Associated Laguerre equation

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... 

When the resulting equation is divided by we obtain an equation that is known as the associated Laguerre equation ... 

Laguerre assumed that the solution of the associated Laguerre equation can be written as... 

This equation is valid only for non-negative values of the integer m, but Eq. (F-47) contains only the square of m, so that we can replace a negative value of mhy m when using this equation. This equation leads to the 0 functions discussed in Chapter 17. The radial function R obeys Eq. (17.3-5), the associated Laguerre equation ... 

Show that your polynomial from Problem 14.7 is a solution to the associated Laguerre equation. 

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

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 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)... 

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... 

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

Formulae of this type are especially useful when looking for solutions of more complicated differential equations related to the simpler classical ones. Two important examples are the associated Legendre and associated Laguerre (18) equations, respectively... 

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). 

Evaluate r l)ni for the hydrogen-like atom using the properties of associated Laguerre polynomials. First substitute equations (6.22) and (6.55) into (6.69) for k = — 1. Then apply equations (F.22) to obtain (6.71). 

On comparing this with Equation 18-37 obtained in the treatment of the r equation for the hydrogen atom by the polynomial method, we see that the two equations are identical when kn+fKp) is identified with L(p) and the parameter X is replaced by its characteristic value n. The polynomials obtained in the solution of the r equation for the hydrogen atom are hence the associated Laguerre polynomials of degree n — l — 1 and of order 21 + 1. Moreover, the wave functions in r are, except for normalizing factors, the functions... 

It is easily shown from Equation 20-1 that the generating function for the associated Laguerre polynomials of order s is1... 

Problem 20-2. Derive relations for the associated Laguerre polynomials and functions corresponding to those of Equations 20-2 and 20-3. 

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... 

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