Laguerre’s polynomials

Here Laguerre s polynomial functions are used. The quantity S can by expressed as shown below, with iC being the force constant of a harmonic oscillator and Qo the coordinate of the equilibrium position of the excited state. 

It eventually becomes the Laguerre s polynomial of degree n for appropriately choosing of = 1 for and = for such that to have the connection... 

In the same way as proceeded with the Laguerre s polynomial, the Hermite equation may be solved by the speeifie eomplex integral representation ... 

This can be done in the same way as proceed with the Laguerre s polynomials, i.e., by considering the product of two Hermite s generating functions ... 

Methods such as Graeffe s root-squaring method, Muller s method, Laguerre s method, and others exist for finding all roots of polynomials with real coefficients. [12 and others]... 

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

The following relations involving the A s can be easily derived from the properties of the associated Laguerre polynomials ... 

The Laguerre polynomials Lkip) are defined by means of the generating function g(p, s)... 

In order to obtain the orthogonality and normalization relations of the assoeiate Laguerre polynomials, we make use of the generating fimction (F. 10). We multiply together g(p, s j), g(p, t J), and the factor pj+ e-f and then integrate over p to give an integral that we abbreviate with the symbol /... 

Thus, the associated Laguerre polynomials form an orthogonal set over the range 0 p ss 00 with a weighting factor p er< . For the case where s and t on the left-hand side have the same exponent, we pick out the term fi = am the summation over (i, giving... 

The sth derivative of the nth Laguerre polynomial is the associated Laguerre polynomial of degree n — s and order s,... 

Since LsT x) — ( )s Lr(x) the generating function for the associated Laguerre polynomials follows as... 

Equation (13) is familiar from before (1.8). It is the associated Laguerre equation with the usual s replaced by 21 + 1 and n by n + l. It follows that the associated Laguerre polynomial L + x) is a solution of (12) and also that... 

The radial functions S i(p) and R i(r) may be expressed in terms of the associated Laguerre polynomials LJk(p), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between S i(p) and L[(p) is to relate S i(p) in equation (6.50) to the polynomial LJk(p) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation. 

The absorption spectrum consists of sequences of transitions from v" = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

In quantum mechanics and other branches of mathematical physics, we repeatedly encounter what are called special functions. These are often solutions of second-order differential equations with variable coefficients. The most famous examples are Bessel functions, which we wiU not need in this book. Our first encounter with special functions are the Hermite polynomials, contained in solutions of the Schrodinger equation. In subsequent chapters we will introduce Legendre and Laguerre functions. Sometime in 2004, theU.S. National Institute of Standards and Tec hnology (NIST) will publish an online Digital Library of Mathematical Functions, http / /dlmf. nist. gov, including graphics and cross-references. 

The differential equation satisfied by the associated Laguerre polynomials may be obtained by repeatedly differentiating equations (F.S) j times... 

While the radial function in Equation (38) with integer nr involves the Laguerre polynomials, the corresponding function in Equation (92) with non-integer vr becomes an infinite series. This poses no difficulty in the accurate evaluation of its zeros as already discussed in Section 2.1, in connection with the s-states. The reader may also take notice of the f-dependence of the function and the eigenenergies in Equations (92) and... 

The exact solutions to the separate equations, which result from this coordinate transformation of the Schrddinger equation for the hydrogen atom, are the sets of functions known as the associated Laguerre polynomials, for the radial equation, and the spherical harmonics, for the angular equation. The quantum numbers, n,l and m arise naturally in the solution of Schrddinger s equation, and so the symbolic form, for the eigenfunction solutions to the H-atom problem, known as atomic orbitals, is... 

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. 

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