Transcription of Angular Momentum 1 Angular momentum in Quantum …
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Angular Momentum 1 Angular momentum in Quantum …
J. BroidaUCSD Fall 2009 Phys 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.
Because J2 and J z commute they may be simultaneously diagonalized, and we denote their (un-normalized) simultaneous eigenfunctions by Yβ α where J2Yβ α = ~ 2αYβ α and J zY β α = ~βY β α. Since J i is Hermitian we have the general result hJ 2 ii = hψ|J2 i ψi = hJψ|Jψi = kJψk ≥ 0 and hence hJ 2i−hJ zi = hJ xi+hJ y 2i ≥ 0.But J2 z Y β α = ~ 2β Yβ α and hence we must
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