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
We shall attempt to represent y , or rather yn , 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 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
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 Ln (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 zn yields
To get the second recursion relation we use both identities of Eqs. (13.61) to verify the third identity,
which, when written similarly in terms of Laguerre polynomials, is seen to be equivalent to
Equation (13.61), modiﬁed to read
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
which yields the special values of Laguerre polynomials
Ln (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
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
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
From the series form of Ln (x ) we verify that the lowest associated Laguerre polynomials are given by
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 zk , we obtain
From this, for x = 0, the binomial expansion
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
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 eρ , 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 En ,wehave
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.