aboutsummaryrefslogtreecommitdiffstats
path: root/buch/papers
diff options
context:
space:
mode:
Diffstat (limited to 'buch/papers')
-rw-r--r--buch/papers/kugel/spherical-harmonics.tex104
1 files changed, 63 insertions, 41 deletions
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}