Spherical Harmonic

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A Spherical Harmonic is a set of angular functions that are obtained from solution to the Laplace's equation in spherical polar coordinates.

  • Context:
    • It is usually denoted by [math]\displaystyle{ Y_\ell^m (\theta, \phi) }[/math], where [math]\displaystyle{ \theta }[/math] is the polar (a.k.a co-latitudinal) coordinate and [math]\displaystyle{ \phi }[/math] is the azimuthal (a.k. longitudinal) coordinate, [math]\displaystyle{ \ell }[/math] is a positive integer and [math]\displaystyle{ m=-\ell, -(\ell-1), \dots, 0, \dots, \ell-1, \ell }[/math].
    • It can be defined as eigenfunctions of the Legendrian Operator, [math]\displaystyle{ L^2 }[/math] with eigenvalues [math]\displaystyle{ \ell(\ell +1) }[/math], i.e.
[math]\displaystyle{ L^2 Y_\ell^m (\theta, \phi)=\ell(\ell +1)Y_\ell^m (\theta, \phi) }[/math]
[math]\displaystyle{ Y_{\ell, m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ P_\ell^{m}(\cos \theta)\ e^{im\phi} }[/math]
  • Example(s):
    • [math]\displaystyle{ Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi} }[/math]
    • [math]\displaystyle{ Y_{1}^{-1}(\theta,\varphi)={1\over 2}\sqrt{3\over 2\pi} \, \sin\theta \, e^{-i\varphi} }[/math]
    • [math]\displaystyle{ Y_{1}^{0}(\theta,\varphi)={1\over 2}\sqrt{3\over \pi}\, \cos\theta }[/math]
    • [math]\displaystyle{ Y_{1}^{1}(\theta,\varphi)={-1\over 2}\sqrt{3\over 2\pi}\, \sin\theta\, e^{i\varphi} }[/math]
    • Small-amplitude oscillations of a spherical object
  • Counter-Example(s):
  • See: Quantum Harmonic Oscillator, Laplace's equation, Spherical Waves Equation.

References

2015

[math]\displaystyle{ P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell }[/math]
and
[math]\displaystyle{ P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 \lt m \le \ell. }[/math]
For each choice of [math]\displaystyle{ \ell }[/math], there are [math]\displaystyle{ 2\ell + 1 }[/math] functions for the various values of [math]\displaystyle{ m }[/math] and choices of sine and cosine. They are all orthogonal in both [math]\displaystyle{ \ell }[/math] and [math]\displaystyle{ m }[/math] when integrated over the surface of the sphere.
The solutions are usually written in terms of complex exponentials:
[math]\displaystyle{ Y_{\ell, m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell. }[/math]
The functions [math]\displaystyle{ Y_{\ell, m}(\theta, \phi) }[/math] are the spherical harmonics, and the quantity in the square root is a normalizing factor.
Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identity
[math]\displaystyle{ Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi). }[/math]
The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series.

2012

2008

1999

1997

[math]\displaystyle{ Y_{\ell, m}(\theta, \phi) = (-1)^m c_{lm} P_\ell^{m}(\cos \theta)\ exp(im\phi) }[/math]
where [math]\displaystyle{ P_\ell^{m} }[/math] is a Legendre function, and the normalization [math]\displaystyle{ c_{lm} }[/math] constant is determined by
[math]\displaystyle{ c_{lm}^2\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!} }[/math]
such that the integral of [math]\displaystyle{ |Y_{\ell, m}(\theta, \phi)|^2 }[/math]over the unit sphere is 1.

1980

  • (J.P. Cox, 1980) ⇒ John P. Cox (1980). “Theory of Stellar Pulsation" , Princeton University Press, Princeton series in Astrophysics Edited by Jeremiah P. Ostriker
    • See : Chapter 17(Section 17.3 Expression of Perturbation variables in Terms of Spherical Harmonics"), pages 218 - 220

1977

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