Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

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Physics : Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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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 Kν (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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

which is equal to

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

FIGURE 13.3 Laguerre polynomial contour.

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

FIGURE 13.4 Laguerre polynomials.

This is exactly what we would obtain from the series

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Table 13.2 Laguerre Polynomials

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Equating coefficients of zn yields

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev               (13.63)

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev           (13.64)

Equation (13.61), modified to read

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

for reasons of economy and numerical stability, is used for computation of numerical values of Ln(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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

which yields the special values of Laguerre polynomials

L(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 Ln (x ) do not by themselves form an orthogonal set. However, if we multiply Eq. (13.52) by e−x  we obtain

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev(13.68)

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev            (13.70)

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

In general,

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

From the last two members of this equation, canceling the common factor zk , we obtain

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

From this, for x = 0, the binomial expansion

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

yields

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev             (13.74)

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Thus, we obtain from differentiating Laguerre’s ODE once

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

and eventually from differentiating Laguerre’s ODE k times

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

 

The associated Laguerre equation (13.77) is not self-adjoint, but it can be put in selfadjoint form by multiplying by Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev which becomes the weighting function (Section 10.1). We obtain

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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.

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Substitution into the associated Laguerre equation yields

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

The corresponding normalization integral Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev


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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev(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 Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev. The radial part, R(r ), satisfies the equation

 

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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 )

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Eq. (13.85) becomes

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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 eρ , which is unacceptable for our physical problem, in which

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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 En ,wehave

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

with

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRevthe Bohr radius.

Thus, the final normalized hydrogen wave function is written as

Hermite and Laguerre Special Functions (Part - 2)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

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