Hilbert space mean value

Set

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

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

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