Quantum Mechanics · Chapter 03

Observables and measurement

The postulates that connect the mathematics to an experiment: which numbers a measurement can return, with what probabilities, and what it leaves the state as. Expectation values and spreads, the uncertainty relation proved from the Cauchy–Schwarz inequality, what happens when measurements are repeated, and which observables can be known together.

A^^{†} = A^

P (a) = ∣ ⟨ a ∣ ψ ⟩ ∣^{2}

⟨ A^⟩ = ⟨ ψ ∣ A^∣ ψ ⟩

Δ A Δ B \geq \frac{1}{2} ∣ ⟨[A^, B^]⟩ ∣

Lessons

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05

The uncertainty relationCan a state be sharp for two observables at once? If they commute, yes. If they do not, their spreads are tied together: the product of the two uncertainties can be no smaller than half the size of the average of their commutator. It follows from nothing more than the Cauchy–Schwarz inequality of the first chapter, and for position and momentum it becomes Heisenberg’s famous limit.

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06Repeated measurementsMeasure a quantity and then measure it again straight away: the second result repeats the first, because the first measurement left the system in the eigenvector it found. Measure something incompatible in between and the memory is wiped: the first quantity is random again. Three polarisers show it vividly — horizontal and vertical together block everything, but a diagonal one slipped between them lets light through.Read lesson →

07Compatible observablesSome pairs of quantities can be known together and some cannot. The dividing line is the commutator: observables that commute share a basis of eigenvectors, so a state can be sharp for both, measuring one does not disturb the other, and the order of measuring them does not matter. A set of commuting observables whose values pin down a single state is the fullest description quantum mechanics allows.Read lesson →

08Degeneracy and measurementWhen several independent states share the same eigenvalue, finding that value does not say which of them the system is in. The measurement then keeps the part of the state inside that eigenvalue’s whole subspace, superposition and all, and throws away the rest. A second, compatible observable can then split the subspace further.Read lesson →

09Chapter practiceBring the chapter together: which results a measurement can give and with what probabilities, the state it leaves, averages and spreads, the uncertainty relation, sequences of measurements, compatible observables and degenerate results. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Observables and measurement

Build the foundation