Lesson 1.5 · 1. Foundations

Mathematical Foundations

This is a reference chapter. We'll cover the essential mathematics you'll need for quantum mechanics, relativity, and beyond. Skim now, return when needed.

Linear Algebra

Linear algebra is the language of quantum mechanics. Everything is vectors and operators.

Vector Spaces

Vector Space

A vector space is a collection of objects (called vectors) that you can add together and scale (multiply by numbers), and the result is always another object in the same collection. The arrows you draw in 3D geometry form a vector space, but so do more abstract things like functions or lists of numbers. The idea is simple: if you can add two things and stretch them, and the rules of arithmetic still work, you have a vector space.

A vector space is a set $V$ where you can:

Add vectors: $\vec{u} + \vec{v} \in V$
Scale vectors: $\alpha \vec{v} \in V$ for any scalar $\alpha$

Familiar examples: $\mathbb{R}^3$ (3D arrows), polynomials, functions.

The key quantum example: the space of all wavefunctions $\psi(x)$.

Inner Products

Inner Product

An inner product is a way to measure two things about vectors: how long they are, and what angle they make with each other. In everyday 3D space, the dot product is an inner product: it tells you how much two arrows point in the same direction. In quantum mechanics, the inner product measures the "overlap" between two states, which directly gives the probability that one state will be found to be in the other.

The inner product $\langle u | v \rangle$ measures "overlap" between vectors.

For functions:

\langle f | g \rangle = \int f^*(x) g(x) \, dx

For finite vectors:

\langle u | v \rangle = \sum_i u_i^* v_i

Properties:

$\langle u | v \rangle = \langle v | u \rangle^*$ (conjugate symmetric)
$\langle u | u \rangle \geq 0$, equals zero only if $u = 0$
$\langle u | \alpha v + \beta w \rangle = \alpha \langle u | v \rangle + \beta \langle u | w \rangle$ (linear in second argument)

Hilbert Spaces

Hilbert Space

A Hilbert space is a vector space equipped with an inner product, and it can have infinitely many dimensions. Think of familiar 3D space, where you can describe any arrow using three numbers. A Hilbert space generalizes this to potentially infinite dimensions: you might need infinitely many numbers to describe a single vector. It is also "complete," meaning you cannot sneak out of the space by taking limits. Quantum states live in Hilbert space, and this mathematical setting is what makes quantum mechanics work.

Technically, a Hilbert space is a vector space with an inner product that is complete (limits of sequences stay in the space).

Quantum mechanics lives in Hilbert space. States are vectors, observables are operators.

Linear Operators

Linear Operator

A linear operator is a mathematical machine that takes in a vector and produces a new vector, while respecting addition and scaling. If you feed it the sum of two vectors, you get the sum of the outputs; if you feed it a scaled vector, you get a scaled output. Think of it like a consistent rule: "rotate every arrow by 30 degrees" or "stretch every arrow by a factor of 2." In quantum mechanics, every measurable quantity (position, momentum, energy) is represented by a linear operator.

An operator $\hat{A}$ acts on vectors to produce vectors:

\hat{A}: V \to V, \quad \hat{A}|\psi\rangle = |\phi\rangle

Examples in quantum mechanics:

Position: $\hat{x}\psi(x) = x\psi(x)$
Momentum: $\hat{p}\psi(x) = -i\hbar\frac{d\psi}{dx}$

Eigenvalues and Eigenvectors

An eigenvector of $\hat{A}$ is a vector that only gets scaled:

\hat{A}|v\rangle = \lambda |v\rangle

The scalar $\lambda$ is the eigenvalue.

The Spectral Theorem

Hermitian operators (those with $\hat{A}^\dagger = \hat{A}$) have:

Real eigenvalues
Orthogonal eigenvectors
A complete basis of eigenvectors

This is why observables in QM are Hermitian: measurements give real numbers.

Dirac Notation

Dirac Notation (Bra-Ket)

Dirac notation is a shorthand invented by physicist Paul Dirac for writing quantum states and the operations on them. Instead of writing a state as a vector with components, you write it as $|\psi\rangle$ (called a "ket"). Its mirror image $\langle\psi|$ (a "bra") represents the dual. Putting a bra and a ket together, $\langle\phi|\psi\rangle$ (a "bracket"), gives the inner product. This notation keeps quantum calculations compact and makes the structure of equations visually clear.

Quantum mechanics uses bra-ket notation:

$|\psi\rangle$: a "ket," a column vector (state)
$\langle\phi|$: a "bra," a row vector (dual state)
$\langle\phi|\psi\rangle$: inner product (bracket)
$|\psi\rangle\langle\phi|$: outer product (an operator)

Complex Analysis Essentials

Complex numbers are unavoidable in quantum mechanics (the $i$ in Schrödinger's equation).

Complex Numbers

A complex number: $z = x + iy$ where $i^2 = -1$.

Real part: $\text{Re}(z) = x$
Imaginary part: $\text{Im}(z) = y$
Complex conjugate: $z^* = x - iy$
Modulus: $|z| = \sqrt{x^2 + y^2} = \sqrt{z z^*}$

Euler's Formula

The most important formula in physics:

e^{i\theta} = \cos\theta + i\sin\theta

Every complex number can be written in polar form:

z = |z| e^{i\theta}

where $\theta = \arg(z)$ is the phase angle.

Why Complex Numbers in QM?

Wavefunctions are complex: $\psi = |\psi|e^{i\phi}$. The phase $\phi$ encodes interference. It's why quantum mechanics is wavelike. Real numbers can't capture this structure.

Contour Integration (Preview)

Many physics integrals are easiest in the complex plane. The key result:

\oint_C f(z) \, dz = 2\pi i \sum_k \text{Res}(f, z_k)

The integral around a closed contour equals $2\pi i$ times the sum of residues at poles inside. This lets you compute real integrals by going to the complex plane.

Differential Geometry Basics

The language of general relativity. Space (and spacetime) can be curved.

Manifolds

A manifold is a space that locally looks like $\mathbb{R}^n$ but may have global structure (curvature, topology).

Examples:

The surface of a sphere (locally flat, globally curved)
Spacetime (4D manifold with Lorentzian geometry)

Vectors and Tensors

On a manifold, vectors live in tangent spaces, the space of "directions" at each point.

A tensor is a multilinear map that transforms correctly under coordinate changes. The metric tensor $g_{\mu\nu}$ tells you how to measure distances:

ds^2 = g_{\mu\nu} dx^\mu dx^\nu

The Metric

In flat space: $ds^2 = dx^2 + dy^2 + dz^2$ (Euclidean).

In special relativity: $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$ (Minkowski).

In general relativity: $g_{\mu\nu}$ varies from point to point. That's gravity.

Curvature

Curvature measures how much parallel transport around a loop rotates a vector.

The Riemann tensor $R^\rho_{\sigma\mu\nu}$ encodes all curvature information. Einstein's equation relates it to matter:

R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4} T_{\mu\nu}

Group Theory

Symmetries form mathematical structures called groups. Groups classify what's possible in physics.

What is a Group?

A group is a set with an operation that satisfies:

Closure: $a \cdot b$ is in the group
Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Identity: There exists $e$ such that $e \cdot a = a$
Inverse: Every $a$ has $a^{-1}$ such that $a \cdot a^{-1} = e$

Important Groups in Physics

Group	What it is	Where it appears
$SO(3)$	3D rotations	Angular momentum, spin
$SU(2)$	2×2 unitary matrices, det = 1	Spin-1/2, weak force
$U(1)$	Phase rotations $e^{i\theta}$	Electromagnetism, charge
$SU(3)$	3×3 unitary matrices, det = 1	Strong force, color charge
Lorentz	Spacetime rotations + boosts	Special relativity
Poincaré	Lorentz + translations	Full spacetime symmetry

Representations

A representation is a way to realize a group as matrices acting on a vector space.

Example: $SO(3)$ (rotations) has representations:

Spin-0: 1×1 matrices (scalars, don't change under rotation)
Spin-1: 3×3 matrices (vectors)
Spin-2: 5×5 matrices (traceless symmetric tensors)

But $SU(2)$ also has half-integer representations:

Spin-1/2: 2×2 matrices (spinors, fermions like electrons)
Spin-3/2: 4×4 matrices

Why Groups Matter

The Standard Model is built from $SU(3) \times SU(2) \times U(1)$. Particles are classified by how they transform under these groups. The mathematics of group theory determines what particles can exist and how they interact.

Lie Algebras

For continuous groups (Lie groups), there's an associated Lie algebra: the infinitesimal generators.

For $SO(3)$, the generators are angular momentum operators $J_x, J_y, J_z$ satisfying:

[J_i, J_j] = i\hbar \epsilon_{ijk} J_k

The commutation relations are the Lie algebra. They determine the group structure.

Fourier Analysis

Decomposing functions into frequencies. Essential for quantum mechanics and wave physics.

Fourier Series

A periodic function can be written as a sum of sines and cosines:

f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}

Fourier Transform

For non-periodic functions, the sum becomes an integral:

\tilde{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx

f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \tilde{f}(k) e^{ikx} dk

Connection: Position and Momentum

In quantum mechanics, position and momentum are Fourier transforms of each other:

$$ \tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}} \int \psi(x) e^{-ipx/\hbar} dx $$

This is why the uncertainty principle exists: a narrow function has a wide Fourier transform.

Summary: The Mathematical Toolkit

Key Takeaways

Linear algebra: Vectors, operators, eigenvalues, the language of QM
Hilbert space: Where quantum states live
Complex numbers: Essential for phases and interference
Differential geometry: Curved spaces, tensors, the language of GR
Group theory: Symmetries determine physics; particles are representations
Fourier analysis: Position ↔ momentum, time ↔ energy

This completes the Foundations phase. You now have the conceptual and mathematical tools for quantum mechanics.

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