Cazul spectrului pur continuu

Pentru cazul spectrului pur continuu (de exemplu cel al operatorului pozitie):

$$\widehat{x}=\int\limits_{\Bbb{R}}x\cdot \widehat{E}(dx)=\int\limits_{\Bbb{R} }x\cdot \vert x\rangle \langle x\vert dx$$

luam in (4.2.14) $\Omega =\Bbb{R}$ si $N(\omega )=N_0=1$. Pentru operatorul fuzzy luam:

$$\widehat{F}(B)=\int\limits_{\Bbb{R}}(f\circ \chi _B)(x)\widehat{E} (dx)=\int\limits_Bd\lambda \int\limits_{\Bbb{R}}f(\lambda -x)\widehat{E}(dx)$$

unde $\int\limits_{\Bbb{R}}f(x)dx=1$ este o conditie similara (4.3.26) si $f(x)>0\quad \forall x\in \Bbb{R}$. Cautam operatorul $\widehat{\mathcal{F }}_c$ pentru care:

$$f(\lambda -x)=\langle x\vert\widehat{\mathcal{F}}_c^{+}\vert\lambda \rangle \langle \lambda \vert\widehat{\mathcal{F}}_c\vert x\rangle$$

Luam cazul real pozitiv:

$$\langle \lambda \vert\widehat{\mathcal{F}}_c\vert x\rangle =\sqrt{f(\lambda -x)}$$ (4.3.42)

si cautam transformarea de stare similara (4.3.36):

$$\widehat{\rho }\mapsto \widehat{\rho }_{post}^{(E,c)}=\int\limits... ...e x\vert\widehat{\rho }\vert x\rangle \langle x\vert\widehat{\mathcal{F}}_c^{+}$$ (4.3.43)

Transformarea de stare OQP este (ecuatiile 4.6.3 si 4.6.4 in [14]):

$$\widehat{\rho }_{post}^{(c)}=\int\limits_{\Bbb{R}}dx\widehat{\mathcal{A}}_x \widehat{\rho }\widehat{\mathcal{A}}_x^{+}$$ (4.3.44)

unde:

$$\langle \lambda \vert\widehat{\mathcal{A}}_x\vert\Psi \rangle =\s... ...(\lambda )\quad \forall \lambda \in \Bbb{{{R},\forall }\Psi {\in \mathcal{H}}}$$

$$\widehat{\mathcal{A}}_x=\widehat{\mathcal{A}}_x^{+}=\int\limits_{... ..._{\Bbb{R}}\langle x\vert\widehat{ \mathcal{F}}_c\vert y\rangle \widehat{E}(dy)$$

in timp ce (4.3.43) se poate scrie in forma:

$$\widehat{\rho }_{post}^{(E,c)}=\int\limits_{\Bbb{R}}dx\widehat{\mathcal{A}} _x^{(E)}\widehat{\rho }\widehat{\mathcal{A}}_x^{(E)+}$$ (4.3.45)

unde:

$$\widehat{\mathcal{A}}_x^{(E)} = \widehat{\mathcal{F}}_c\vert x\rangle \langle x\vert$$

$$\widehat{\mathcal{A}}_x^{(E)+} = \vert x\rangle \langle x\vert\widehat{\mathcal{F}} _c^{+}$$