Polynomials, Laguerre

The quantum-mechanical problem of a particle moving in a central field is represented by a three-dimensional Schrodinger equation with a spherically symmetric potential V(r)  

As in the case of Helmholtz s equation, we have separabity in spherical polar coordinates r, 0,4 ) = R(.r)Yem(0,4 )- In convenient units with ft = m = 1, 

We consider the electron in a hydrogen atom or hydrogenlike ion (He, Li +, ) orbiting around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units e /ATt Q = 1) can be written as 

It is again useful to find asymptotic solutions to the differential equation. When r oo, the equation is approximated by 

Chapter 12 Partial Differential Equations and Special Functions 

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The functions P " are associated Legendre polynomials of order m and degree I, and are associated Laguerre polynomials of degree (v — l)/2 in... 

These functions are universally known as Slater type orbitals (STOs) and are just the leading term in the appropriate Laguerre polynomials. The first three Slater functions are as follows ... 

The Sonine polynomials are related to the associated Laguerre polynomials (see Margenau and Murphy, op. eft., p. 128) by... 

It is shown in Appendix 6 that the generalized Laguerre polynomials are eigenfunctions of the integral operator (3.26) with kernel (3.52). Let us search for the solution of (3.26) in the form of expansion over these eigenfunctions... 

As the isotropic Raman and CARS spectra may be expressed via ao(co) by virtue of the Laguerre polynomials orthogonality, we have... 

Now it is possible to employ the eigenfunctions given in [121], which are proportional to the generalized Laguerre polynomials. The corresponding eigenvalues are equidistant ... 

Undetermined coefficients 4 can be easily found using the initial condition (6.3) and orthogonality of the Laguerre polynomials [37]... 

In a general case parameters re, XdP and y must be determined by self-consistent two-parameter fitting. Owing to the property of orthogonality of Laguerre polynomials, one has for the spectral band shapes... 

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

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... 

G is then a generating function for these integrals, which occur as coefficients in its expansion in powers of u and and it can he evaluated with the use of the generating function for the associated Laguerre polynomials, given in equation (19). Thus we have... 

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

Radial functions in terms of associated Laguerre polynomials... 

The radial functions Sni p) and R i(r) may be expressed in terms of the associated Laguerre polynomials L p), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Sniip) and L p) is to relate Sni p) in equation (6.50) to the polynomial L 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 differential equation satisfied by the associated Laguerre polynomials is given by equation (F.16) as... 

The normalized radial functions Rniir) may be expressed in terms of the associated Laguerre polynomials by combining equations (6.22), (6.23), and (6.54)... 

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

Combining equations (F.3) and (F.4), we obtain the formula for the Laguerre polynomials... 

Since the Laguerre polynomial Lk p) divided by k is the coefficient of 5 in the expansion (F.l) of the generating function, we have... 

The associated Laguerre polynomials L p) are defined in terms of the Laguerre polynomials by... 

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