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hilbert_space_mean_value [2013/08/31 23:43] nikolaj |
hilbert_space_mean_value [2014/03/21 11:11] |
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| - | ===== Hilbert space mean value ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $V$...Hilbert space | | ||
| - | | @#FFBB00: $\overline{\cdot}_{-}:\mathrm{Observable}(V)\times V\to\mathbb R$ | | ||
| - | | @#FFBB00: $\overline{A}_{\psi}:=\frac{\langle \psi | A\ \psi \rangle}{\Vert \psi \Vert^2}$ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | One can rewrite this in many ways using: | ||
| - | |||
| - | * $\langle \psi | A\ \psi \rangle=\langle A \rangle_\psi$ | ||
| - | |||
| - | * $\Vert \psi \Vert^2=\langle \psi | \psi \rangle=\langle 1 \rangle_\psi$ | ||
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| - | |||
| - | For any vector $\phi$ we have... | ||
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| - | * $\Delta_\psi A = \left(\overline{\left(A-\overline A\right)^2}\right)^\frac{1}{2} = \overline{A^2}-\overline{A}^2=\frac{\Vert(A-\overline A)\psi\Vert}{\Vert\psi\Vert}$ 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. | ||
| - | |||
| - | $AB=BA\implies \gamma\in [-1,1]$. | ||
| - | |||
| - | ==== Context ==== | ||
| - | === Requirements === | ||
| - | [[Hilbert space]] | ||
