Spherical Harmonics In P Dimensions

Christopher Frye, Costas Efthimiou

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814596718
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

The current book makes several useful topics from the theory of special functions, in particular the theory of spherical harmonics and Legendre polynomials in arbitrary dimensions, available to undergraduates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before exploring the main subject matter.

Contents:

• Introduction and Motivation
• Working in p Dimensions
• Orthogonal Polynomials
• Spherical Harmonics in p Dimensions
• Solutions to Problems

Readership: Undergraduate and graduate students in mathematical physics and differential equations.

Chapter 1

Introduction and Motivation

Many important equations in physics involve the Laplace operator, which is given by

in two and three dimensions¹, respectively. We will see later (Proposition 2.1) that the Laplace operator is invariant under a rotation of the coordinate system. Thus, it arises in many physical situations in which there exists spherical symmetry, i.e., where physical quantities depend only on the radial distance r from some center of symmetry

. For example, the electric potential V in free space is found by solving the Laplace equation,

which is rotationally invariant. Also, in quantum mechanics, the wave function ψ of a particle in a central field can be found by solving the time-independent Schrödinger equation,

where ħ is Planck’s constant, m is the mass of the particle, V(r) is its potential energy, and E is its total energy.

We will give a brief introduction to these problems in two and three dimensions to motivate the main subject of this discussion. In doing this, we will get a preview of some of the properties of spherical harmonics — which, for now, we can just think of as some special set of functions — that we will develop later in the general setting of ℝ^p.

1.1 Separation of Variables

Two-Dimensional Case

Since we are interested in problems with spherical symmetry, let us rewrite the Laplace operator in spherical coordinates, which in ℝ² are just the ordinary polar coordinates²,

Alternatively,

x = r cos ϕ, y = r sin ϕ.

Using the chain rule, we can rewrite the Laplace operator as

In checking this result, perhaps it is easiest to begin with (1.6) and recover (1.1). First we compute

which implies that

and

which gives

Inserting these into (1.6) gives us back (1.1).

Thus, (1.3) becomes

To solve this equation it is standard to assume that

Φ(r, ϕ) = χ(r) Y(ϕ),

where χ(r) is a function of r alone and Y (ϕ) is a function of ϕ alone. Then

Multiplying by r²/χY and rearranging,

We see that a function of r alone (the left side) is equal to a function of ϕ alone (the right side). Since we can vary r without changing ϕ, i.e., without changing the right side of the above equation, it must be that the left side of the above equation does not vary with r either. This means the left side of the above equation is not really a function of r but a constant. As a consequence, the right side is the same constant. Thus, for some –λ we can write

We solve Y′′ = λY to get the linearly independent solutions

but we must reject some of these solutions. Since (r₀, ϕ₀) represents the same point as (r₀, ϕ₀ + 2πk) for any k ∈ ℤ, we require Y (ϕ) to have period 2π. Thus, we can only accept the linearly independent periodic solutions 1,

must be an integer. Then, let us replace λ with −m² and write our linearly independent solutions to Y″ = −m²Y as

where³ n ∈ ℕ₀ and m ∈ ℕ. Notice from (1.6) that ∂²/∂ϕ² is the angular part of the Laplace operator in two dimensions and that the solutions given in (1.9) are eigenf...

Table of contents

• Cover page
• Title page
• Copyright page
• Contents
• Preface
• Acknowledgements
• List of Symbols
• 1 Introduction and Motivation
• 2 Working in p Dimensions
• 3 Orthogonal Polynomials
• 4 Spherical Harmonics in p Dimensions
• 5 Solutions to Problems
• Bibliography
• Index