Observables · Quantum Mechanics

Intuition

A measurable quantity — a polarisation, an energy, a position — is represented in quantum mechanics by a Hermitian operator, called an observable. The numbers a measurement can return are its eigenvalues and nothing else, which is where quantisation comes from: a quantity whose operator has a discrete set of eigenvalues can only ever be found at those values.

An observable is a question with a fixed list of possible answers, written on a dial that has only certain marks. However the needle is nudged, it can only come to rest on a mark. The marks are the eigenvalues; which mark it stops at is a matter of probability.

The spectrum of an observable with eigenvalues $-2$ , $0.5$ and $2$ . Every measurement of it returns one of these three numbers; no value between them is ever read, however the system was prepared.

The first postulate of measurement

Each measurable quantity of a system corresponds to a Hermitian operator $\hat{A}$ on its state space, its observable. The possible results of measuring it are the eigenvalues of $\hat{A}$ , the set of which is its spectrum. The chapter on operators showed why a Hermitian operator is the right object: its eigenvalues are real, and its eigenvectors form an orthonormal basis of states that can be told apart with certainty.

\hat{A}^{\dagger}=\hat{A},\qquad \hat{A}\ket{a_{n}}=a_{n}\ket{a_{n}},\qquad \text{results}\in\{a_{n}\}

What the postulate says

A state with a definite value of $\hat{A}$ is an eigenvector: measuring $\hat{A}$ on $\ket{a_{n}}$ gives with certainty.

Applications

Practice

Quantities Are Operators

In quantum mechanics a measurable quantity is represented by a Hermitian operator, its observable. The results a measurement can give are the eigenvalues of that operator.

Try it

Which mathematical object represents a measurable quantity in quantum mechanics?