Laguerre functions, generalize

Section 6.5.1. The functions and more generally are known as the Laguerre functions, a term that is also used for the generalization of L (x) to nonintegral complex indices n and or. 

To illustrate the convergence properties of the Laguerre functions, we shall use these functions to expand the numerical Hartree—Fock orbitals of the ground state of the carbon atom. However, to see how such expansions are obtained, we shall first consider in general terms the expansion of a function /(x) in a set of basis functions Thus, we wish to determine an expansion in... 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

Laguerre polynomial, spherical harmonic, and generalized hypergeometric function, respectively. Equation (29) yields the leading asymptotics (v/([Pg.152]

The general solution of the radial equation is not straightforward. We are interested in solutions for which the electron is in a stationary state bound to the nucleus. This gives a boundary condition that r) 0 at large r values. The set of functions that satisfy Equation (A9.46) is then a product of an associated Laguerre polynomial, pr), and a decaying exponential that ensures this boundary condition is met ... 

For Kr l, no general explicit formula can be given. But the integration has become considerably easier because ni and U2 do not, as with the real hydrogen functions, appear in the arguments of the Laguerre polynomials. In relation to the integration I will restrict myself to the remark that x 2 must be expanded as functions on the sphere in t9 ... 

Thus we have demonstrated how the L j /(p) polynomials can be generated and that they do satisfy the general associated Laguerre polynomial equation. Schrodinger worked out the Hydrogen orbitals from these functions in his third revolutionary paper [7] and perhaps we can appreciate the patience required to carry the derivation to useful results ... 

The Linear Differential Equation of the Second Order, 48. The Legendre Polynomials, 62. The Associated Legendre Polynomials, 52. The General Solution of the Associated Legendre Equation, 53. The Functions 0j.r ( ) and 57. Recursion Formulae for the Legendre Polynomials, 59. The Hermite Polynomials, 60. The Laguerre Polynomials, 63. 

The boundary conditions are in general of the mixed type involving a combination of the function value and derivative at the two boundaries taken here to occur tx = a andx = b. Special cases of this equation lead to many classical functions such as Bessel functions, Legendre polynomials, Hemite polynomials, Laguerre polynomials and Chebyshev polynomials. In addition the Schrodinger time independent wave equation is a form of the Sturm-Liouville problem. 

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