Rigid Rotator and Angular Momentum

Angular momentum is a vector quantity, and for a moving point mass it is the cross-product of a vector r, the vector from the axis of about which the particle is rotating to the particle, and the linear momentum vector p. 

For a system of several point-mass particles, the total angular momentum is the vector sum of the angular momenta of each of the particles. 

Quantum mechanical operators corresponding to the components of an angular momentum vector can be found directly from the familiar rectilinear position and momentum operators, for example. 

With these explicit forms for the operators, it is straightforward to show that the commutator of any pair of them produces ih times the third one  

This transformation enables us to write the angular momentum component operators in either system. Chain rule differentiation provides the substitution for a differential operator in Equation 8.47. For instance. 

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