## Observable

### Set

context | $V$…Hilbert space |

definiendum | $\mathrm{Observable}(V)\equiv\mathrm{SelfAdjoint}(V)\cap\mathrm{End}(V)$ |

#### Discussion

Observables are the linear self-adjoint operators.

- $\langle\psi|A\ \phi\rangle\in \mathbb C$ is called
*transition amplitude*.

- $\frac{|\langle\psi|A\ \phi\rangle|^2}{\Vert\psi\Vert^2\Vert\psi\Vert^2}\ge 0$ is called
*transition probability*.

- $\langle \psi | A\ \psi \rangle\in \mathbb R$ is called expectation value.

- $\frac{\langle \psi | A\ \psi \rangle}{\Vert\psi\Vert^2}\in \mathbb R$ is called mean value.

#### Reference

Wikipedia: Self-adjoint operator, Observable