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Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

(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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

13.2.6 Expand e−ax in a series of associated Laguerre polynomials Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET 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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

13.2.7 Show that

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hint. Note that

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Yet the radial integral has the (misleading) form

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

13.2.15 
             
Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

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 - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

13.2.20 Write a subroutine to calculate the numerical value of Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET for specified values of n, k , and x . Require that n and k be nonnegative integers and x ≥ 0.

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET

The document Hermite and Laguerre Special Functions - 3 | Physics for IIT JAM, UGC - NET, CSIR NET is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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FAQs on Hermite and Laguerre Special Functions - 3 - Physics for IIT JAM, UGC - NET, CSIR NET

1. What are Hermite and Laguerre special functions?
Ans. Hermite and Laguerre special functions are mathematical functions that are commonly used in physics to solve various problems related to quantum mechanics, statistical mechanics, and other areas of physics. Hermite functions are solutions to Hermite's differential equation and are used to describe the wave functions of quantum harmonic oscillators. Laguerre functions are solutions to Laguerre's differential equation and are used to describe various physical systems such as hydrogen atom orbitals and quantum mechanical tunneling.
2. How are Hermite and Laguerre special functions applied in physics?
Ans. Hermite and Laguerre special functions find applications in various areas of physics. In quantum mechanics, Hermite functions are used to describe the wave functions of quantum harmonic oscillators, which are fundamental systems in quantum mechanics. Laguerre functions, on the other hand, are used to describe the wave functions of hydrogen atom orbitals and other physical systems. These special functions are crucial in solving differential equations that arise in physics and are used to calculate probabilities, energy levels, and other properties of physical systems.
3. Can you provide examples of physical systems where Hermite and Laguerre special functions are used?
Ans. Yes, there are several examples of physical systems where Hermite and Laguerre special functions are used. Some examples include: - Quantum harmonic oscillator: Hermite functions are used to describe the wave functions and energy levels of a quantum harmonic oscillator, which is a fundamental system in quantum mechanics. - Hydrogen atom: Laguerre functions are used to describe the wave functions and energy levels of the electron in a hydrogen atom. - Quantum mechanical tunneling: Laguerre functions are used to describe the wave functions of particles tunneling through potential barriers. - Quantum mechanical scattering: Hermite functions are used to describe the scattering of particles by potentials.
4. Are Hermite and Laguerre special functions only applicable in physics?
Ans. No, Hermite and Laguerre special functions are not limited to physics and find applications in various branches of science and engineering. They are widely used in mathematics, statistics, signal processing, and other fields. In mathematics, these special functions are used to solve differential equations and express solutions in a series form. In statistics, they find applications in probability distributions and the study of random variables. In signal processing, Hermite functions are used in the analysis and synthesis of signals.
5. Are there any real-world applications of Hermite and Laguerre special functions?
Ans. Yes, Hermite and Laguerre special functions have real-world applications beyond theoretical physics. Some examples include: - Quantum chemistry: These special functions are used to describe the electronic structure and properties of molecules, which are important in drug discovery, materials science, and chemical reactions. - Electromagnetic wave propagation: Hermite and Laguerre functions are used to solve Maxwell's equations and describe the behavior of electromagnetic waves in different media, including optical fibers and waveguides. - Image and signal processing: These special functions are used in the analysis and processing of images and signals, such as noise reduction, compression, and feature extraction. - Financial modeling: Hermite and Laguerre functions find applications in financial mathematics, such as option pricing models and risk management.
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