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.
Frequently asked questions
Simply head over to the account section in settings and click on “Cancel Subscription” - it’s as simple as that. After you cancel, your membership will stay active for the remainder of the time you’ve paid for. Learn more here.
At the moment all of our mobile-responsive ePub books are available to download via the app. Most of our PDFs are also available to download and we're working on making the final remaining ones downloadable now. Learn more here.
Both plans give you full access to the library and all of Perlego’s features. The only differences are the price and subscription period: With the annual plan you’ll save around 30% compared to 12 months on the monthly plan.
We are an online textbook subscription service, where you can get access to an entire online library for less than the price of a single book per month. With over 1 million books across 1000+ topics, we’ve got you covered! Learn more here.
Look out for the read-aloud symbol on your next book to see if you can listen to it. The read-aloud tool reads text aloud for you, highlighting the text as it is being read. You can pause it, speed it up and slow it down. Learn more here.
Yes, you can access Spherical Harmonics In P Dimensions by Christopher Frye, Costas Efthimiou in PDF and/or ePUB format, as well as other popular books in Biological Sciences & Science General. We have over one million books available in our catalogue for you to explore.
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 ℝp.
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
where3n ∈ 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...
