What are spherical harmonic functions?
By Rachel Newton
What are spherical harmonic functions?
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series.
How many spherical harmonics are there?
Arfken, G. “Spherical Harmonics” and “Integrals of the Products of Three Spherical Harmonics.” §12.6 and 12.9 in Mathematical Methods for Physicists, 3rd ed.
What is M in spherical harmonics?
The indices ℓ and m indicate degree and order of the function. The spherical harmonic functions can be used to describe a function of θ and φ in the form of a linear expansion. Completeness implies that this expansion converges to an exact result for sufficient terms.
Is the spherical harmonic even for every angular QM number?
Start with acting the parity operator on the simplest spherical harmonic, l = m = 0: Now we can scale this up to the Y 2 0 ( θ, ϕ) case given in example one: It appears that for every even, angular QM number, the spherical harmonic is even.
What are spherical harmonics in physics?
S 1). S^1). S 1). Spherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. As a result, they are extremely convenient in representing solutions to partial differential equations in which the Laplacian appears.
What is Legendre’s equation for a spherical harmonic?
Unsurprisingly, that equation is called “Legendre’s equation”, and it features a transformation of cos θ = x. As the general function shows above, for the spherical harmonic where l = m = 0, the bracketed term turns into a simple constant. The exponential equals one and we say that:
What is spherical harmonic polynomial representation?
Harmonic polynomial representation. For any , the space of spherical harmonics of degree is just the space of restrictions to the sphere of the elements of . As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).
