The Kinetic Moment²

The results listed below go to the eigenfunctions and eigenvalues of the squared kinetic moment and z-component of the kinetic moment. One can find the detailed derivation demanding knowledge of the commutativity relations here. One can obtain for the eigenvalues:
 

The calculation here shows m can only be integer. Since the smallest m value (m = - l) corresponds to the sum of the highest possible m value (m = l)  l can be either half-integer or integer. Given l there are 2l + 1 values of m. The half-integer values correspond to spin. The corresponding eigenfunctions are not the space coordinates functions and they can't be derived from the Schroedinger equation. The half-integer eigenvalues of the knetic moment can only be obtained by the kinetic moment operator conversion. Firstly we would like to have a look at the integer eigenvalues, i.e. l can be 0,1,2,3,... and m begins from m = -l and lasts till m = +l.

The eigenfunctions of the kinetic moment z-component depend only on angle j therefore we can choose the following separation of variables:
 

The functions of angle J for m = 0 is called the Legendre Polynomial (P_l(cosJ)) and for m ¹ 0 it is also called the associated Legendre Polynomial (P_l,m(cosJ); it's sometimes written by using superscript index m: P_l^m(cosJ)).
All the Legendre Polynomials P_l^m(cosJ) till l =3 are represented in the polar coordinates on the next figure. Moreover we obtain the function absolute sum which depends on the angle as the radius vector of this angle relative to the z-axis. For your recalling:

It is y = sinα in the Cartesian system:

and the same in polar coordinates:

Fig.1a-k: The polar coordinate J-dependence illustration Legendre Polynomial (a,d,g) and associated Legendre Polynomial.

see Fig.1g-k for P3m

Fig.2: The space quantization: a number of possible kinetic moment L alignments having the kinetic moment quantum number l = 2.

The total function Y_l,m(J,j) = P(J)·f(j)  is called the Ball area function.
The normalization of ball area function originates from the integration over corresponding volume ("angle") dW = sinJ dJ dj:
 

The normalized ball area functions for l = 0, 1, 2, 3, 4
 

The ball area function equations are very hard to plot because of the imaginary j-dependence in the exponent:

Therefore we will use special linear combinations of functions:

It's very interesting! Why can we do it in so arbitrary way? We can do it in this way because of the following

Now we take the two linear combinations: 

and obtain real functions after normalization 
 

So we can produce real functions which are utterly useful for chemical analysis. One can find functions for l and |m_l| in the next table:
 

States having different l are characterized by the following letters (according to historical reasons):

  l = 0,1,2,3,4,5,6,7,8,...
    s,p,d,f,g,h,i,k,l,...

The lowest s-state (l = 0) is isotropic; if there is spherical symmetry. It means on the polar diagram that we obtain the spherical shell since the s-function meaning is equal in all directions (J, j). It's clear because there is no selected orientation for zero electron motion.

For l = 1, or p-state, there are three trigonometric functions p_x, p_y, p_z that describe possible electron movement along 3 coordinate axes and therefore they are important for chemical bounding description. 

For l = 2, or d-state, there are 5 trigonometric functions which are a bit complicated to be plotted. For higher l values (g, h, j...) such figure would be much harder to plot. 

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