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| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 65 |
1 files changed, 49 insertions, 16 deletions
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index 2a00754..4f393d4 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -220,7 +220,7 @@ and $\lambda = n(n+1)$, we obtain what is known in the literature as the \emph{associated Legendre equation of order $m$}: \nocite{olver_introduction_2013} \begin{equation} \label{kugel:eqn:associated-legendre} - (1 - z^2)\frac{d^2 Z}{dz} + (1 - z^2)\frac{d^2 Z}{dz^2} - 2z\frac{d Z}{dz} + \left( n(n + 1) - \frac{m^2}{1 - z^2} \right) Z(z) = 0, \quad @@ -236,7 +236,7 @@ This reduces the problem because it removes the double pole, which is always tricky to deal with. In fact, the reduced problem when $m = 0$ is known as the \emph{Legendre equation}: \begin{equation} \label{kugel:eqn:legendre} - (1 - z^2)\frac{d^2 Z}{dz} + (1 - z^2)\frac{d^2 Z}{dz^2} - 2z\frac{d Z}{dz} + n(n + 1) Z(z) = 0, \quad @@ -275,7 +275,7 @@ Further, there are a few more interesting but not very relevant forms to write $P_n(z)$ such as \emph{Rodrigues' formula} and \emph{Laplace's integral representation} which are \begin{equation*} - P_n(z) = \frac{1}{2^n} \frac{d^n}{dz^n} (x^2 - 1)^n, + P_n(z) = \frac{1}{2^n n!} \frac{d^n}{dz^n} (z^2 - 1)^n, \qquad \text{and} \qquad P_n(z) = \frac{1}{\pi} \int_0^\pi \left( z + \cos\vartheta \sqrt{z^2 - 1} @@ -312,8 +312,8 @@ obtain the \emph{associated Legendre functions}. \label{kugel:def:ferrers-functions} The functions \begin{equation} - P^m_n (z) = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z) - = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n + P^m_n (z) = (1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z) + = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n \end{equation} are known as Ferrers or associated Legendre functions. \end{definition} @@ -486,7 +486,8 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result. \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz = \frac{2(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'} = \begin{cases} - \frac{2(m + n)!}{(2n + 1)(n - m)!} & \text{if } n = n', \\ + \frac{2(m + n)!}{(2n + 1)(n - m)!} + & \text{if } n = n' \text{ and } m = m', \\ 0 & \text{otherwise}. \end{cases} \end{equation*} @@ -497,16 +498,26 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result. derivative is a pain to deal with.} \end{proof} -An interesting fact to observe in lemma -\ref{kugel:thm:associated-legendre-ortho} is that the orthogonality is only -affected in the lower index, while varying $m$ only changes the constant in -front of the Kronecker delta. By having the orthogonality relations of the -Legendre functions we can finally show that spherical harmonics are also -orthogonal. +By having the orthogonality relations of the Legendre functions we can finally +show that spherical harmonics are also orthogonal under the following inner +product: -\begin{lemma} For the spherical harmonics - \kugeltodo{Fix horizontal spacing, inner product definition is missing.} +\begin{definition}[Inner product in $S^2$] + For 2 complex valued functions $f(\vartheta, \varphi)$ and $g(\vartheta, + \varphi)$ on the surface of the sphere the inner product is defined to be + \begin{equation*} + \langle f, g \rangle + = \int_{-\pi}^\pi \int_0^{2\pi} + f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)} + \sin \vartheta \, d\varphi \, d\vartheta. + \end{equation*} +\end{definition} + + +\begin{theorem} For the (unnormalized) spherical harmonics \label{kugel:thm:spherical-harmonics-ortho} + \kugeltodo{Why do I get a minus in front of $4\pi$??? It should not be there + right?} \begin{equation*} \langle Y^m_n, Y^{m'}_{n'} \rangle = \int_{-\pi}^\pi \int_0^{2\pi} @@ -518,7 +529,7 @@ orthogonal. 0 & \text{otherwise}. \end{cases} \end{equation*} -\end{lemma} +\end{theorem} \begin{proof} We will begin by doing a bit of algebraic maipulaiton: \begin{align*} @@ -558,7 +569,29 @@ orthogonal. \subsection{Normalization} -\kugeltodo{Discuss various normalizations.} +At this point we have shown that the spherical harmonics form an orthogonal +system, but in many applications we usually also want a normalization of some +kind. For example the most obvious desirable property could be for the spherical +harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n, +Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add a +normalization factor in front of the previous definition +\ref{kugel:def:spherical-harmonics} as follows. + +\begin{definition}[Orthonormal spherical harmonics] + \label{kugel:def:spherical-harmonics-orthonormal} + The functions + \begin{equation*} + Y^m_n(\vartheta, \varphi) + = \sqrt{\frac{2n + 1}{4\pi} \frac{(n-m)!}{(m+n)!}} + P^m_n(\cos \vartheta) e^{im\varphi} + \end{equation*} + where $m, n \in \mathbb{Z}$ and $|m| < n$ are the orthonormal spherical + harmonics. +\end{definition} + +Orthornomality is very useful indeed, but it is not the only common +normalization that is found in the literature. In physics, quantum mechanics to +be more specific, it is common to use the so called Schmidt semi-normalization. \if 0 As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline |