Observable

Remarks

The spectrum $\sigma(A)$ of an observable is a closed subset of $\R$; its points are the possible results of a measurement of $A$. By the spectral theorem, $A$ has a unique projection-valued measure $P^A$ on the Borel sets of $\R$ with $A = \int_\R \lambda \, dP^A(\lambda)$. The eigenvectors of $A$ are the states in which a measurement of $A$ gives one value with certainty.

Example: the matrix $\sigma_z = \mathrm{diag}(1, -1)$ on $\C^2$ is self-adjoint, so it is an observable. Its spectrum is $\set{1, -1}$, and $P^{\sigma_z}(\set{1}) = \mathrm{diag}(1, 0)$.

Non-example: $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ on $\C^2$ is not an observable, since $N^* = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \ne N$. It has the single eigenvalue $0$ but no basis of eigenvectors, so it has no projection-valued measure.

For unbounded operators, being symmetric, $\ip{A\psi}{\varphi} = \ip{\psi}{A\varphi}$ on $D(A)$, is weaker than self-adjoint, and only self-adjoint operators have a spectral theorem. Generalised measurements are described by positive operator-valued measures instead of single observables.