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 hilbert_space_mean_value [2013/08/31 23:43]nikolaj hilbert_space_mean_value [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/08/31 23:43 nikolaj 2013/08/31 23:43 nikolaj 2013/08/31 23:10 nikolaj 2013/08/31 23:10 nikolaj 2013/08/31 23:09 nikolaj 2013/08/31 21:52 nikolaj 2013/08/31 21:51 nikolaj 2013/08/31 21:40 nikolaj created Next revision Previous revision 2013/08/31 23:43 nikolaj 2013/08/31 23:43 nikolaj 2013/08/31 23:10 nikolaj 2013/08/31 23:10 nikolaj 2013/08/31 23:09 nikolaj 2013/08/31 21:52 nikolaj 2013/08/31 21:51 nikolaj 2013/08/31 21:40 nikolaj created Line 1: Line 1: ===== Hilbert space mean value ===== ===== Hilbert space mean value ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $V$...Hilbert space | + | @#55CCEE: context ​    | @#55CCEE: $V$...Hilbert space | - | @#FFBB00: $\overline{\cdot}_{-}:​\mathrm{Observable}(V)\times V\to\mathbb R$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\overline{\cdot}_{-}:​\mathrm{Observable}(V)\times V\to\mathbb R$ | - | @#FFBB00: $\overline{A}_{\psi}:​=\frac{\langle \psi | A\ \psi \rangle}{\Vert \psi \Vert^2}$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\overline{A}_{\psi}:​=\frac{\langle \psi | A\ \psi \rangle}{\Vert \psi \Vert^2}$ | ==== Discussion ==== ==== Discussion ==== Line 19: Line 19: * $\gamma=\overline{(A-\overline{A})(B-\overline{B})}/​(\Delta A\cdot \Delta B)=(\overline{AB}-\overline{A}\overline{B})/​(\Delta A\cdot \Delta B)$ is called the correlation coefficient. * $\gamma=\overline{(A-\overline{A})(B-\overline{B})}/​(\Delta A\cdot \Delta B)=(\overline{AB}-\overline{A}\overline{B})/​(\Delta A\cdot \Delta B)$ is called the correlation coefficient. + + === Theorems === $AB=BA\implies \gamma\in [-1,1]$. $AB=BA\implies \gamma\in [-1,1]$. - ==== Context ​==== + ==== Parents ​==== - === Requirements ​=== + === Context ​=== [[Hilbert space]] [[Hilbert space]]