Orbital angular momentum and spherical harmonics

Orbital angular momentum is associated with rotational motion in three-dimensional 

Here 6 and 4 are the spherical polar angles (only two angles are required to define the orientation of the vector r in space). Since these operators are the same as the infinitesimal rotation operators, all the results of the previous sections apply. The eigenfunctions of and Lz axe known as the spherical harmonics, 

For a single-valued solution, m must be an integer (and so therefore must ). The functions 0fa(6 ) can be found by solving 

The spherical harmonics are normalised with respect to integration 

The explicit forms of the first few spherical harmonics are given in table 5.1. It is sometimes more convenient to use modified spherical harmonics C j, defined by 

Orbital angular momentum is associated with rotational motion in three-dimensional space. In terms of the operators representing the position r and linear momentum p of a particle, we have the important expression for the orbital angular momentum 

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