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Commutation Relations of the Angular Momentum Operators

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Consider a particle described by the Cartesian coordinates (x,y,z) = X and their conjugate momenta (px, py, pz) = p. The classical definition of the orbital angular momentum of such a particle about the origin is L = X x p.

Let us assume that the operators (Lx, Ly, Lz) = L which represent the components of orbital angular momentum in quantum mechanics can be defined in an analogous manner to the corresponding components of classical angular momentum. In other words, we are going to assume that the above equations specify the angular momentum operators in terms of the position and linear momentum operators. Notes that Lx, Ly and Lz are Hermitian, so they represent things which can, in principle, be measure. Note, also there is no ambiguity regarding the order in which operators appear in products on the right-hand sides of the equations because all product consists of operators that commute.....

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The solution shows that [Lx,Ly] = ihLz in two methods. One is the direct calculus approach while the other uses commutation relations of the coordinate and linear momentum operators.

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