations

From finite dimensions to $L^2$ , position, momentum, and the Fourier transform

This module extends finite-dimensional linear algebra to infinite-dimensional Hilbert spaces, focusing on $L^2(\mathbb{R})$ , the position and momentum bases, the Fourier transform as a change of basis, delta-function normalization, and the Gaussian state as a worked example.

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Mathematical Foundations of Quantum Mechanics

Prerequisites

High-school algebra and trigonometry
Basic calculus (derivatives and integrals)
Willingness to work through proofs and derivations