Spherical harmonics

In mathematics, spherical harmonics Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^m_\ell} are an orthogonal and complete set of functions of the spherical polar angles θ and φ. The name "spherical harmonics" is due to Lord Kelvin. In quantum mechanics they appear as eigenfunctions of orbital angular momentum. Spherical harmonics are ubiquitous in atomic and molecular physics. They are important in the representation of the gravitational field, geoid, and magnetic field of planetary bodies, characterization of the cosmic microwave background radiation and recognition of 3D shapes in computer graphics.

Definition

The notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^m_\ell} will be reserved for functions normalized to unity. It is convenient to introduce first non-normalized functions that are proportional to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^m_\ell} . Several definitions are possible, we start with one that is common in quantum mechanically oriented texts. The spherical polar angles are the colatitude angle θ and the longitudinal (azimuthal) angle φ. The numbers l and m are integral numbers and l is positive or zero.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\ell^m(\theta,\varphi) \equiv i^{m+|m|}\; \left[\frac{(\ell-|m|)!}{(\ell+|m|)!}\right]^{1/2} P^{(|m|)}_\ell(\cos\theta) e^{im\varphi}, \qquad -\ell \le m \le \ell, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{(m)}_\ell(\cos\theta)} is a (phaseless) associated Legendre function. The m dependent phase is known as the Condon & Shortley phase:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{m+|m|} = \begin{cases} (-1)^m & \quad\hbox{if}\quad m > 0 \\ 1 & \quad\hbox{if}\quad m \le 0 \end{cases} }

An alternative definition uses the fact that the associated Legendre functions can be defined (via the Rodrigues formula) for negative m,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_\ell^m(\theta,\varphi) \equiv (-1)^m \left[\frac{(\ell-m)!}{(\ell+m)!}\right]^{1/2} P^{(m)}_\ell(\cos\theta) e^{im\varphi}, \qquad -\ell \le m \le \ell, }

The two definitions obviously agree for positive and zero m, but for negative m this is less apparent. It is also not immediately clear that the choices of phases yield the same function. However, below we will see that the definitions agree for negative m as well. Hence, for all l ≥ 0,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_\ell^m(\theta,\varphi) \equiv C_\ell^m(\theta,\varphi), \quad\hbox{for}\quad m=-\ell,\ldots,\ell. }

Complex conjugation

Noting that that the associated Legendre function is real and that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big(i^{m+|m|}\Big)^* = (-1)^m\, i^{-m+|m|}, \, }

we find for the complex conjugate of the spherical harmonic in the first definition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\ell^m(\theta,\varphi)^* = (-1)^m\, i^{-m+|m|}\; \left[\frac{(\ell-|m|)!}{(\ell+|m|)!}\right]^{1/2} P^{(|m|)}_\ell(\cos\theta) e^{-im\varphi} = (-1)^m C_\ell^{-m}(\theta,\varphi). }

Complex conjugation gives for the functions of positive m in the second definition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_\ell^{|m|}(\theta,\varphi)^* \equiv (-1)^m \left[\frac{(\ell-|m|)!}{(\ell+|m|)!}\right]^{1/2} P^{(|m|)}_\ell(\cos\theta) e^{-i|m|\varphi}. }

Use of the following non-trivial relation (that does not depend on any choice of phase):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{(|m|)}_\ell(\cos\theta) = (-1)^m \frac{(\ell+|m|)!}{(\ell-|m|)!} P^{(-|m|)}_\ell(\cos\theta). }

gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_\ell^{|m|}(\theta,\varphi)^* = \left[\frac{(\ell+|m|)!}{(\ell-|m|)!}\right]^{1/2} P^{(-|m|)}_\ell(\cos\theta) e^{-i|m|\varphi}= (-1)^m\tilde{C}_\ell^{-|m|}(\theta,\varphi). }

Since the two definitions of spherical harmonics coincide for positive m and complex conjugation gives in both definitions the same relation to functions of negative m, it follows that the two definitions agree. From here on we drop the tilde and assume both definitions to be simultaneously valid.

Note

If the m-dependent phase would be dropped in both definitions, the functions would still agree for non-negative m. However, the first definition would satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\ell^m(\theta,\varphi)^* = C_\ell^{-m}(\theta,\varphi), }

whereas the second would still satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_\ell^{m}(\theta,\varphi)^* = (-1)^m\tilde{C}_\ell^{-m}(\theta,\varphi), }

from which follows that the functions would differ in phase for negative m.

Normalization

It can be shown that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\pi} \int_{0}^{2\pi} C_\ell^m(\theta, \varphi)^* C_{\ell'}^{m'}(\theta, \varphi) \;\sin\theta d\theta d\varphi = \delta_{\ell\ell'}\delta_{mm'} \frac{4\pi}{2\ell+1}. }

The integral over φ gives 2π and a Kronecker delta on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m'} . Thus, for the integral over θ it suffices to consider the case m=m'. The necessary integral is given here. The (non-unit) normalization of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,C^m_\ell} is known as Racah's normalization or Schmidt's semi-normalization. It is often more convenient than unit normalization. Unit normalized functions are defined as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\ell^{m}(\theta,\varphi) \equiv \sqrt{\frac{2\ell+1}{4\pi}} C_\ell^{m}(\theta,\varphi). }

Condon-Shortley phase

One source of confusion with the definition of the spherical harmonic functions concerns the phase factor. In quantum mechanics the phase, introduced above, is commonly used. It was introduced by Condon and Shortley. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre functions, or to prefix it to the definition of the spherical harmonic functions, as done above. There is no requirement to use the Condon-Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions.

Orbital angular momentum

In quantum mechanics the following operator, the orbital angular momentum operator, appears frequently

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L} = -i \hbar \mathbf{r} \times \mathbf{\nabla}, }

where the cross stands for the cross product of the position vector r and the gradient ∇. From here on we take Planck's reduced constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} equal to unity. The components of L satisfy the angular momentum commutation relations.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L_i, L_j] = i\sum_{j=1}^3 \epsilon_{ijk} L_k,\qquad i,j,k = x,y,z, }

where ε_ijk is the Levi-Civita symbol. In angular momentum theory it is shown that these commutation relations are sufficient to prove that L² has eigenvalues l(l+1),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (L_x^2+L_y^2+L_z^2) \Psi \equiv L^2 \Psi = \ell(\ell+1) \Psi, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} is a natural number. The operator L² expressed in spherical polar coordinates is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2 = - \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial \theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2}\right]. }

The eigenvalue equation can be simplified by separation of variables. We substitute

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi = \Theta(\theta) \Phi(\varphi) }

into the eigenvalue equation. After dividing out Ψ and multiplying with sin²θ we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\frac{1}{\Theta(\theta)}\sin\theta \frac{\partial}{\partial\theta} \sin\theta \frac{\partial \Theta(\theta)}{\partial \theta} + \ell(\ell+1)\sin^2\theta \right] + \left[\frac{1}{\Phi(\varphi)} \frac{\partial^2 \Phi(\varphi)}{\partial\varphi^2}\right] = 0 }

In the spirit of the method of separation of variables, we put the terms in square brackets equal to plus and minus the same constant, respectively. Without loss of generality we take m² as this constant (m can be complex) and consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2 \Phi(\varphi)}{\partial\varphi^2} = -m^2 \Phi(\varphi) }

This has the solutions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(\varphi) = N e^{\pm i m \varphi} }

The requirement that exp[i m (φ + 2π)] = exp[i m φ] gives that m is integral. Substitution of this result into the eigenvalue equation gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial \Theta(\theta)}{\partial \theta} + \ell(\ell+1) - \frac{m^2}{\sin^2\theta} \right]\Theta(\theta) = 0 . }

Upon writing x = cos θ the equation becomes the associated Legendre equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2) \frac{d^2 \Theta }{dx^2} -2x\frac{d \Theta}{dx} + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] \Theta = 0 . }

This equation has two classes of solutions: the associated Legendre functions of the first and second kind. The functions of the second kind are non-regular for x = ±1 and do not concern us further. The functions of the first kind are the associated Legendre functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta(\theta) \propto P^{(\pm m)}_{\ell}(\cos\theta). }

It follows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2 \Psi = \ell(\ell+1) \Psi \Longrightarrow \Psi = P^{(\pm m)}_{\ell}(\cos\theta) e^{\pm i m \varphi}. }

The eigenvalue equation is invariant under choice of phase and normalization, so that these choice must be imposed separately. This was done earlier in this article. Finally, noting that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z = -i \frac{\partial}{\partial \varphi} }

we summarize the two important relations holding for spherical harmonics:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2 Y^{m}_\ell(\theta, \varphi) = \ell(\ell+1) Y^{m}_\ell(\theta, \varphi) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z Y^{m}_\ell(\theta, \varphi) = m Y^{m}_\ell(\theta, \varphi) }

Laplace equation

The Laplace equation ∇² Ψ = 0 reads in spherical polar coordinates

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial \Psi}{\partial r} + \frac{1}{r^2\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial\Psi}{\partial \theta} + \frac{1}{r^2\sin^2\theta} \frac{\partial^2\Psi}{\partial\varphi^2} = 0. }

Clearly, this can be rewritten as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial \Psi}{\partial r} - \frac{L^2}{r^2} \Psi = 0. }

Making the Ansatz Ψ = R(r) Y^m_l the equation can be solved readily. The solutions are known as solid harmonics. See solid harmonics for more details.

Connection with full 3D rotation group

The group of proper (no reflections) rotations in three dimensions is SO(3). It consists of all 3 x 3 orthogonal matrices with unit determinant. A unit vector is uniquely determined by two spherical polar angles and conversely. Hence we write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^m_\ell(\hat{\mathbf{r}}) \quad\hbox{with} \quad \hat{\mathbf{r}} \equiv \frac{\mathbf{r}}{|\mathbf{r}|}. }

Let R be a unimodular (unit determinant) orthogonal matrix, then we define a rotation operator by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{R} Y^m_\ell(\hat{\mathbf{r}}) \equiv Y^m_\ell(\mathbf{R}^{-1} \hat{\mathbf{r}}). }

The inverse matrix appears here (acting on a column vector) in order to assure that this map of rotation matrices to rotation operators is a homomorphism. Since this point was discussed at some length in Wigner's famous book on group theory, it is known as Wigner's convention. Some authors omit the inverse and find accordingly that the multiplication order of operators and matrices is reverse.

It can be shown that the rotation operator is an exponential operator in the components of the orbital angular momentum operator L. It can also be shown that the action of these operators on the spherical harmonics do no change l. That is, the linear space spanned by 2l+1 spherical harmonics of same l and different m is invariant under L, and therefore also under rotations,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{R} Y^m_\ell(\hat{\mathbf{r}}) = \sum_{m'=-\ell}^{\ell} Y^{m'}_\ell(\hat{\mathbf{r}}) D^{(\ell)}(\mathbf{R})_{m'm}. }

The square 2l+1 dimensional matrix that appears here is known as Wigner's D-matrix. Obviously, the set of matrices of fixed l form a representation of the group SO(3). It can be shown that they form an irreducible representation of this group. The rotation operator is unitary and the spherical harmonics are orthonormal, hence the Wigner rotation matrix is a unitary matrix:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\mathbf{D}^{(\ell)}\right)^\dagger \mathbf{D}^{(\ell)} = \mathbf{E}_\ell \Longleftrightarrow \sum_{m=-\ell}^{\ell} \big(D^{(\ell)}_{mm'}\big)^* D^{(\ell)}_{m m''} = \delta_{m' m''}, }

where E_l is the 2l+1 dimensional identity matrix. From this unitarity follows the following useful invariance

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{m=-\ell}^{\ell} Y^m_\ell(\hat{\mathbf{r}})^* \;Y^m_\ell(\hat{\mathbf{r}}) = \sum_{m=-\ell}^{\ell} Y^m_\ell(\mathbf{R}\hat{\mathbf{r}})^* \;Y^m_\ell(\mathbf{R}\hat{\mathbf{r}})\quad\hbox{for any}\quad \mathbf{R} \in \mathrm{SO(3)}. }

(To be continued: spherical harmonic addition theorem, real spherical harmonics, explicit espressions)