Angular Momentum in Cartesian Coordinates

The angular momentum has three components —, and — along the X-, y-, and z-axes in a Cartesian coordinate system. These components satisfy the eigenvalue-eigenfunction equation, with eigenvalue matrix, A  

When an angular momentum is directed toward arbitrary directions that have direction cosines cos a, cos P, and cos y, where a, P, and y are Euler angles, angular momentum M(a, P, y) may be written as follows  

Eigenfunetion matrix ( )(a,p,y) and its inverse, ( ) (a,p,y), are obviously dependent on the Euler angles. As a result, projeetion matriees P(a,p,y) are also dependent on the direetion. 

Note that when one of the direetion eosines, for example, cos y, is unity in eq. (23), the remaining terms cos a and cos P should be simultaneously zero. The following relations must therefore be introduced for the sake of removing infinite 

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The meaning of the integer m remains to be investigated. To do this we consider the classical z component of angular momentum in Cartesian coordinates. This has the form... 

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