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