Laguerre function

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Hol0ien, E., Phys. Rev. 104, 1301, Radial configurational interaction in He and similar atomic systems." An orthonormal set of associated Laguerre functions is used. 

Stewart, A. L., Proc. Phys. Soc. A70, 756, "Wave functions for He and similar atomic systems. Perturbation treatment of second order, based on complete set of Laguerre functions. 

Hol0IEN, E., Proc. Phys. Soc. A71, 357, "The 2s2 XS state solution of the non-relativistic Schrodinger equation for He and H. " Associated Laguerre functions. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

Another example, of trivial nature, is the Stark effect, where inspection shows that there are only two nonvanishing terms remaining in (38), to yield of course the exact result, though i Ijas been thought that the set (21) involving the Laguerre functions... 

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 ... 

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. 

We will be interested in the properties of so-called Laguerre functions (van Nie, 1964) as well. They have the functional form... 

The following integral from Gradshteyn and Ryzhik (1980, pp. 843-848) is useful in the evaluation of the Fourier transform of a Laguerre function as well as its convolution with a complex-valued Gaussian (Martin and Bowen, 1993) ... 

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 functions S(m are tesseral (i.e., real combinations of spherical) harmonics, Lf are Laguerre functions, and T(a) are gamma functions (Powell and Craseman, 1961) k is restricted to 0 k n and it must have the same parity as n. The constant A, in the case of a finite basis, can be used to optimize this basis. The matrix elements required in this basis can be easily computed from Eq. (14) and the relation... 

Figure 3 shows the eigenvalues for this (s-wave) problem In a basis of 50 Laguerre functions of the form... 

The natures of the Lanczos functions and various properties of Stleltjes orbitals are best demonstrated by specific example. In Figure 3 are shown the first ten radial Lanczos functions for the ls->kp spectrum of a hydrogen atom (18). These are obtained from solution of Equations (4) for j=l to 10 employing the appropriate hydrogenic Hamiltonian in the operator A(H) and the Is ground state orbital in the test function. In this case. Equation (4b) can be solved for the v. in terms of Laguerre functions with constant expo-... 

These functions are called the associated Laguerre functions. We shall discuss them in detail in succeeding sections. 

In Appendix VII, it is shown that the normalization integral for the associated Laguerre function has the value... 

In this section we shall examine the competition between singularity and localisation effects in Rayleigh-Ritz variational calculations performed by John Loeser and Dudley Herschbach [22] on heliumlike ions for a wide range of D and Z, using a Pekeris-type basis of products of generalised Laguerre functions... 

Fortunately, this diagnosis suggests an obvious cure one should use properly (anti-)symmetrised products of generalised Laguerre functions with three flexible length scales ... 

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