Notes and essays on AI, ML, and physics.https://www.aoyangqin.com/https://www.aoyangqin.com/blog/magnet-4/https://www.aoyangqin.com/blog/magnet-4/We now know one electron's magnetism is relativistic. But why do billions of them in an iron bar line up — when the force aligning them is not magnetic at all? The finale: exchange interaction, the Ising model, and the symmetry breaking that links a fridge magnet to the origin of mass.Sun, 05 Apr 2026 00:00:00 GMTphysicselectromagnetismhttps://www.aoyangqin.com/blog/magnet-3/https://www.aoyangqin.com/blog/magnet-3/Part 2 conjured spin by switching on a field. But spin is intrinsic — it should be written into how the electron transforms when we rotate space itself. Part 3: the group theory of SU(2), SO(3), and the Lorentz group, and why the electron has no choice but to be a spinor.Sun, 29 Mar 2026 00:00:00 GMTphysicselectromagnetismhttps://www.aoyangqin.com/blog/magnet-2/https://www.aoyangqin.com/blog/magnet-2/Demand a wave equation that is first order in time and respects relativity, and the electron is forced to grow four components, an internal axis, and a magnetic moment out of thin air. Part 2: how spin — and magnetism — fall out of Dirac's equation.Sun, 22 Mar 2026 00:00:00 GMTphysicselectromagnetismhttps://www.aoyangqin.com/blog/magnet-1/https://www.aoyangqin.com/blog/magnet-1/A magnet pulls a nail across the table — surely the simplest demonstration in physics. Yet the magnetic force does no work, and classical physics turns out to forbid magnetism outright. Part 1 of a series: setting up the paradox.Sun, 15 Mar 2026 00:00:00 GMTphysicselectromagnetismhttps://www.aoyangqin.com/blog/classical-mechanics-4/https://www.aoyangqin.com/blog/classical-mechanics-4/Invariant tori, the KAM theorem, and the chaos that grows in the cracks; then the leap from a single trajectory to a thermodynamic law. The finale: how the whole journey resolves into one geometric picture of physics.Sun, 25 Jan 2026 00:00:00 GMTphysicsclassical-mechanicshttps://www.aoyangqin.com/blog/classical-mechanics-3/https://www.aoyangqin.com/blog/classical-mechanics-3/Strip mechanics down to its bones and what remains is a closed, non-degenerate 2-form on phase space. Part 3 is the geometry of that form — Poisson brackets as a Lie algebra, canonical transformations, Liouville and Poincaré, and the Hamilton–Jacobi equation that almost became quantum mechanics.Sun, 18 Jan 2026 00:00:00 GMTphysicsclassical-mechanicshttps://www.aoyangqin.com/blog/classical-mechanics-2/https://www.aoyangqin.com/blog/classical-mechanics-2/Every continuous symmetry hides a conserved quantity, and — turned around — symmetry alone can dictate the action. Part 2 follows that idea from Noether's theorem through relativistic invariance to the electromagnetic coupling and its quantum echo.Sun, 11 Jan 2026 00:00:00 GMTphysicsclassical-mechanicshttps://www.aoyangqin.com/blog/classical-mechanics-1/https://www.aoyangqin.com/blog/classical-mechanics-1/Newton hands us forces; we will hand them back as a single number attached to a path. Part 1 builds the foundations — potential and force, differential forms and Stokes, and the least-action principle that quietly reorganizes all of mechanics.Sun, 04 Jan 2026 00:00:00 GMTphysicsclassical-mechanicshttps://www.aoyangqin.com/blog/quantum-computing-2/https://www.aoyangqin.com/blog/quantum-computing-2/How do you pull a classical answer out of a quantum state? Measurement is the subtlest piece of the whole story — and the moment we understand it, superdense coding and teleportation fall right out. Part 2, the finale.Sun, 30 Nov 2025 00:00:00 GMTquantum-computinghttps://www.aoyangqin.com/blog/quantum-computing-1/https://www.aoyangqin.com/blog/quantum-computing-1/Before qubits and entanglement, there is a quieter question: why must computation be reversible at all? Part 1 of a two-part series builds the foundations — gates, qubits, dynamics, and the first taste of entanglement — ending with real Qiskit code.Sun, 23 Nov 2025 00:00:00 GMTquantum-computinghttps://www.aoyangqin.com/blog/statistical-mechanics-2/https://www.aoyangqin.com/blog/statistical-mechanics-2/Run the whole argument backwards: write an energy model, sum a single function, and let entropy fall out of it. Part 2 follows that reversal into quantum statistics — to particles that lose their names, Fermi seas, and Bose–Einstein condensates, and the deepest poem in physics.Sun, 26 Oct 2025 00:00:00 GMTphysicsstatistical-mechanicshttps://www.aoyangqin.com/blog/statistical-mechanics-1/https://www.aoyangqin.com/blog/statistical-mechanics-1/Concede that you cannot track $10^{23}$ particles, agree only that Nature plays no favorites, and watch entropy, temperature, pressure, and free energy fall out one after another like dominoes. Part 1 of a two-part series: from counting to the architecture of thermodynamics.Sun, 19 Oct 2025 00:00:00 GMTphysicsstatistical-mechanicshttps://www.aoyangqin.com/blog/quantum-chaos/https://www.aoyangqin.com/blog/quantum-chaos/Classical chaos lives and dies by the butterfly effect. But the Schrödinger equation is linear, so two close quantum states stay close forever. So where does chaos hide in the quantum world?Sun, 14 Sep 2025 00:00:00 GMTquantum-mechanicschaosphysicschaos-theorycomplex-systems