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Angular Momentum 1 Angular momentum in Quantum …

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J. BroidaUCSD Fall 2009Phys 130B QM IIAngular Momentum1 Angular Momentum in Quantum MechanicsAs is the case with most operators in Quantum mechanics, we start from the clas-sical definition and make the transition to Quantum mechanical operators via thestandard substitutionx xandp i~ . Be aware that I will not distinguisha classical quantity such asxfrom the corresponding Quantum mechanical operatorx. One frequently sees a new notation such as xused to denote the operator, butfor the most part I will take it as clear from the context what is meant. I will alsogenerally usexandrinterchangeably; sometimes I feel that one is preferable overthe other for clarity , Angular Momentum is defined byL=r in QM we have[xi,pj] =i~ ijit follows that [Li,Lj]6= 0. To find out just what this commutation relation is, firstrecall that components of the vector cross product can be written (see the handoutSupplementary Notes on Mathematics)(a b)i= I am using a sloppy summation convention where repeatedindices are summedover even if they are both in the lower position, but this is standard when it comesto Angular Momentum .

Angular Momentum 1 Angular momentum in Quantum Mechanics As is the case with most operators in quantum mechanics, we start from the clas-sical definition and make the transition to quantum mechanical operators via the standard substitution x → x and p → −i~∇. Be aware that I will not distinguish

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