Editing Spherical harmonics (section)

==History==
[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|[[Pierre-Simon Laplace]],  1749–1827]]
Spherical harmonics were first investigated in connection with the [[Newtonian potential]] of [[Newton's law of universal gravitation]] in three dimensions. In 1782, [[Pierre-Simon de Laplace]] had, in his ''Mécanique Céleste'', determined that the [[gravitational potential]] <math>\R^3 \to \R</math> at a point {{math|'''x'''}} associated with a set of point masses {{math|''m''<sub>''i''</sub>}} located at points {{math|'''x'''<sub>''i''</sub>}} was given by

<math display="block">V(\mathbf{x}) = \sum_i \frac{m_i}{|\mathbf{x}_i - \mathbf{x}|}.</math>

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, [[Adrien-Marie Legendre]] had investigated the expansion of the Newtonian potential in powers of {{math|1=''r'' = {{abs|'''x'''}}}} and {{math|1=''r''<sub>1</sub> = {{abs|'''x'''<sub>1</sub>}}}}. He discovered that if {{math|''r'' ≤ ''r''<sub>1</sub>}} then

<math display="block">\frac{1}{|\mathbf{x}_1 - \mathbf{x}|} = P_0(\cos\gamma)\frac{1}{r_1} + P_1(\cos\gamma)\frac{r}{r_1^2} + P_2(\cos\gamma)\frac{r^2}{r_1^3}+\cdots</math>

where {{math|''γ''}} is the angle between the vectors {{math|'''x'''}} and {{math|'''x'''<sub>1</sub>}}. The functions <math>P_i: [-1, 1] \to \R</math> are the [[Legendre polynomials]], and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle {{math|''γ''}} between {{math|'''x'''<sub>1</sub>}} and {{math|'''x'''}}. (See {{slink|Legendre polynomials#Applications}} for more detail.)

In 1867, [[William Thomson, 1st Baron Kelvin|William Thomson]] (Lord Kelvin) and [[Peter Guthrie Tait]] introduced the [[Solid harmonics|solid spherical harmonics]] in their ''[[Treatise on Natural Philosophy]]'', and also first introduced the name of "spherical harmonics" for these functions. The [[solid harmonics]] were [[homogeneous function|homogeneous]] polynomial solutions <math>\R^3 \to \R</math> of [[Laplace's equation]]
<math display="block">\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.</math>
By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See [[#Harmonic polynomial representation|Harmonic polynomial representation]].) The term "Laplace's coefficients" was employed by [[William Whewell]] to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the [[zonal spherical harmonics]] that had properly been introduced by Laplace and Legendre.

The 19th century development of [[Fourier series]] made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the [[heat equation]] and [[wave equation]]. This could be achieved by expansion of functions in series of [[trigonometric function]]s. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a [[vibrating string|string]], the spherical harmonics represent the fundamental modes of [[Wave equation#Spherical waves|vibration of a sphere]] in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as [[Euler's formula|complex exponentials]], spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing [[spherical symmetry]], such as those of celestial mechanics originally studied by Laplace and Legendre.

The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of [[quantum mechanics]]. The (complex-valued) spherical harmonics <math>S^2 \to \Complex</math> are [[eigenfunction]]s of the square of the [[angular momentum operator|orbital angular momentum]] operator
<math display="block">-i\hbar\mathbf{r}\times\nabla,</math>
and therefore they represent the different [[angular momentum quantization|quantized]] configurations of [[atomic orbitals]].