Quantum theory, one experiment at a time

A guided course in the foundations of quantum theory. It follows the first three chapters of Asher Peres’s Quantum Theory: Concepts and Methods, a graduate text, but starts from school physics and builds everything else as it goes.

There are thirteen lessons. Each is built around an experiment you run yourself on the page, asks you to commit to a prediction before it shows you the answer, and ends with questions to check you have it. By the last one you will be working with state vectors, inner products, observables and density matrices, and you will know which experiments forced each of them on us.

Start with lesson 1.1Where classical physics ran out

Where it starts. A polariser prepares each photon; a crystal of calcite sends it up or down. Turn the dial and the proportions change. What never changes is that nothing tells you where the next photon will go. Lesson 1.2 takes this apart.

Part I. What the experiments say

Four bench-top experiments with polarised light and silver atoms. Each one is easy to describe and impossible to explain with the physics of 1900. Peres, Chapter 1, Introduction to Quantum Physics.

1.1Where classical physics ran out12 minA hot empty box that should hold infinite energy, light that arrives in lumps, and why a lump can’t be both small and pure in colour.1.2One photon at a time15 minSend single photons at a crystal that splits light in two. Each goes one way or the other, and nothing in the world says which.1.3Superposition is not a mixture14 minLet the two beams overlap. What comes out is a new kind of light, not half of each, and an experiment can tell the difference.1.4The question you cannot ask16 minTry to measure the polarisation of a single photon. You can’t, and the reason gives quantum theory its vocabulary: preparations and tests.1.5Measuring is not reading13 minSilver atoms through a magnet land in two spots. A puzzle with three magnets shows the result was not sitting in the atom beforehand.

Part II. The rules of the game

What a quantum state actually is, what happens when one test follows another, and why probabilities stop adding up. The chapter ends with the one idea everything else rests on: amplitudes. Peres, Chapter 2, Quantum Tests.

2.1States and complete tests15 minA state is a list of odds. A pure state is the most that can be known, a random mixture the least, and they are not the same as order and disorder.2.2One test after another14 minTwo magnets in a row give a table of probabilities. Its rows and its columns each add up to one, and it reads the same forwards and backwards.2.3When paths go dark13 minThree magnets, eight beams. Switch the middle one off and four of the beams vanish instead of merging. Probabilities do not add.2.4Amplitudes: arrows that add18 minBehind every probability is an arrow. Multiply arrows along a path, add them across paths, square at the end. That is the whole calculus.

Part III. The mathematics that fits

The rules of Part II turn out to be the geometry of arrows in a space with complex coordinates. Here the standard machinery appears: state vectors, inner products, observables and density matrices. Peres, Chapter 3, Complex Vector Space.

3.1States as vectors16 minA test is a choice of axes and a state is an arrow among them. The rule that probabilities sum to one is Pythagoras’s theorem.3.2The overlap rule18 minOne formula gives every probability in quantum theory. It also says how sure you can be of two different tests at once, and how to find an unknown state.3.3Observables17 minWrite numbers on the outcomes of a test and you have a measurement. The numbers and the test together are a matrix, and its eigenvectors are the states with a sure answer.3.4Mixtures and the density matrix16 minMost light is not in a pure state. One small matrix holds everything that can be known about it, and quite different recipes give the same one.

The rules, A to K

Peres builds the theory from eleven lettered statements, each drawn out of an experiment. Here they are in plain words, with the lesson that reaches each one. Come back to this list when you have finished: it is the whole of the first three chapters.

AStatistical determinism2.1If a system is prepared so that one complete test has a certain outcome, the odds of every other test are fixed too, however the preparation was done.BEquivalent tests2.1Two complete tests that are certain for the same preparations are the same test: they give the same odds for every preparation.CRandom mixtures2.1A system with N distinguishable states can be prepared so that every complete test gives each of its N outcomes with probability 1/N.DReciprocity2.2The chance that a system prepared in one pure state passes a test for another is the same in both directions.EInterference2.3When several paths lead from a preparation to an outcome, its probability is not, in general, the sum of the probabilities of the paths.FComposition of amplitudes2.4Every path has a complex amplitude. Amplitudes multiply along a path and, when nothing records which path was taken, add across paths.GSuperposition3.1Every complex vector other than zero stands for a pure state that can be prepared, and every orthonormal basis for a complete test that can be done.HThe expectation rule3.2A system prepared in state u passes a test for state v with probability |⟨u, v⟩|². Rules I and J add that real apparatus, obeying ordinary physical law, must be able to do this.KCompleteness3.4The density matrix says everything there is to say about how a system was prepared. Two recipes with the same matrix cannot be told apart.

About the book

Asher Peres, Quantum Theory: Concepts and Methods (Kluwer, 1993). Peres starts from what is done in a laboratory, preparations and tests, and lets the mathematics follow. That is why it can be taught from the ground up. The explanations, figures and questions here are new; the order of the ideas and the lettering of the rules are his. If these lessons go well, read the book: its exercises go much further than ours.