Laguerre polynomials table

Substituting from the table of associated Laguerre polynomials (1.17) the first few normalized radial wave functions are ... 

In the table of hydrogenlike radial wave functions the polynomial contained in parentheses represents for each function the associated Laguerre polynomial as defined by... 

It should be observed that in contrast to the Hermite and Legendre polynomials, the Laguerre polynomials contain both odd and even powers of x. The first few are given in Table 22.1. 

The Is atomic orbital for the hydrogen atom results as an exact solution, for the choice of the first Laguerre polynomial (n = 1) for the radial wave function and the lowest spherical harmonic (/ = 0) Yqo, for the angular wave function. Thus, from Table 1.1, the normalized Is atomic orbital for the hydrogen atom is. 

FIGURE 3.7 The representations of the electronic probability density of existence (wave-functions) for Hydrogenic atoms, for the superior (excited) levels (or shells, quantified by the number n) with the respective sub-sheUs (or orbitals, quantified by the mixed numbers nl), employing the derived radial wave functions in terms of respective Laguerre polynomials of Table 3.1. 

The lowest power in a Laguerre polynomial is n - (number of terms) = if. Thus, all s orbitals have a constant term in the polynomial, that is R(0) 0 all p orbitals are linear in r for small r all d orbitals are quadratic in r for small r, etc. AO with if = 1 cannot have a constant term in the polynomial in Equation 2.14, since Equation 2.13 is then unsatisfied. AO with if = 2 also cannot have a linear term. The radial functions are thus summarized in Table 2.1. 

The solution which ensures F, is everywhere bounded is the generalized Laguerre polynomial i (w), where / and m are the modal subscripts appearing in Table 14-1. The general expression for F, and specific forms for fundamental and low-order modes are presented in Table 14-2, from which the modal fields are determined by substituting... 

The im (9) functions are related to the associated Legendre polynomials, and the first few are listed in table 6.1. The R i(r) are the radial wave functions, known as associated Laguerre functions, the first few of which are listed in table 6.2. The quantities n, l and m in (6.8) are known as quantum numbers, and have the following allowed values ... 

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