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numerical-range:generalizations:restricted-numerical-range:coherent-numerical-range

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 numerical-range:generalizations:restricted-numerical-range:coherent-numerical-range [2018/05/22 13:53]plewandowska numerical-range:generalizations:restricted-numerical-range:coherent-numerical-range [2018/10/08 08:09] (current)plewandowska [Definition of $SU(2)$ coherent states] Both sides previous revision Previous revision 2018/10/08 08:09 plewandowska [Definition of $SU(2)$ coherent states] 2018/05/22 13:53 plewandowska 2013/06/03 14:37 lpawela 2013/06/03 14:35 lpawela 2013/06/03 14:34 lpawela created 2018/10/08 08:09 plewandowska [Definition of $SU(2)$ coherent states] 2018/05/22 13:53 plewandowska 2013/06/03 14:37 lpawela 2013/06/03 14:35 lpawela 2013/06/03 14:34 lpawela created Line 12: Line 12: $SU(2)$ coherent states are related to the $SU(2)$ algebra of the $SU(2)$ coherent states are related to the $SU(2)$ algebra of the - components of the angular momentum operator $J = \{J_x , J_y , J_z \}$. Let us choose a reference state $\ket{\kappa}$,​ usually taken as the maximal eigenstate $\ket{j,j}$ of the component $J_z$ acting on $\mathcal{H}_N$ , $N = 2j + 1$, $j = 1/2, 1/3, \ldots$. This state, pointing toward the ‘north pole’ of the sphere, enjoys the minimal uncertainty equal to $j$ . Then, the vector coherent state is defined by the Wigner rotation matrix $R_{\vartheta,​ \varphi}$ + components of the angular momentum operator $J = \{J_x , J_y , J_z \}$. Let us choose a reference state $\ket{\kappa}$,​ usually taken as the maximal eigenstate $\ket{j,j}$ of the component $J_z$ acting on $\mathcal{H}_d$ , $d = 2j + 1$, $j = 1/2, 1/3, \ldots$. This state, pointing toward the ‘north pole’ of the sphere, enjoys the minimal uncertainty equal to $j$ . Then, the vector coherent state is defined by the Wigner rotation matrix $R_{\vartheta,​ \varphi}$  \ket{\vartheta,​\varphi} = R_{\vartheta,​\varphi} \ket{\kappa} = (1 + |\gamma|^2)^{−j} e^{\gamma J_−} \ket{j, j} \ket{\vartheta,​\varphi} = R_{\vartheta,​\varphi} \ket{\kappa} = (1 + |\gamma|^2)^{−j} e^{\gamma J_−} \ket{j, j}  where $R_{\vartheta,​\varphi} = \exp[( \ii \vartheta( \cos\varphi J_x − \sin \varphi J_y)]$ , $J_− = J_x − \ii J_y$ and $\gamma = \tan(\vartheta/​2)e^{\ii\varphi}$,​ for $(\vartheta,​\varphi) \in S^2$ (we use the spherical coordinates). where $R_{\vartheta,​\varphi} = \exp[( \ii \vartheta( \cos\varphi J_x − \sin \varphi J_y)]$ , $J_− = J_x − \ii J_y$ and $\gamma = \tan(\vartheta/​2)e^{\ii\varphi}$,​ for $(\vartheta,​\varphi) \in S^2$ (we use the spherical coordinates).