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The Effect Of Angular Momentum On Classical Mechanics

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INTRODUCTION

We know the importance of Angular Momentum in Classical Mechanics; the total angular momentum of an isolated physical system is a constant of the motion. For example, if a point particle P, of mass m, is moving in a central potential (one which depends only on the distance between P and a fixed point O), the force to which P is subjected is always directed towards O. Its moment with respect to O is consequently zero, and the angular momentum theorem implies that derivative of L (Angular momentum of P with respect to O) with respect to time is zero.
This fact has important consequences: the motion of the particle P is limited to a fixed plane (the plane passing through O and perpendicular to the angular momentum L); moreover, this motion obeys the law of constant areal velocity (Kepler’s Second Law).
All these properties have their equivalents in Quantum Mechanics. With the angular momentum L of a classical system is associated with an observable L, actually a set of three observables, Lx, Ly, and Lz, which correspond to the three components of L in a Cartesian Frame. These three observables commute with the Hamiltonian H for a particle in the central potential V(r). This property simplifies the determination and classification of eigenstates of H.

Quantization of Angular Momentum: the component, along a fixed axis, of the intrinsic angular momenta are quantized, which enables us to understand atomic magnetism, the Zeeman Effect, etc. We shall denote by

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