Transcription of Chapter 9 Angular Momentum Quantum Mechanical …
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Chapter 9 Angular MomentumQuantum Mechanical Angular Momentum OperatorsClassical Angular Momentum is a vector quantity denoted~L=~rX~p. A common mnemonicto calculate the components is~L= ^i^j^kxyzpxpypz = ypz zpy ^i+ zpx xpz ^j+ xpy ypx ^j=Lx^i+Ly^j+Lz^j:Let's focus on one component of Angular Momentum , sayLx=ypz the rightside of the equation are two components of position and two components of linear mechanically, all four quantities are operators. Since the product of two operators is anoperator, and the di erence of operators is another operator, we expect the components of angularmomentum to be operators. In other words, Quantum mechanicallyLx=YPz ZPy;Ly=ZPx X Pz;Lz=X Py YPx:These are the components. Angular Momentum is the vector sum of the components. The sumof operators is another operator, so Angular Momentum is an operator. We have not encounteredan operator like this one, however, this operator is comparable to a vector sum of operators; it isessentially a ket with operator components.
generally three dimensional. The generalization to three dimensions2;3 is £ X i; X j ⁄ = 0; (9¡3) 2 Cohen-Tannoudji, Quantum Mechanics (John Wiley & Sons, New York, 1977), pp 149 { 151. 3 Sakurai, Modern Quantum Mechanics (Addison{Wesley Publishing Company, Reading, Mas-sachusetts; 1994), pp 44 { 51. 301
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