Hyperspherical Harmonics and Their Physical Applications

James Emil Avery, John Scales Avery;;;

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813229310
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

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Hyperspherical harmonics are extremely useful in nuclear physics and reactive scattering theory. However, their use has been confined to specialists with very strong backgrounds in mathematics. This book aims to change the theory of hyperspherical harmonics from an esoteric field, mastered by specialists, into an easily-used tool with a place in the working kit of all theoretical physicists, theoretical chemists and mathematicians. The theory presented here is accessible without the knowledge of Lie-groups and representation theory, and can be understood with an ordinary knowledge of calculus. The book is accompanied by programs and exercises designed for teaching and practical use.

--> Contents:

• Preface
• Harmonic Functions
• Generalized Angular Momentum
• Gegenbauer Polynomials
• Fourier Transforms in d Dimensions
• Fock's Treatment of Hydrogenlike Atoms and Its Generalization
• D-Dimensional Hydrogenlike Orbitals in Direct Space
• Generalized Sturmians
• Choosing Appropriate Hyperspherical Representations
• Molecular Integrals from Hyperspherical Harmonics
• Lagrangians for Particles and Fields
• Coordinate Transformations for N Bodies
• Some Illustrative Examples
• Appendices:
  • The D-Dimensional Harmonic Oscillator
  • Molecular Integrals for Slatertype Orbitals
• Bibliography
• Index

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--> Readership: Scientists and researchers in theoretical physics, theoretical chemistry, and mathematics. -->
Keywords:Harmonic Functions;Reactive Scattering Theory; Nuclear Physics;Gegenbauer Polynomials;Generalized Sturmians;Slatertype OrbitalsReview: Key Features:

• Exercises are included at the end of each chapter
• The e-version of the exercises and solutions can be found in the supplementary tab

Chapter 1

HARMONIC FUNCTIONS

1.1Definitions

Let us consider a d-dimensional Euclidean space with Cartesian coordinates x₁, x₂, x₃, . . . , x_d. In this space, we can define a hyperradius r by the Euclidean radius:

We can also define the generalized Laplacian operator Δ by

A homogeneous polynomial of order n in the coordinates x₁, x₂, x₃, . . . , x_d is defined to be a polynomial of the form

where a, b, c, . . . , are the coefficients of the polynomial, and

and where the numbers

, are positive integers or zero. (For example, f₂ = x² + 2xy is a homogeneous polynomial of order 2.)

A polynomial f_n is homogeneous and of order n if and only if the following equation holds

(See exercise 10). We can see that (1.5) is true for a homogeneous polynomial, because for each monomial,

But from (1.4) we know that n₁ + n₂ + … + n_d = n, and if each term of f_n has this property, then it holds for the whole polynomial.

It is interesting to notice that equation (1.5) holds for a wider class of functions than homogeneous polynomials. Let f_n be a homogeneous polynomial of order n, and let a be a number, not necessarily integral or positive or real. Then, making use of the chain rule and equation (1.5),

Now let f_n be a homogeneous polynomial of order n and g_n′ be a homogeneous polynomial of order n′, and let a and b be any two numbers, not necessarily integers or positive or real. Then

In general, we will call a function f_s a homogeneous function of order s for some complex number s, if

Homogeneous polynomials are special cases of homogeneous functions. Several properties of homogeneous functions flow from the definition in equation (1.9): The product of any two homogeneous functions is another homogeneous function whose order is the sum of the orders of the two terms in the product, since

The quotient of two homogeneous functions is a homogeneous function whose order is the difference between the orders of the two terms in the quotient, because

Finally, the sum of (or difference between) two homogeneous functions of the same order is a homogeneous function of that order:

Besides homogeneous functions, we can also define harmonic functions, which are homogeneous functions that also satisfy the generalized Laplace equation.

Harmonic polynomials are spec...