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**Exercises**

**13.2.1 **Show with the aid of the Leibniz formula that the series expansion of L_{n} (x ) (Eq. (13.60)) follows from the Rodrigues representation (Eq. (13.59)).

**13.2.2 (a) **Using the explicit series form (Eq. (13.60)) show that

(b) Repeat without using the explicit series form of L_{n} (x ).

**13.2.3** From the generating function derive the Rodrigues representation

**13.2.4** Derive the normalization relation (Eq. (13.79)) for the associated Laguerre polynomials.

**13.2.6 **Expand e−ax in a series of associated Laguerre polynomials ﬁxed and n ranging from 0 to ∞.

(a) Evaluate directly the coefﬁcients in your assumed expansion.

(b) Develop the desired expansion from the generating function.

**13.2.7 Show that**

Hint. Note that

**13.2.8** Assume that a particular problem in quantum mechanics has led to the ODE

for nonnegative integers n, k . Write y(x) as

y(x) = A(x)B(x)C(x),

with the requirement that

(a) A(x ) be a negative exponential giving the required asymptotic behavior of y(x) and

(b) B(x ) be a positive power of x giving the behavior of y(x) for 0 ≤ x ≪ 1.

Determine A(x ) and B(x ). Find the relation between C(x) and the associated Laguerre polynomial.

**13.2.9** From Eq. (13.91) the normalized radial part of the hydrogenic wave function is

The quantity { r } is the average displacement of the electron from the nucleus, whereas {r^{ −1}} is the average of the reciprocal displacement.

**13.2.11** The hydrogen wave functions, Eq. (13.91), are mutually orthogonal, as they should be, since they are eigenfunctions of the self-adjoint Schrödinger equation

Yet the radial integral has the (misleading) form

which appears to match Eq. (13.83) and not the associated Laguerre orthogonality relation, Eq. (13.79). How do you resolve this paradox?

ANS. The parameter α is dependent on n. The ﬁrst three α,previously shown, are 2Z/ n_{1} a_{0} . The last three are 2Z/ n_{2} a_{0 }.For n_{1} = n_{2} Eq. (13.83) applies. For n_{1} = n_{2} neither Eq. (13.79) nor Eq. (13.83) is applicable

**13.2.12 **A quantum mechanical analysis of the Stark effect (parabolic coordinate) leads to the ODE

Here F is a measure of the perturbation energy introduced by an external electric ﬁeld.

Find the unperturbed wave functions (F = 0) in terms of associated Laguerre polynomials.

**13.2.13 **The wave equation for the three-dimensional harmonic oscillator is

**13.2.14 **Write a computer program that will generate the coefﬁcients as in the polynomial form of the Laguerre polynomial

**13.2.15 **

**13.2.16 **Tabulate L_{10} (x ) for x = 0.0(0.1)30.0. This will include the 10 roots of L_{10} . Beyond x = 30.0,L_{10} (x ) is monotonic increasing. If graphic software is available, plot your results.

Check value. Eighth root = 16.279.

**13.2.17** Determine the 10 roots of L_{10} (x ) using root-ﬁnding software. You may use your knowledge of the approximate location of the roots or develop a search routine to look for the roots. The 10 roots of L_{10} (x ) are the evaluation points for the 10-point Gauss–Laguerre quadrature. Check your values by comparing with AMS-55, Table 25.9.

**13.2.18 **Calculate the coefﬁcients of a Laguerre series expansion (L_{n} (x ), k = 0) of the exponential e^{−x} . Evaluate the coefﬁcients by the Gauss–Laguerre quadrature (compare Eq. (10.64)). Check your results against the values given in Exercise 13.2.6.

Note. Direct application of the Gauss–Laguerre quadrature with f(x )L_{n} (x )e^{−x} gives poor accuracy because of the extra e^{−x} . Try a change of variable, y = 2x , so that the function appearing in the integrand will be simply L_{n} (y /2).

**13.2.19 (a) **Write a subroutine to calculate the Laguerre matrix elements

**13.2.20** Write a subroutine to calculate the numerical value of for speciﬁed values of n, k , and x . Require that n and k be nonnegative integers and x ≥ 0.

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