Hermite and Laguerre Special Functions - 2 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

LAGUERRE FUNCTIONS

Differential Equation — Laguerre Polynomials 

If we start with the appropriate generating function, it is possible to develop the Laguerre polynomials in analogy with the Hermite polynomials. Alternatively, a series solution may be developed by the methods of Section 9.5. Instead, to illustrate a different technique, let us start with Laguerre’s ODE and obtain a solution in the form of a contour integral, as we did with the integral representation for the modified Bessel function K_ν (x ) .

From this integral representation a generating function will be derived.

Laguerre’s ODE (which derives from the radial ODE of Schrödinger’s PDE for the hydrogen atom) is

We shall attempt to represent y , or rather y_n , since y will depend on the parameter n, a nonnegative integer, by the contour integral

and demonstrate that it satisfies Laguerre’s ODE. The contour includes the origin but does not enclose the point z = 1. By differentiating the exponential in Eq. (13.53a) we obtain

Substituting into the left-hand side of Eq. (13.52), we obtain

which is equal to

If we integrate our perfect differential around a closed contour (Fig. 13.3), the integral will vanish, thus verifying that y_n (x ) (Eq. (13.53a)) is a solution of Laguerre’s equation.
It has become customary to define L_n (x ), the Laguerre polynomial (Fig. 13.4), by⁵

FIGURE 13.3 Laguerre polynomial contour.

FIGURE 13.4 Laguerre polynomials.

This is exactly what we would obtain from the series

if we multiplied g(x , z) by z⁻ⁿ⁻¹ and integrated around the origin. As in the development of the calculus of residues (Section 7.1), only the z⁻¹ term in the series survives. On this basis we identify g(x , z) as the generating function for the Laguerre polynomials.

With the transformation

the new contour enclosing the point s = x in the s -plane. By Cauchy’s integral formula (for derivatives),

giving Rodrigues’ formula for Laguerre polynomials. From these representations of L_n (x ) we find the series form (for integral n),

and the specific polynomials listed in Table 13.2 (Exercise 13.2.1). Clearly, the definition of Laguerre polynomials in Eqs. (13.55), (13.56), (13.59), and (13.60) are equivalent.
Practical applications will decide which approach is used as one’s starting point. Equation (13.59) is most convenient for generating Table 13.2, Eq. (13.56) for deriving recursion relations from which the ODE (13.52) is recovered.

By differentiating the generating function in Eq. (13.56) with respect to x and z,we obtain recurrence relations for the Laguerre polynomials as follows. Using the product rule for differentiation we verify the identities

Table 13.2 Laguerre Polynomials

Writing the left-hand and right-hand sides of the first identity in terms of Laguerre polynomials using Eq. (13.56) we obtain

Equating coefficients of z_n yields

To get the second recursion relation we use both identities of Eqs. (13.61) to verify the third identity,

               (13.63)

which, when written similarly in terms of Laguerre polynomials, is seen to be equivalent to

           (13.64)

Equation (13.61), modified to read

for reasons of economy and numerical stability, is used for computation of numerical values of L_n(x ). The computer starts with known numerical values of L₀(x ) and L₁(x), Table 13.2, and works up step by step. This is the same technique discussed for computing Legendre polynomials,
Also, from Eq. (13.56) we find

which yields the special values of Laguerre polynomials

L_n (0) = 1.            (13.66)

As is seen from the form of the generating function, from the form of Laguerre’s ODE, or from Table 13.2, the Laguerre polynomials have neither odd nor even symmetry under the parity transformation x →−x .
The Laguerre ODE is not self-adjoint, and the Laguerre polynomials L_n (x ) do not by themselves form an orthogonal set. However, if we multiply Eq. (13.52) by e^−x  we obtain

This orthogonality is a consequence of the Sturm–Liouville theory The normalization follows from the generating function. It is sometimes convenient to define orthogonalized Laguerre functions (with unit weighting function) by

(13.68)

Our new orthonormal function, ϕn (x ), satisfies the ODE

which is seen to have the (self-adjoint) Sturm–Liouville form. Note that the interval (0 ≤ x< ∞) was used because Sturm–Liouville boundary conditions are satisfied at its endpoints.

Associated Laguerre Polynomials

In many applications, particularly in quantum mechanics, we need the associated Laguerre polynomials defined by

            (13.70)

From the series form of L_n (x ) we verify that the lowest associated Laguerre polynomials are given by

In general,

A generating function may be developed by differentiating the Laguerre generating function k times to yield

From the last two members of this equation, canceling the common factor z^k , we obtain

From this, for x = 0, the binomial expansion

yields

             (13.74)

Recurrence relations can be derived from the generating function or by differentiating the Laguerre polynomial recurrence relations. Among the numerous possibilities are

Thus, we obtain from differentiating Laguerre’s ODE once

and eventually from differentiating Laguerre’s ODE k times

Adjusting the index n → n + k , we have the associated Laguerre ODE

When associated Laguerre polynomials appear in a physical problem it is usually because that physical problem involves Eq. (13.77). The most important application is the bound states of the hydrogen atom, which are derived in upcoming Example 13.2.1.
A Rodrigues representation of the associated Laguerre polynomial

The associated Laguerre equation (13.77) is not self-adjoint, but it can be put in selfadjoint form by multiplying by  which becomes the weighting function (Section 10.1). We obtain

Equation (13.79) shows the same orthogonality interval (0, ∞) as that for the Laguerre polynomials, but with a new weighting function we have a new set of orthogonal polynomials, the associated Laguerre polynomials.

Substitution into the associated Laguerre equation yields

The corresponding normalization integral 

Example 13.2.1 THE HYDROGEN ATOM

The most important application of the Laguerre polynomials is in the solution of the Schrödinger equation for the hydrogen atom. This equation is

(13.84)

in which Z = 1 for hydrogen, 2 for ionized helium, and so on. Separating variables, we find that the angular dependence of ψ is the spherical harmonics . The radial part, R(r ), satisfies the equation

For bound states, R → 0as r →∞, and R is finite at the origin, r = 0. We do not consider continuum states with positive energy. Only when the latter are included do hydrogen wave functions form a complete set.
By use of the abbreviations (resulting from rescaling r to the dimensionless radial variable ρ )

Eq. (13.85) becomes

in which k is replaced by 2L + 1 and n by λ − L − 1, upon using

We must restrict the parameter λ by requiring it to be an integer n, n = 1, 2, 3,.... This is necessary because the Laguerre function of nonintegral n would diverge as ρⁿ e^ρ , which is unacceptable for our physical problem, in which

This restriction on λ, imposed by our boundary condition, has the effect of quantizing the energy,

The negative sign reflects the fact that we are dealing here with bound states ( E< 0), corresponding to an electron that is unable to escape to infinity, where the Coulomb potential goes to zero. Using this result for E_n ,wehave

with

the Bohr radius.

Thus, the final normalized hydrogen wave function is written as

Regular solutions exist for n ≥ L + 1, so the lowest state with L = 1 (called a 2P state) occurs only with n = 2.