Quantum Mechanics · Chapter 02

Operators

Everything that can be done to a quantum state is a linear operator. Adjoints, eigenvalues, the Hermitian operators that will stand for measurable quantities and the unitary ones that will move states in time, projectors, the spectral decomposition that writes an operator through its eigenvalues, and the commutator that measures how badly two operators fail to commute.

A^∣ ψ ⟩

A^^{†}

[A^, B^]

A^∣ a ⟩ = a ∣ a ⟩

Lessons

Build the foundation

Unitary operatorsSome operators change states without destroying any information: every overlap between two states, and so every probability built from overlaps, comes out the same. They are called unitary, and they are how quantum systems change in time, how a polariser is turned, how a quantum gate acts. Undoing one is easy: its inverse is its adjoint.

Read lesson →

06ProjectorsA projector keeps the part of a state that lies in a chosen subspace and throws the rest away. Applying it twice does nothing more than applying it once. The ket-bra of a normalised state is the simplest example, and projectors are how the next chapter will say what a measurement does to a state.Read lesson →

07Spectral decompositionA Hermitian operator does nothing but multiply each of its eigen-directions by its eigenvalue. So it can be written as a sum: each eigenvalue times the projector onto its eigenvectors. Written this way, any function of the operator is the same sum with the function applied to the eigenvalues — squares, square roots and exponentials of operators become as easy as those of numbers.Read lesson →

08CommutatorsTurning a page and then flipping the book over is not the same as flipping it and then turning the page. Operators are like that, and the commutator — one order minus the other — measures by how much. It will decide which quantities can have sharp values together, and in the form position times momentum minus momentum times position it carries the whole of quantum mechanics’ break with classical physics.Read lesson →

09Chapter practiceBring the chapter together: matrices of operators and their products, adjoints, eigenvalues, the Hermitian and unitary operators with their special spectra, projectors, the spectral decomposition and commutators. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Operators

Build the foundation