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 is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore, you can choose the format that is most suitable to ...
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.
Addition of Angular Momentum in a Helium Atom
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Consider a Helium atom with two electrons. Suppose you know that one of the electrons is in the l1=3 state, while the other is in the l2 = 2 state. What are the possible values of l1z and l2z? So how many different quantum states describing the orbital angular momentum configuration of the two electrons are possible? Suppose L = L1 + L2 represents the total orbital angular momentum. What are the possible values of l, the quantum number associated with the total orbital angular momentum of the two electrons. For each possible value of l, list the possible values of lz, the total z-component of the orbital angular momentum. Show that counting the states in this l, lz basis agrees with that of the product basis.