Angular momentum component quantum

The quantum number K gives the rotational angular-momentum component along a molecule-fixed axis of the spherical top. From (5.34) and... 

Since the possible numerical values of the angular momentum components are integers it appears reasonable that the modulus L, which classically is the maximum possible for any of the components, should obey a rule of the same form, i.e. L = lh, l integer. This rule however, does not hold for quantum systems. Since... 

Show that the x and z components of the angular momentum have quantum mechanical operators that do not commute and find their commutator ... 

K i the quantum number of the angular momentum component in the direction of the... 

The terms of an atom with a valence electron are characterised by four quantum numbers, n, Z, j, m. Here I corresponds to the angular momentum of the electronic orbit in the model, j to the total angular momentum of the atom, and m to its component in the direction of an externed field. For the formal representation we can ascribe the angular momentum component which contributes to j apart from... 

Another useful operator can be found from the angular momentum component operators. The square of the angular momentum of a classical system, or of a quantum mechanical system, is a scalar quantity, the dot product of L with itself. The dot product of any vector with itself equals the square of the length of the vector. Thus, L Lis the square of the magnitude of the angular momentum vector. The quantum mechanical operator that corresponds to this dot product is designated U, and its explicit form can be derived from the component expression for a dot product. Thus,... 

Here i denote the axes of a coordinate system attached to the body in which the moment of inertia tensor is diagonal. These diagonal elements are Ij and Lf are the attendant angular momentum components squared. Quantum mechanically we have... 

The components of the quantum mechanical angular momentum operators along the three principal axes are ... 

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