What is Observables? — Physics

Definition 8.3 University Physics — Year 3 · Chapter 8 — The Formalism of Quantum Mechanics

An observable is a linear operator $\hat A$ on $\mathcal H$ that is Hermitian: $\braket{\varphi}{\hat A\psi} = \braket{\hat A\varphi}{\psi}$ for all states (in matrix language, $A = A^{*\top}$ ). Its eigenvectors and eigenvalues, $\hat A\ket{a} = a\ket{a}$ , carry the physics: the $a$ are the possible measured values. Familiar cases: position ( $\hat x$ : multiplication by $x$ ), momentum ( $\hat p = -\iu\hbar\,\dd/\dd x$ ), energy ( $\hat H = \hat p^2/2m + V(\hat x)$ ) — and, in two dimensions, any Hermitian $2\times2$ matrix.

Examples

Example 8.5 (A two-level observable)

On the polarization space, in the $(\ket H, \ket V)$ basis, consider

\hat A = \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix} :

Hermitian; eigenvalues $\pm1$ ; eigenvectors $(\ket H \pm \ket V)/\sqrt2$ — the polarizations at $\pm45^\circ$ . Measuring $\hat A$ means asking “diagonal or antidiagonal?”, and the eigenbasis is the pair of questions’ answers. Every $45^\circ$ polarizer in the opening experiment is this matrix in glass.

Read in context →