Quantum Mechanics

An interactive book on the postulates, one-dimensional problems, spin and angular momentum, and identical particles

Primary goal

One question runs through this book: what can we know about a quantum system, and with what probability?

A state is a vector, an observable is a Hermitian operator, and a measurement returns one of its eigenvalues. So the book keeps asking three things, first of the postulates and then of each system it meets:

What are the possible results? The eigenvalues: the energy levels of a well or an oscillator, $ℏ m$ for a component of angular momentum, $\pm ℏ/2$ for spin.
With what probability? The squared components of the state in the eigenbasis, $∣ b_{n} ∣^{2} = ∣ ⟨ u_{n} ∣ ψ ⟩ ∣^{2}$ .
What can be known together, and how does it change in time? Commutators, the uncertainty principle, and the Schrödinger equation.

Part 1 is the mathematics the course uses. Part 2 is the postulates, part 3 the one-dimensional problems, part 4 spin and angular momentum, each in Centini’s order and his notation. Part 5, identical particles and helium, is from his 2024/25 whiteboards, as are the parts of 3 and 4 this year’s lectures have not reached yet; each section says where it comes from under its title.

How to read it

Drag things. Every section has figures you can move. Points, arrows and sliders with a handle can be dragged with a pointer or a finger; with the keyboard, focus a figure, press Enter to pick a handle and the arrow keys to move it (Shift for larger steps). Things snap where something interesting happens, and the text under the figure says what.
Colours mean something, the same in the text and in the figures: $red$ , $green$ , $blue$ for the eigenstates $u_{1}, u_{2}, u_{3}$ and their levels, $violet$ for the state $ψ$ and $∣ ψ ∣^{2}$ , $orange$ for the eigenvalue or eigenvector being looked for, $teal$ and $pink$ for real and imaginary parts, $grey$ for the potential (Notation and colours).
Dotted links open in place. A reference like Definition 1.4.1 shows what it names right here; click again to close it.
Proofs and solutions are folded. Try first, then open.
Only a few definitions are essential, marked as such. Everything else is built from them (Index of terms).
“Beyond the boards” marks what this book adds to Centini’s whiteboards: skipped steps, examples, pictures, and results the boards state without proof. Where a board has a slip, the text says so and gives the corrected version.

Part 1. Mathematical tools

Complex numbers and phasors, waves and wave packets, operators and their eigenvalues, Hermitian operators and Dirac notation, and the Fourier transform: the language the postulates are written in.

Part 2. The postulates

What a state is, what a measurement returns and with what probability, when two observables can be known together, how a state evolves in time, and why $Δ x Δ p \geq ℏ/2$ .

Part 3. One-dimensional problems

The time-independent Schrödinger equation solved region by region: the infinite and finite wells, the barrier and tunnelling, and the harmonic oscillator with its ladder operators.

Part 4. Spin and angular momentum

Spin one half and the Pauli matrices, angular momentum from its commutators alone, the orbital case and the spherical harmonics, and how two angular momenta add.

Part 5. Identical particles

Why identical particles must be symmetric or antisymmetric under exchange, what that does to how they share states, and the helium atom. From the 2024/25 boards only; the most likely part to be dropped this year.