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chapter7 - Chapter 5 ANGULAR MOMENTUM AND ROTATIONS ~ In...

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Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum ~ L of an isolated system about any …xed point is conserved. The existence of a conserved vector ~ L associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents L i of the total angular momentum operator ~ L of an isolated system are also constants of the motion. The di¤erent components of ~ L are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator ~ L is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. Indeed, it is this lack of commutivity that imparts to angular momentum observables their rich characteristic structure and makes them quite useful, e.g., in classi- fying the bound states of atomic, molecular, and nuclear systems containing one or more particles, and in decomposing the scattering states of such systems into components as- sociated with di¤erent angular momenta. Just as importantly, the existence of internal “spin” degrees of freedom, i.e., intrinsic angular momenta associated with the internal structure of fundamental particles, provides additional motivation for the study of angu- lar momentum and to the general properties exhibited by dynamical quantum systems under rotations. 5.1 Orbital Angular Momentum of One or More Particles The classical orbital angular momentum of a single particle about a given origin is given by the cross product ~ ` = ~ r £ ~p (5.1) of its position and momentum vectors. The total angular momentum of a system of such structureless point particles is then the vector sum ~ L = X ® ~ ` ® = X ® ~ r ® £ ~ p ® (5.2)
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162 Angular Momentum and Rotations of the individual angular momenta of the particles making up the collection. In quantum
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