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authorNao Pross <np@0hm.ch>2022-08-18 17:23:40 +0200
committerNao Pross <np@0hm.ch>2022-08-18 19:03:31 +0200
commitc3261041f9bcf77a90ee0aa3e2dc73bf71edb923 (patch)
tree0703d7a63f79d3b4088653ee65a2360691882f04 /buch
parentMerge remote-tracking branch 'upstream/master' (diff)
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kugel: Corrections in orthogonality
Diffstat (limited to '')
-rw-r--r--buch/papers/kugel/spherical-harmonics.tex65
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