Polynomials, associated Laguerre

Laguerre polynomials, 202 Laguerre polynomials, associated, 201 Langevin dynamics, 371, 383... 

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

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

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

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... 

Since n and I are integers, equation (G.51) is identical to the associated Laguerre differential equation (F.16) with k = n + I and j = 21 + 1. Thus, the solutions u(p) are proportional to the associated Laguerre polynomials (p), whose properties are discussed in Appendix F... 

As the integers if and / both begin at zero, y = 1,2,3... can of course be identified as the principal quantum number n for the hydrogen atom (see Section 6.6.1). Thus, the quantization of the energy is due to the termination of the series, a condition imposed to obtain an acceptable solution. The associated Laguerre polynomials provide quantitative descriptions of the radial part of the wave functions for the hydrogen atom, as described in Appendix IV. 

The radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

Develop the indicia equation for the associated Laguerre polynomials [Eq. (133)]. 

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... 

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

Atomic Size The associated Laguerre polynomial (x) is a polynomial of degree nr = n — l — 1, which has nr radial nodes (zeros). The radial distribution function therefore exhibits n — l maxima. Whenever n = l + 1 and the orbital quantum number, l has its largest value, there is only one maximum. In this case nT = 0 and from (14) follows... 

It is readily shown from equation (42.1) which defines the generating function for Laguerre polynomials that the associated Laguerre polynomials may be defined by the equation... 

This identity can then he used to derive recurrence relations for the associated Laguerre polynomials similar to those of equations (42.8) and (42.9) (cf. Examples C(ii), (iii) below). 

The energy Ea is a quantum term associated with the proton reaction coordinate coupling to the Q vibration, Ea = h1 /2m. and Co is the tunneling matrix element for the transfer from the 0th vibrational level in the reactant state to the 0th vibrational level in the product state. The term AQe is the shift in the oscillator equilibrium position and F L(Eq, Ea, Laguerre polynomial. For a thorough discussion of Eq. (8), see [13],... 

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