From 4e29e512c4f4f0f1244cbe38c804e46bafda225d Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Fri, 19 Aug 2022 21:57:24 +0200 Subject: kugel: Corrections and normalizations --- buch/papers/kugel/spherical-harmonics.tex | 104 ++++++++++++++++++------------ 1 file changed, 63 insertions(+), 41 deletions(-) (limited to 'buch') diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index 9d055e0..72f7402 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -493,9 +493,12 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result. \end{equation*} \end{lemma} \begin{proof} - \kugeltodo{Is this correct? And Is it worth showing? IMHO no, it is mostly the - same as Lemma \ref{kugel:thm:legendre-poly-ortho} with the difference that the - $m$-th derivative is a pain to deal with.} + To show that the expression equals zero when $n \neq n'$ we can perform + exactly the same steps as in the proof of lemma + \ref{kugel:thm:legendre-poly-ortho}, so we will not repeat them here and prove + instead only the case when $n = n'$. + \kugeltodo{Finish proof, or not? I have to look and decide if it is + interesting enough.} \end{proof} By having the orthogonality relations of the Legendre functions we can finally @@ -516,19 +519,19 @@ product: \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*} + \begin{align*} \langle Y^m_n, Y^{m'}_{n'} \rangle - = \int_{0}^\pi \int_0^{2\pi} + &= \int_{0}^\pi \int_0^{2\pi} Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)} \sin \vartheta \, d\varphi \, d\vartheta - = \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'} + \\ + &= \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'} = \begin{cases} - \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} & \text{if } n = n', \\ + \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} + & \text{if } n = n' \text{ and } m = m', \\ 0 & \text{otherwise}. \end{cases} - \end{equation*} + \end{align*} \end{theorem} \begin{proof} We will begin by doing a bit of algebraic maipulaiton: @@ -563,38 +566,15 @@ product: used lemma \ref{kugel:thm:associated-legendre-ortho}. We are allowed to use the lemma because $m = m'$. - Now we just need look at the case when $m \neq m'$. Fortunately this is easy: - the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, or in - other words we are integrating a complex exponetial over the entire period, - which always results in zero. Thus, we do not need to do anything and the - proof is complete. + Now we just need look at the case when $m \neq m'$. Fortunately this is + easier: the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, + or in other words we are integrating a complex exponetial over the entire + period, which always results in zero. Thus, we do not need to do anything and + the proof is complete. \end{proof} -\subsection{Normalization} - -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. +\kugeltodo{Briefly mention that we could have skipped the tedious proofs by +showing that the (associated) Legendre equation is a Sturm Liouville problem.} \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 @@ -620,7 +600,49 @@ Inoltre, possiamo provare l'ortogonalità di $\Theta(\vartheta)$ utilizzando \eq Ora, visto che la soluzione dell'eigenfunction problem è formata dalla moltiplicazione di $\Phi_m(\varphi)$ e $P_{m,n}(x)$ \fi -\subsection{Properties} + +\subsection{Normalization and the Phase Factor} + +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 an +ugly 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, but it is not the only common normalization that +is found in the literature. In physics, geomagnetism to be more specific, it is +common to use the so called Schmidt semi-normalization (or sometimes also called +quasi-normalization). + +\begin{definition}[Schmidt semi-normalized spherical harmonics] + \label{kugel:def:spherical-harmonics-schmidt} + The Schmidt semi-normalized spherical harmonics are + \begin{equation*} + Y^m_n(\vartheta, \varphi) + = \sqrt{2 \frac{(n - m)!}{(n + m)!}} + P^m_n(\cos \vartheta) e^{im\varphi} + \end{equation*} + where $m, n \in \mathbb{Z}$ and $|m| < n$. +\end{definition} + +However, for our purposes we will mostly only need the orthonormal spherical +harmonics. So from now on, unless specified otherwise, when we say spherical +harmonics or write $Y^m_n$, we mean the orthonormal spherical harmonics of +definition \ref{kugel:def:spherical-harmonics-orthonormal}. \subsection{Recurrence Relations} -- cgit v1.2.1