Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

The document Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

Exercises
 

13.2.1 Show with the aid of the Leibniz formula that the series expansion of Ln (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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

(b) Repeat without using the explicit series form of Ln (x ).

13.2.3 From the generating function derive the Rodrigues representation

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

13.2.6 Expand e−ax in a series of associated Laguerre polynomials Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev 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.

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

13.2.7 Show that

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hint. Note that

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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.

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Yet the radial integral has the (misleading) form

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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/ n1 a0 . The last three are 2Z/ n2 a.For n1 = n2 Eq. (13.83) applies. For n1 = n2 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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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.

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

13.2.15 
             
Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

13.2.16 Tabulate L10 (x ) for x = 0.0(0.1)30.0. This will include the 10 roots of L10 . Beyond x = 30.0,L10 (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 L10 (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 L10 (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 (Ln (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 )Ln (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 Ln (y /2).

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

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

13.2.20 Write a subroutine to calculate the numerical value of Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev for specified values of n, k , and x . Require that n and k be nonnegative integers and x ≥ 0.

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics, UGC -NET Physics Physics Notes | EduRev

Dynamic Test

Content Category

Related Searches

pdf

,

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics

,

Exam

,

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics

,

ppt

,

Summary

,

practice quizzes

,

UGC -NET Physics Physics Notes | EduRev

,

Important questions

,

UGC -NET Physics Physics Notes | EduRev

,

mock tests for examination

,

video lectures

,

past year papers

,

Hermite and Laguerre Special Functions (Part - 3)- Mathematical Methods of Physics

,

Previous Year Questions with Solutions

,

Sample Paper

,

UGC -NET Physics Physics Notes | EduRev

,

Viva Questions

,

shortcuts and tricks

,

Extra Questions

,

study material

,

Objective type Questions

,

Semester Notes

,

Free

,

MCQs

;