Postulate 2: Observables

Postulate 1 told us where states live. Postulate 2 tells us how to represent the measurable quantities of a system — energy, spin, position, polarization. The central claim is that every such observable corresponds to a particular kind of linear operator.

The observable postulate

Postulate 2. Every measurable physical quantity (an observable) is represented by a Hermitian operator $\hat{A}$ acting on the state space $\mathcal{H}$ . The possible results of measuring the observable are the eigenvalues of $\hat{A}$ .

A Hermitian operator satisfies

\hat{A}^{\dagger} = \hat{A},

where $\dagger$ denotes the conjugate transpose (adjoint). In a matrix representation this means $A_{ij} = A_{ji}^{*}$ .

Why Hermitian? Real spectra

Measurement results are real numbers — a voltmeter never reads $3 + 2i$ . Hermiticity guarantees this. If $\hat{A}|a\rangle = a|a\rangle$ with $|a\rangle \neq 0$ , then

a\,\langle a|a\rangle = \langle a|\hat{A}|a\rangle = \langle a|\hat{A}^{\dagger}|a\rangle = \overline{\langle a|\hat{A}|a\rangle} = \overline{a}\,\langle a|a\rangle.

Since $\langle a|a\rangle > 0$ , we conclude $a = \overline{a}$ , i.e. every eigenvalue is real. This is exactly why the eigenvalues are admissible measurement outcomes.

The spectral theorem and the eigenbasis

A second crucial property of Hermitian operators (on a finite-dimensional space) is the spectral theorem: eigenvectors belonging to distinct eigenvalues are orthogonal, and one can always choose an orthonormal basis of eigenvectors. Writing the (real) eigenvalues $a_k$ with orthonormal eigenvectors $|a_k\rangle$ , the operator decomposes as

\hat{A} = \sum_k a_k\, |a_k\rangle\langle a_k|.

This spectral decomposition is the workhorse of quantum measurement: the projectors $\hat{P}_k = |a_k\rangle\langle a_k|$ pick out the part of the state associated with outcome $a_k$ , and we will use them in the next lessons to state the measurement and Born rules cleanly.

Example: the Pauli observables for a qubit

For a single qubit the three Pauli matrices are Hermitian observables:

\hat{X} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \hat{Y} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \hat{Z} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Each is its own adjoint. Each has eigenvalues $+1$ and $-1$ — the two possible outcomes of a spin- $\tfrac12$ measurement along the corresponding axis (in units of $\hbar/2$ ). The eigenvectors of $\hat{Z}$ are $|0\rangle$ and $|1\rangle$ ; the eigenvectors of $\hat{X}$ are $|\pm\rangle = \tfrac{1}{\sqrt2}(|0\rangle \pm |1\rangle)$ . Knowing an observable means knowing both its outcomes (eigenvalues) and the states that yield those outcomes with certainty (eigenvectors).

What an operator does not, by itself, tell you

Postulate 2 connects an observable to its spectrum, but it is silent about which eigenvalue you will actually get on a given run and with what probability. A state is generally a superposition of eigenvectors, so the outcome is not determined. Supplying the probabilities — and the rule for how the state changes afterward — is the job of Postulate 3, the measurement postulate.