Functions and Surface Harmonics

There are consequently 21 + 1 independent wave functions with given values of n and l, and n2 independent wave functions with a given value of n, that is, with the same energy value. The 21 + 1 wave functions with the same n and l are said to form a completed subgroup, and the n2 wave functions with the same n a completed group. The wave functions will be described in the following sections of this chapter. 

A similar treatment applied to the wave equation with W positive leads to the result that there exist acceptable solutions for all positive values of the energy, as indicated by the general discussion of Section 9c. It is a particularly pleasing feature of the quantum mechanics that a unified treatment can be given the continuous as well as the discrete spectrum of energy values. Because of the rather complicated nature of the discussion of the wave functions for the continuous spectrum (in particular their orthogonality and normalization properties) and of their minor importance for most chemical problems, we shall not treat them further.1 

The functions of which we have obtained by solution of the equation are well known to mathematicians under the name of associated Legendre functions.2 The functions of d and p are 

1 The functions of 0 for m =0 are called Legendre functions. The associated Legendre functions include the Legendre functions and additional 

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