===== Hilbert space mean value =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V$...Hilbert space |

| @#FFBB00: definiendum | @#FFBB00: $\overline{\cdot}_{-}:\mathrm{Observable}(V)\times V\to\mathbb R$ |
| @#FFBB00: definiendum | @#FFBB00: $\overline{A}_{\psi}:=\frac{\langle \psi | A\ \psi \rangle}{\Vert \psi \Vert^2}$ |

==== Discussion ====
One can rewrite this in many ways using: 

  * $\langle \psi | A\ \psi \rangle=\langle A \rangle_\psi$

  * $\Vert \psi \Vert^2=\langle \psi | \psi \rangle=\langle 1 \rangle_\psi$


For any vector $\phi$ we have...

  * $\Delta_\psi A = \left(\overline{\left(A-\overline A\right)^2}\right)^\frac{1}{2} = \overline{A^2}-\overline{A}^2=\frac{\Vert(A-\overline A)\psi\Vert}{\Vert\psi\Vert}$ is called non-negative mean fluctuation. 

  * $\gamma=\overline{(A-\overline{A})(B-\overline{B})}/(\Delta A\cdot \Delta B)=(\overline{AB}-\overline{A}\overline{B})/(\Delta A\cdot \Delta B)$ is called the correlation coefficient.

=== Theorems ===

$AB=BA\implies \gamma\in [-1,1]$.

==== Parents ====
=== Context ===
[[Hilbert space]]