Explicit Cartesian expressions for the complex solid harmonics

3 EXPLICIT CARTESIAN EXPRESSIONS FOR THE COMPLEX SOLID HARMONICS 

The complex solid harmonics are eigenfiinctirxis of the total angular-momentum operator, which in (6.3.4) is given in polar coordinates. The form of this operator is rather complicated, however, making the solution of the associated eigenvalue problem (in order to arrive at the solid harmonics) 

The solid harmonics are easily seen to be solutions to Laplace s equation [14]. Thus, from the expressions (6.3.3) and (6.4.1), we find 

Moreover, if we have a solution to Laplace s equation in the form r f(6, p), then the angular part /(6 , p) must be an eigenfunction of with eigenvalue /(/ -f 1) - in other words, f(B, p) must then be a linear combination of the spherical harmonics Yi with l m 1. 

Let us determine, in Cartesian coordinates, the solutions to Laplace s equation that correspond to the solid harmonics for a given angular momentum 1. AccortUng to the preceding discussion, the solution must be a polynomial in x, y and z of degree /  

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