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suivant: Autres relations utiles faisant monter: Généralisation. précédent: Généralisation.

Commutateurs de $J_+$ et $J_-$.

Les opérateurs $J_+$ et $J_-$ satisfont les relations de commutation suivantes :
$\displaystyle \lbrack J_+,J_z \rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_z \rbrack + \imath \lbrack J_y,J_z \rbrack = -\imath\hbar J_y - \hbar J_x = -\hbar J_+$  
$\displaystyle \lbrack J_-,J_z \rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_z \rbrack - \imath \lbrack J_y,J_z \rbrack = -\imath\hbar J_y + \hbar J_x = \hbar J_-$  
$\displaystyle \lbrack J_+,J_x\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_x\rbrack + \imath \lbrack J_y,J_x\rbrack = \hbar J_z$  
$\displaystyle \lbrack J_-,J_x\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_x\rbrack - \imath \lbrack J_y,J_x\rbrack = -\hbar J_z$  
$\displaystyle \lbrack J_+,J_y\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_y\rbrack + \imath \lbrack J_y,J_y\rbrack = \imath\hbar J_z$  
$\displaystyle \lbrack J_-,J_y\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J_y\rbrack - \imath \lbrack J_y,J_y\rbrack = \imath\hbar J_z$  
$\displaystyle \lbrack J_+,J^2\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J^2\rbrack + \imath \lbrack J_y,J^2 \rbrack = 0$  
$\displaystyle \lbrack J_-,J^2\rbrack$ $\textstyle =$ $\displaystyle \lbrack J_x,J^2\rbrack - \imath \lbrack J_y,J^2 \rbrack = 0$  
$\displaystyle \lbrack J_+,J_-\rbrack$ $\textstyle =$ $\displaystyle - \lbrack J_-,J_+\rbrack = \lbrack J_+,J_x\rbrack - \imath \lbrack J_+,J_y\rbrack = 2 \hbar J_z$ (2.20)


Bernard Silvi 2005-03-16