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\title{Formelsammlung zur Theoretischen Physik}
\author{Alexander Wagner}
\date{27 Mar 2000 }

\begin{document}
\maketitle
\pagestyle{plain}
\markright{Theoretische Physik}

\chapter{Physikalische Gr��en}

\section{Naturkonstanten}

\begin{tabular} {|l|c|l|}
\hline
Name & Symbol & Wert \\
\hline \hline
Plancksches Wirkungsquantum & h & $6,62618\cdot 10^{-34} J s$ \\
     & $\hbar = {h \over 2 \pi}$ &  $1,05459\cdot 10^{-34} J s$\\
Vakuumlichtgeschwindigkeit  & c & $ 2,99792 \cdot 10^{8} {m\over s}$\\
Magnetische Feldkonstante & $\mu_0$ & $4 \pi \cdot 10^{-7} {H\over m} = 1,25664\cdot 10^{-6} {H \over m}$ \\
Elektrische Feldkonstante & $\varepsilon_0$ & $8,85419\cdot 10^{-12} {F \over m}$ \\
\hline
Elementarladung & e & $1,60219 \cdot 10^{-19}C$ \\
Feinstrukturkonstante & $\alpha = {e^2 \over 4\pi \varepsilon_0 \hbar c}$ &
${1 \over 137,036}  = 7,29735 \cdot 10^{-3}$ \\
Stefan-Boltzmann-Konstante & $\sigma={2\pi^5 k^4 \over 15 c^2 h^3}$ & $5,6703\cdot 10^-8 {W\over m^2 K^4}$\\
Wiensche Konstante & $b=\lambda_{max} T$ & $2,897756\cdot 10^{-3}m K $ \\
\hline
Gravitationskonstante & G & $6,672\cdot 10^{-11} {N\cdot m^2 \over kg}$\\
\hline
Avogadro-Zahl     & $N_A$ & $6,02205 \cdot 10^{23} {1\over mol}$ \\
Faraday-Konstante & $F = N_A e$ & $9,64846\cdot 10^{4} {C \over mol}$ \\
Boltzmann-Konstante & $k_B$ & $1,38066\cdot 10^{-23} {J\over K}$ \\
Gaskonstante        & $R = k_B N_A$  & $8,31441 {J\over mol K}$ \\
\hline
Atomare Masseneinheit & $1 u ={1 \over 12} M_{^{12}C}$ & $1,66057 \cdot 10^{-27} kg$ \\
Elektronenmasse & $m_e$          & $9,10953\cdot 10^{-31} kg = 5,48580\cdot 10^{-4}u$\\
Protonenmasse   & $m_p$          & $1,67265\cdot 10^{-27} kg = 1,007276 u$ \\
Neutronenmasse  & $m_n$          & $1,67492\cdot 10^{-27} kg = 1,008665 u$ \\
                & $m_p\over m_e$ & $1836,15$ \\
                & $|e|\over m_e$ & $1,75880\cdot 10^{11} {C\over kg}$ \\
Comptonwellenl�nge des Elektrons & $\lambda_c = {h \over m c}$ & $2,42631\cdot 10^{-12} m$ \\
klassischer Elektronenradius     & $r_0 = {e^2 \over 4\pi \varepsilon_0 m c^2}$ & $2,81794\cdot 10^{-15} m$ \\
\hline
\end{tabular}

\begin{tabular} {|l|c|l|}
\hline
Name & Symbol & Wert \\
\hline \hline
Bohrradius & $a_0 = {4\pi \varepsilon_0 \hbar^2 \over m e^2}$ &
$5,29177\cdot 10^{-11} m$ \\
Ionisationsenergie H-Atoms & $I_P^H(\infty) = {e^2\over 8\pi
\varepsilon_0 a_0}$& $2,17991\cdot
10^{-18} J$\\
 && $= 13,6058 eV$ \\
&   $= {\alpha^2 m c^2 \over 2}$ & \\
Rydbergkonstante ($m = \infty$)  & $R_{\infty}= {m e^4 \over 8 \varepsilon_0^2 h^3 c}$ & $1,09737\cdot 10^{7} {1\over m}$ \\
& $ = {\alpha \over 4 \pi a_0}$ & \\
Rydbergkonstante (H-Atom) & $R_H$ & $1,09678\cdot 10^7 {1\over m}$\\
\hline
Bohrsches Magneton & $\mu_B = {e \hbar \over 2 m_e}$ & $9,27408\cdot 10^{-24} {J\over T}$\\
Kernmagneton & $\mu_K = {e \hbar \over 2 m_p}$ & $5,05082\cdot 10^{-27} {J \over T}$\\
Mag. Moment des $e^-$ & $M_e$ & $9,28483\cdot 10^{-24} {J\over T}$\\
 && $= 1,00116 \mu_B$ \\
Mag. Moment des $p^+$   & $M_p$ & $1,41062\cdot 10^{-26} {J\over T}$\\
 && $ = 2,79285 \mu_K$ \\
Mag. Moment des $n$  & $M_n$ & $-0,96630\cdot 10^{-26} {J\over T}$ \\
 && $= -1,91315 \mu_K$ \\ \hline
gyromagnetisches Verh�ltnis & $\gamma_p$ & $2,67522128\cdot 10^8 {rad \over T s}$ \\
Quantenhallwiderstand & $R_{Hall}$ & $25812,8056 \Omega$ \\
Mag. Flu�quantum & $\Phi_0 = {h\over 2e}$ & $2,06783461\cdot 10^{-15}
Wb$\\ \hline
\end{tabular}

\chapter {Vektoranalysis, Feldtheorie}

\section{Allgemeine Beziehungen}

Im folgenden einige grundlegende Gleichungen der Vektorrechnung und
Beziehungen zwischen den Vektorprodukten\footnote{s. a. {\sc
Bronstein} S. 149ff {\it Vektoralgebra und analytische Geometrie}}

\begin{eqnarray}
\lefteqn{\vec{A} \cdot \vec{B} \times \vec{C}  =   \vec{A} \times
\vec{B} \cdot \vec{C} \equiv (\vec{A},\vec{B},\vec{C})  =
(\vec{B},\vec{C},\vec{A})  = (\vec{C},\vec{A},\vec{B})  =}\nonumber \\
\lefteqn{-(\vec{A},\vec{C},\vec{B}) = -(\vec{C},\vec{B},\vec{A})
= -(\vec{B},\vec{A},\vec{C})}
\end{eqnarray}

\begin{equation}
\vec{A} \times (\vec{B}\times \vec{C}) = (\vec{A} \cdot \vec{C})
\vec{B} - (\vec{A} \cdot \vec{B}) \vec{C}
\end{equation}

\begin{equation}
\vec{A} \times (\vec{B} \times \vec{C}) + \vec{B} \times (\vec{C} \times \vec{A})+\vec{C} \times (\vec{A} \times \vec{B}) =
0
\end{equation}

\begin{equation}
(\vec{A}\times \vec{B})\cdot (\vec{C} \times D) = \vec{A} \cdot (\vec{B} \times (\vec{C}\times D) ) = (\vec{A}
\cdot \vec{B})(\vec{B} \cdot D) - (\vec{A} \cdot D) (\vec{B}\cdot D)
\end{equation}

\begin{equation}
(\vec{A} \times \vec{B}) \times (\vec{C} \times D) = (\vec{A}\times \vec{B} \cdot D)\vec{C} - (\vec{A} \times \vec{B}
\cdot \vec{C}) D
\end{equation}

Der vollkommen antisymmetrische Tensor $\varepsilon_{ijk}$
\footnote{s. a. {\sc Bronstein} S. 234}

\begin{eqnarray}
(\vec{a} \times \vec{b})_i & = & \varepsilon_{ijk} a_j b_k \\
\varepsilon_{ijk} & = &\varepsilon_{kji} \ \ \ \ \
\varepsilon_{ijk}\mbox{ zyklisch} \\
\varepsilon_{ijk} & = & - \varepsilon_{jik} \ \ \ \ \
\varepsilon_{ijk} \mbox{ antisymmetrisch} \\
\varepsilon_{kij} \varepsilon_{klm} & = & \delta_{il} \delta_{jm} -
\delta_{im} \delta_{jl}
\end{eqnarray}

\section{Definition von $grad, div, rot \mbox{ und } \Delta$ in
verschiedenen Koordinatensystemen}

Definition der Vektoroperatoren und deren Darstellung in
den gebr�uchlichsten Koordinatensystemen \footnote{s. a. {\sc
Bronstein} S. 523ff {\it Vektoranalysis und Feldtheorie } sowie
insbesondere auch S. 539}

\begin{itemize}
\item Definition: Divergenz $div$
\begin{equation}
div \vec{A}(\vec{r}) = \lim_{\Delta V \rightarrow 0} {1 \over
{|\Delta V|}} \oint_{\Delta F} d\vec{F} \cdot \vec{A} =
\oint_{\Delta F} d\vec{F} \cdot \vec{A}
\end{equation}
\item Definition: Rotation $rot$
\begin{equation}
{{\Delta F} \over {|\Delta F|}} \cdot rot \vec{A}(\vec{r}) =
\lim_{\Delta F \rightarrow 0}  \oint_{\Delta C} d\vec{r} \cdot
\vec{A} = \oint_{\Delta C} d\vec{r} \cdot \vec{A}
\end{equation}
\end{itemize}

\begin{equation}
grad \Phi = \left\{
\begin{array}{ll}
\displaystyle
{{\partial \Phi}\over {\partial x}} \vec{e_x} +
{{\partial \Phi}\over {\partial y}} \vec{e_x} +
{{\partial \Phi}\over {\partial z}} \vec{e_z} &
\mbox{(Kartesische Koordinaten)} \\ \\
\displaystyle
{{\partial \Phi}\over {\partial \rho}} \vec{e_{\rho}} +
{1\over \rho}{{\partial \Phi}\over {\partial \phi}} \vec{e_{\phi}} +
{{\partial \Phi}\over {\partial z}} \vec{e_{z}} &
\mbox{(Zylinderkoordinaten)} \\ \\
\displaystyle
{{\partial \Phi}\over {\partial r}} \vec{e_{r}} +
{1\over r}{{\partial \Phi}\over {\partial \theta}} \vec{e_{\phi}} +
{1\over {r \sin(\theta)}}{{\partial \Phi}\over {\partial \theta}}
\vec{e_{\theta}} & \mbox{(Kugelkoordinaten)} \\
\end{array} \right.
\end{equation}

\begin{equation}
div \vec{A} = \left\{
\begin{array}{l}
\displaystyle
{{\partial A_x}\over {\partial x}} +
{{\partial A_y}\over {\partial y}} +
{{\partial A_z}\over {\partial z}}  \\ \\
\displaystyle
{1\over \rho}{{\partial (\rho A_{\rho})}\over {\partial \rho}} +
{1\over \rho}{{\partial A_{\phi}}\over {\partial \phi}} +
{{\partial A_z}\over {\partial z}} \\ \\
\displaystyle
{1\over {r^2}} {{\partial (r^2 A_{r})}\over {\partial r}} +
{1\over {r \sin(\theta)}}{{\partial (sin(\theta)A_{\theta})}\over
{\partial \theta}} +
{1\over {r \sin(\theta)}}{{\partial A_{\phi}}\over {\partial \phi}} \\
\end{array}\right.
\end{equation}

\begin{equation}
rot \vec{A} =  \left\{
\begin{array}{l}
\displaystyle
\left(
{{\partial A_z}\over {\partial y}} -
{{\partial A_y}\over {\partial z}}
\right) \vec{e_x} +
\left(
{{\partial A_x}\over {\partial z}} -
{{\partial A_z}\over {\partial x}}
\right) \vec{e_y} + \\ \\
\displaystyle
+ \left(
{{\partial A_y}\over {\partial x}} -
{{\partial A_x}\over {\partial y}}
\right) \vec{e_z} \\ \\ \\
\displaystyle
\left( {1\over \rho}
{{\partial A_z}\over {\partial \phi}} -
{{\partial A_{\phi}}\over {\partial z}}
\right) \vec{e_{\rho}} + \left(
{{\partial A_{\rho}}\over {\partial z}} -
{{\partial A_z}\over {\partial \rho}}
\right) \vec{e_{\phi}} + \\ \\
\displaystyle
+ \left( {1\over \rho}
{{\partial (\rho A_{\phi})}\over {\partial \rho}} -
{{\partial A_{\rho}}\over {\partial \phi}}
\right) \vec{e_z} \\ \\ \\
\displaystyle
\left( {1\over {r \sin(\theta)}}
{{\partial (\sin(\theta)A_{\phi})}\over {\partial \theta}} -
{{\partial A_{\theta}}\over {\partial \phi}}
\right) \vec{e_{r}} + \\ \\
\displaystyle
+ {1\over r} \left( {1 \over {\sin(\theta)}}
{{\partial A_{r}}\over {\partial \phi}}
- {{\partial(r A_{\phi}) }\over {\partial r}}
\right) \vec{e_{\theta}} + \\ \\
\displaystyle
+ {1\over r}\left( {{\partial (r A_{\theta})}\over {\partial r}} -
{{\partial A_r}\over {\partial \theta}}
\right) \vec{e_{\phi}} \\ \\
\end{array}
\right.
\end{equation}

\begin{equation}
\Delta \Phi = \left\{
\begin{array}{l}
\displaystyle
{{\partial^2 \Phi}\over {\partial x^2}} +
{{\partial^2 \Phi}\over {\partial y^2}} +
{{\partial^2 \Phi}\over {\partial z^2}} \\ \\
\displaystyle
{1 \over \rho} {\partial \over {\partial \rho}}
\left(\rho {{\partial \Phi} \over {\partial \rho}} \right) +
{1 \over \rho^2} {{\partial^2 \Phi}\over {\partial \phi^2}} +
{{\partial^2 \Phi}\over {\partial z^2}} \\ \\
\displaystyle
{1 \over r} {{\partial^2 (r \Phi)}\over {\partial r^2}} +
{1 \over {r^2 \sin(\theta)}} {\partial \over {\partial \theta}}
\left(\sin(\theta) {{\partial \Phi}\over {\partial \theta}}\right)+
{1 \over {r^2 \sin^2(\theta)}} {{\partial^2 \Phi}\over {\partial
\phi^2}} \\ \\
\end{array}
\right.
\end{equation}

\section{Gradient, Divergenz, Rotation, Nabla ($\nabla$)}

\begin{equation}
grad (\phi \psi) \equiv \nabla(\phi \psi) = \phi \nabla \psi + \psi
\nabla \phi
\end{equation}

\begin{equation}
div (\phi \vec{A}) \equiv \nabla \cdot (\phi \vec{A}) = \vec{A} \cdot \nabla \phi + \phi
\nabla \cdot \vec{A}
\end{equation}

\begin{equation}
rot (\phi \vec{A}) \equiv \nabla \times (\phi \vec{A}) = \phi \nabla \times \vec{A} - \vec{A}
\times \nabla \phi
\end{equation}

\begin{equation}
div (\vec{A} \times \vec{B}) \equiv \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla
\times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})
\end{equation}

\begin{equation}
rot (\vec{A} \times \vec{B}) \equiv \nabla \times (\vec{A} \times \vec{B}) = \vec{A} (\nabla \cdot
\vec{B}) - \vec{B} (\nabla \cdot \vec{A}) + (\vec{B} \cdot \nabla) \vec{A} - (\vec{A} \cdot \nabla) \vec{B}
\end{equation}

\begin{equation}
grad (\vec{A}\cdot \vec{B}) \equiv \nabla (\vec{A} \cdot \vec{B}) = \vec{A} \times (\nabla \times \vec{B})
+ \vec{B} \times (\nabla \times \vec{A}) + (\vec{B} \cdot \nabla) \vec{A} + (\vec{A}\cdot \nabla)\vec{B}
\end{equation}

\begin{equation}
\nabla^2 \phi = \nabla \cdot \nabla \phi = \Delta \phi
\end{equation}

\begin{equation}
\nabla^2 \vec{A} = \nabla(\nabla \cdot \vec{A}) - \nabla \times (\nabla \times \vec{A})
\end{equation}

\begin{equation}
\nabla \times \nabla \phi = 0
\end{equation}

\begin{equation}
\nabla \cdot (\nabla \times \vec{A}) = 0
\end{equation}

\section {Verschiedene Beziehungen des Ortsvektors}

Sei $\vec{r}$ ein Ortsvektor der L�nge r vom Ursprung zu $(x, y, z)$
und sei $\vec{J}$ ein beliebiger konstanter Vektor. Dann gilt

\begin{equation}
\nabla \cdot \vec{r} = 3
\end{equation}

\begin{equation}\nabla \times \vec{r}  = 0\end{equation}

\begin{equation}\nabla \vec{r} = {\vec{r} \over {|r|}}\end{equation}

\begin{equation}\nabla {1 \over {|r|}} = {{-\vec{r}}\over {|r|}}\end{equation}

\begin{equation} \nabla \cdot {{\vec{r}}\over {|r|^3}} = -\nabla^2 {1 \over {|r|}} =
0 \;\;\; \mbox{wenn $r \neq 0$}
\end{equation}

\begin{equation}
\nabla \cdot {{\vec{J}} \over r} = \vec{J} \cdot (\nabla {1 \over r})
= - {{(\vec{J}\cot \vec{r})}\over {|r|^3}} \end{equation}

\begin{equation}
\nabla^2 {{\vec{J}}\over {|r|}} = \vec{J}\nabla^2 {1 \over {|r|}} = 0
\;\;\; \mbox{wenn $r \neq 0$}
\end{equation}

\begin{equation}
\nabla \times (\vec{J}\times \vec{B})= \vec{J} (\nabla \cdot B) +
\vec{J} \times (\nabla \times \vec{B}) - \nabla (\vec{J} \cdot
\vec{B})
\end{equation}

\section {Integrals�tze}

Sei V ein durch die geschlossene Oberfl�che S begrenztes Volumen. dS
sei positiv, wenn es vom Volumen nach au�en zeigt (Normalenvektor).
\footnote{Zu den Integrals�tzen s. a. {\sc Bronstein} S. 545ff {\it
Integrals�tze}} Dann gilt:

{\em Satz von Gau�}
\begin{equation}
\oint_S \vec{A} \cdot \vec{dS} = \int_V (\nabla \cdot \vec{A}) dV =
\int_V div \vec{V} dV
\end{equation}

In 4 Dimensionen:
\begin{equation}
  \int_{\partial B} d\sigma_\lambda A^\lambda = \int_B dx
  \partial^\lambda A_\lambda
\end{equation}

Als Folgerungen aus dem Satz von Gau� ergegen sich die
Integrals�tze von Green

{\em 1. Greenscher Satz}
\begin{equation}
\int_{V} (U_1 \Delta U_2 + grad U_2 grad U_1) dv = \oint_S U_1 grad
U_2 \cdot \vec{dS}
\end{equation}

{\em 2. Greenscher Satz}
\begin{equation}
 \int_V (U_1 \Delta U_2 - U_2 \Delta U_1) dv  =  \oint_S
(U_1 grad U_2 - U_2 grad U_1) \cdot \vec{dS}
\end{equation}

Speziell f�r $U_1 = 1$ wird der 2. Greensche Satz zu

{\em 3. Greenscher Satz}
\begin{equation}
\int_V \Delta U dv = \oint_S grad U \cdot \vec{dS}
\end{equation}

\begin{equation}
\oint_S \phi dS = \int_V (\nabla\phi) dV
\end{equation}

\begin{equation}
\oint_S dS \times \vec{A} = \int_V (\nabla \times \vec{A}) dV
\end{equation}

Sei nun S eine offene Fl�che die durch die C begrenzt wird, dessen
Linienelement ds sei. Es gilt:

{\em Satz von Stokes}

\begin{equation}
\oint_C \vec{A} \cdot ds = \int_S (\nabla \times A) dS
\end{equation}


\begin{equation}
\oint_C \phi ds = \int_S dS \times \nabla \phi
\end{equation}

\chapter {Die Dirac'sche $\delta$-Funktion}

Die $\delta$-Funktion ist erkl�rt durch:

\begin{equation} (\delta(x-x_0), \varphi(x))=\varphi(x_0) \end{equation}
also
\begin{equation} \int dx \delta (x-x_0) \varphi(x)=\varphi(x_0) \end{equation}

\section {Limesdarstellungen}

\begin{equation}
\delta(x) \;\;=\;\; {1 \over \pi} \lim_{\varepsilon\rightarrow 0} {\varepsilon
\over {x^2+\varepsilon^2}}\;\; =\end{equation}
\begin{equation}{1 \over {2 \sqrt{\pi}}}
\lim_{\varepsilon\rightarrow 0} {1 \over
{\sqrt{\varepsilon}}} \exp{({-x^2 \over {4\varepsilon}})}\;\; =
\end{equation}
\begin{equation}
=\;\;{1 \over
\pi}
\lim_{\nu\rightarrow \infty} {\sin(\nu x) \over x}\;\; = \;\;{1 \over \pi}
\lim_{\nu\rightarrow\infty}
{{\sin^2(\nu x)} \over {\nu x^2}}
\end{equation}

\section {Die $\delta$-Funktion als Ableitung}

Stufenfunktionen:
\begin{equation}
\Theta(x)=\left\{
\begin{array}{ll}
1 & x>0 \\
0 & x<0 \\
\end{array}\right.
\end{equation}
\begin{equation}
\varepsilon(x)=\left\{
\begin{array}{ll}
1 & x>0  \\
-1 & x<0   \\
\end{array}\right.
\end{equation}

Verschiedene Darstellungen der $\delta$-Funktion:

\begin{equation} \varepsilon(x)=\Theta(x)-\Theta(-x)={d \over dx}|x| \end{equation}
\begin{equation} \Theta(x)={1 \over 2} (1+\varepsilon(x)) \end{equation}
\begin{equation}
\delta(x)\;\;=\;\;{d \over dx} \Theta(x)\;\;=\;\;{1 \over 2} {d \over dx}
\varepsilon(x)\;\; =\;\;{1 \over 2} {d^2 \over dx^2} |x|
\end{equation}

\section{Formeln mit $\delta$}

\begin{eqnarray}
\lefteqn{\delta(x)\;\; = \;\;\delta(-x)\;\;=\;\;|a|\delta(ax)
\varphi(x)\delta(x-a)\;\;=} \nonumber \\
\lefteqn{\varphi(a)\delta(x-a) \varphi(x\pm x)\delta(x)
\;\;=} \nonumber \\
\lefteqn{\varphi(a)\delta(x) x \delta(x)\;\; = } \nonumber \\
\lefteqn{x^2 \delta(x)\;\;=\cdots\;\;=\;\;0}
\end{eqnarray}

\begin{equation}
\delta(\psi(x))=\sum_{N}{ {\delta(x-x_N)} \over |{\psi'(x_N)}|}
\end{equation}

Wenn $\psi$ nur einfache Nullstellen $x_N$ hat.

\begin{equation}
\int_{-\infty}^{+\infty}dx \varphi(x) {d^n \over
dx^n}\delta(x-y)=(-1)^n {d^n \over dy^n \varphi(y)}
\end{equation}

\section{Dreidimensionale $\delta$-Funktion}

\begin{eqnarray}
\lefteqn{\delta(\vec{x}-\vec{x'}) = \delta(x_1-x_1')\delta(x_2-
x_2')\delta(x_3-x_3') =  }\nonumber \\
\lefteqn{{1 \over r^2}\delta(r-r') \delta(\cos(\theta)-
\cos(\theta')) \delta(\varphi- \varphi') = }\nonumber \\
\lefteqn{-{1 \over {4 \pi}} \Delta {1 \over {|\vec{x}-\vec{x'}|}}}
\end{eqnarray}

\begin{equation}
\vec{x}=(x_1, x_2, x_3)
\end{equation}

bzw. in Kugelkoordinaten

\begin{equation}
\vec{x}=(r \cos(\varphi)\sin(\theta), r \sin(\varphi)\sin(\theta),
r\cos(\theta))
\end{equation}

\section {Logarithmus, Hauptwert, $\delta_{\pm}$}

F�r den komplexen Logarithmus ist

\begin{equation}
\lim_{\varepsilon\rightarrow 0} \ln(x \pm i \varepsilon)=\ln|x| \pm
i \pi \Theta(x) \end{equation}

Durch diffenenzieren folgt:
\begin{equation}
\lim_{\varepsilon \rightarrow 0} {1 \over {x \pm i\varepsilon}}= P {1
\over x} \mp i \pi \delta(x) =: \mp 2 \pi i \delta_{\pm}(x)
\end{equation}

Dabei bedeutet P den Cauchy'schen Hauptwert:

\begin{equation}
(P{1 \over x}, \varphi(x)) = \lim_{\varepsilon\rightarrow 0}
\left(\int_{-\infty}^{-\varepsilon} +
\int_{\varepsilon}^{\infty}\right) {\varphi(x) \over x} dx
\end{equation}

\begin{equation} \delta_{\pm}(-x)=\delta_{\mp}(x) \end{equation}

\begin{equation} x\delta_{\pm}(x)=\mp {1 \over {2 \pi i}} \end{equation}

\begin{equation} \delta_{+}(x)+\delta_{-}(x)= {1 \over {2 \pi i}} \lim_{\varepsilon
\rightarrow 0}({1 \over {x-i \varepsilon}}-{1 \over {x+i
\varepsilon}})=\delta(x) \end{equation}

\begin{equation} \delta_{+}(x)-\delta_{-}(x)=-{1 \over {2 \pi i}} \lim_{\varepsilon
\rightarrow 0}({1 \over {x-i \varepsilon}}+{1 \over {x+i
\varepsilon}})={i \over \pi} P{1 \over x} \end{equation}

\begin{equation} P{1 \over x} =\lim_{\varepsilon \rightarrow 0} {x \over
{x^2+\varepsilon^2}} \end{equation}

\section {Fourierintegrale}

\begin{equation}
\delta(x) = {1 \over {2 \pi}} \int_{-\infty}^{+\infty} e^{i k x} dk
\end{equation}

\begin{equation} \delta_{\pm}(x) = {1 \over {2 \pi}} \int_{-\infty}^{+\infty} e^{i k
x} \Theta(\pm k) dk
\end{equation}

\begin{equation}
P{1 \over x}={1 \over {2 i}} \int_{-\infty}^{+\infty} e^{i k x}
\varepsilon(k) dk
\end{equation}

\begin{equation}
\Theta(\pm x)= \pm {1 \over {2 \pi i}} \int_{-\infty}^{+\infty}
{{e^{i k x}} \over {k \mp i \varepsilon}}, \mbox{ wobei  }
\varepsilon(x) = {1 \over {i \pi}} \int_{-\infty}^{+\infty} e^{i k x}
 P {1 \over k} dk
\end{equation}

\begin{equation}
\delta'(x)= \int_{-\infty}^{+\infty} i k e^{i k x} dk \mbox{ etc.}
\end{equation}

\begin{equation}
\delta(\vec{x})= {1 \over {(2\pi)^3}} \int e^{i \vec{k}
\vec{x}} d^3k
\end{equation}

\begin{equation} \delta(r-a)={a \over {2 \pi^2}} \int e^{i \vec{k}\vec{x}}
{\sin(a k) \over k} d^3k \mbox{ mit  }k=\sqrt{\vec{k}^2} \end{equation}

\chapter {Orthogonale Funktionensysteme}

F�r orthogonale Funktionen gilt allgemein:

\begin{equation} \int_{a}^{b}=\overline{f_n(x)} f_m(x) w(x) dx = {1 \over N_n^2}
\delta_{mn}\end{equation}
\begin{equation} \sum_{n} N_n^2 f_n(x) \overline{f_n(y)} = {1 \over {w(x)}}
\delta(x-y) \end{equation}

\section {Trigonometrische Funktionen}

\begin{equation}
f_n=e^{i n x} \mbox{ mit } -\infty \leq n \leq +\infty
\end{equation}

\begin{equation}
N_n^2 = {1 \over {2\pi}}, a=0, b=2\pi, w(x)=1
\end{equation}

\section {Hermite'sche Polynome  }

Energieeigenfunktionen des harmonischen Oszillators in der
Quantenmechanik\footnote{s. a. {\sc Bronstein} S. 402
und 435 bzw. unter {\it Harmonischer Oszillator} S. 433ff}

\begin{equation}
f_n=H_n(x) \mbox{ mit } 0\leq n \leq \infty
\end{equation}

\begin{equation}
N_n^2={1\over {2^n n! \sqrt{\pi}}}\quad a=-\infty,\quad b=+\infty
\quad w(x)=e^{-x^2}
\end{equation}

\begin{equation}H_0(x) = 1 \;\;\;\; H_1(x)=2x \;\;\;\; H_2(x)=4x^2-2 \;\;\;\;
H_3(x)=8x^3-12x\end{equation}

allgemein:
\begin{equation} H_n(x) = (-1)^n e^{x^2} {d^n \over {dx^n}} e^{-x^2} \end{equation}


\begin{equation} H_{2n+1}(0)=0 \;\;\;\; H_{2n}(0)={{(-1)^n(2n)!} \over n!} \end{equation}

Rekursionsformel zur Berechnung:

\begin{equation}H_{n+1}(x)=2x H_n(x)-2n H_{n-1}(x) \end{equation}

\begin{equation}\left({d^2 \over {dx^2}} -2x {d \over {dx}} + 2n \right) H_n(x) = 0
\end{equation}



\section {Laguerre'sche Funktionen}

Partikul�re L�sungen der Laguerre'schen Differentialgleichung, einer
speziellen hypergeometrischen Differentialgleichung. \footnote{s. a.
{\sc Bronstein} S. 402}

\begin{equation}
f_n(x)=L_n^{(k)}(x) \mbox{ mit } 0\leq n \leq \infty
\end{equation}

\begin{equation}
N_n^2={{n!} \over {\Gamma(n+k+1)}}, \;\;\;\; a=0, \;\;\;\; b=\infty,
\;\;\;\; w(x)=x^k e^{-x}
\end{equation}
k rell, f�r ganze k ergeben sich die Laguerre-Polynome

\begin{equation}L_0(x)=1\;\;\;\;L_1^{(k)}(x)=k+1-x\end{equation}
\begin{equation}L_2^{(k)}(x)={1 \over 2} \left[(k+1)(k+2)-2(k+2)x+x^2 \right]\end{equation}
\begin{equation}L_n^{(k)}={1 \over n!} x^{-k} e^x {{d^n} \over {dx^n}}(e^{-x}
x^{n+k}) = \sum_{\nu=0}^n \left(
\begin{array}{cc}
{n+k} \\
{n-\nu}
\end{array}
\right){{(-x)^{\nu}}
\over {\nu!}} \end{equation}

\begin{equation} L_k^{-k}(x)=(-1)^k {{x^k} \over {k!}} \;\;\;\;
L_n^{(k)}(0)=\left(
\begin{array} {cc}
{n+k}\\
{k}
\end{array}
\right)\end{equation}

\begin{equation}\left(x{{d^2} \over {dx^2}} +(k+1-x){d \over
dx}+n \right)\;\;L_n^{(k)}(x)=0
\end{equation}

\begin{eqnarray}
\lefteqn{{L_n^{(k)}}'(x) = -L_{n-1}^{(k+1)}(x)
= L_n^{(k)}(x)-L_n^{(k+1)}(x) = }\nonumber \\
\lefteqn{{1 \over x}(n L_n^{(k)}(x)-(n+k) L_{n-1}^{(k)}(x))}
\end{eqnarray}

Fortsetzung f�r k nicht ganz:

\begin{equation}L_n^{(k)}(z)=\left(
\begin{array}{cc}
{n+k}\\{n}
\end{array}
\right)\; _1F_1(-n;k+1;z)\end{equation}

($_1F_1$ ist die konfluente hypergeometrische Funktion)

\section {Sph�rische Besselfunktionen}

Stellen die partikul�ren L�sungen der Besselschen
Differentialgleichung dar. \footnote{s.a. {\sc Bronstein } S. 398ff}

\begin{equation}
j_l(x)\;\;=\;\;\sqrt{{\pi \over {2x}}} J_{l+{1 \over
2}}(x)\;\;=\;\;x^l\left(-{1 \over x} {d \over {dx}}\right)^l
{{\sin(x)} \over x} \end{equation}

\begin{equation} j_0(x)={{\sin(x)}\over x} \;\;\;\; j_1(x)={{\sin(x)}\over
{x^2}}-{{\cos(x)}\over x} \cdots\end{equation}

\begin{equation}\left\{x^2{d^2 \over {dx^2}} + 2x {d \over dx} +
\left[x^2-l(l+1)\right]\right\}\;\;j_l(x)=0\end{equation}

\begin{equation}j_l'(x)=j_{l-1}(x)-{{l+1} \over x} j_l(x)\end{equation}


\begin{equation}\int_0^\infty y^2 j_l(yx) j_l(yx')dy={\pi \over
{2x^2}}\delta(x-x')\end{equation}

\section {Legendre'sche Polynome}

Eigenfunktionen des Drehimpulsoperators $\vec{L}^2$ und $\vec{L_z}$ in
der Quantenmechanik \footnote{s. a. {\sc Bronstein  }
Eigenschaften S. 400f, bzw. S. 458  (Tabelle)}

\begin{equation}
f_n(x)=P_n(x) \mbox{ mit } 0 \leq n \leq \infty, N_n^2 =
n+{1 \over 2}, a=-1 b=1, w(x)=1
\end{equation}

\begin{equation} P_0(x)=1 \;\;\;\; P_1(x)=x \;\;\;\;  P_2(x)={1 \over 2}(3x^2-1)
\;\;\;\; P_3(x)={1 \over 2}(5x^3-3x) \end{equation}

allgemein:
\begin{equation}P_n(x)={1 \over {2^n n!}} {d^n \over {dx^n}} (x^2-1)^n \end{equation}

\begin{equation}P_{2n+1}(0)=0\;\;\;\; P_{2n}(0)=(-1)^n {{(2n)!}\over{2^{2n}(n!)^2}}\;\;\;\;
P_n(\pm 1)=(\pm 1)^n\end{equation}

Rekursionsformel:
\begin{equation}(n+1)P_{n+1}(x)=(2n+1) x P_n(x)-n P_{n-1}(x)\end{equation}

\begin{equation}((1-x^2){d^2 \over {dx^2}}-2x{d\over{dx}}+n(n+1))P_n(x)=0\end{equation}
\begin{equation}(1-x^2)P_n'(x)=n(P_{n-1}(x)-xP_n(x))={{n(n+1)} \over
{2n+1}}(P_{n-1}(x)-P_{n+1}(x))\end{equation}

\begin{equation}
P_l(\cos(\theta)) = {4\pi \over {2 l + 1}} \sum_{m=-l}^l
\overline{Y_l^m(\theta_1, \phi_1)} Y_l^m(\theta_2, \phi_2)
\end{equation}

\begin{equation}
{1 \over {|\vec{r_1}-\vec{r_2}|}} = {1 \over \sqrt{r_1^2+r_2^2+2 r_1 r_2 \cos(\gamma)} }
= \sum_{l=0}^{\infty} {r_<^l \over r_>^{l+1}} P_l(\cos(\gamma))
\end{equation}


\section {Kugelfl�chenfunktionen (Spherical harmonics)}

Diese bilden ein vollst�ndiges Orthogonalsystem in zwei Dimensionen.
Kugelfl�chenfunktionen bilden die Eigenfunktionen der
Drehimpulsoperatoren $\vec{L}^2$ und $L_z$. \footnote{s. a.
Legendre-Polynome sowie deren zugeordnete Funktionen ({\sc Bronstein}
S. 400f, 458) , und {\sc Bronstein} S. 430ff, und 433 }

F�r eine Liste der Kugelfl�chenfunktionen siehe unter [\ref{sec_Ylm}]


\begin{equation} \int d\Omega=\int_0^{2\pi}d\varphi \int_0^\pi \sin(\theta) d\theta=
\int_0^{2\pi}d\varphi \int_{-1}^{+1}d(cos(\theta)\end{equation}

\begin{equation}\int d\Omega \overline{Y_l^m(\theta, \varphi)} Y_{l'}^{m'}(\theta',
\varphi')=\delta_{l,l'}\delta_{m,m'}\end{equation}

\begin{equation}\sum_{l=0}^\infty \sum_{m=-l}^l Y_l^m(\theta, \varphi)
\overline{Y_l^m(\theta', \varphi')}
=\delta(\varphi-\varphi')\delta(\cos(\theta)-cos(\theta'))\end{equation}

\begin{equation}Y_l^m(\theta, \varphi)=\sqrt{{{(2l+1)(l-|m|)!} \over
{4\pi(l+|m|)!}}} e^{i m \varphi} P_l^m cos(\theta)\end{equation}

\begin{eqnarray}
\lefteqn{P_l^m(cos(\theta)) = \varepsilon_m sin^{|m|}(\theta)
{\left(d \over {d(\cos(\theta)}\right)}^m P_l(\cos(\theta))}
\nonumber \\&& \mbox{mit }
\varepsilon_m=\left\{
\begin{array}{ll}
(-1)^m & x\leq 0 \\
1 & x<0 \\
\end{array}\right.
\end{eqnarray}

\begin{equation}
Y_l^{-m}(\theta, \varphi)=(-1)^m \overline{Y_l^m(\theta, \varphi)}
\end{equation}

\begin{equation}
Y_l^0(\theta, \varphi)=\sqrt{{{2l+1} \over
{4\pi}}}P_l(\cos(\theta))
\end{equation}

\begin{eqnarray}
\lefteqn {Y_l^m(0,\varphi) =  }  \nonumber \\
\lefteqn{ \delta_{m,0}\sqrt{{{2l+1}\over {4\pi}}} \left({{\partial^2}
\over {\partial\theta^2}}+ \cot(\theta){\partial
\over {\partial\theta}} + {1 \over {\sin^2(\theta)}}
{{\partial^2}\over {\partial \varphi^2}} +
l(l+1)\right)\;\;Y_l^m(\theta, \varphi)} \nonumber \\
 & = & 0  \nonumber \\
\end{eqnarray}

\begin{equation}
\cos(\theta) Y_l^m=\sqrt{{{(l+m)(l-m)} \over
{(2l-1)(2l+1)}}}\;Y_{l-1}^m+\sqrt{{{(l+m+1)(l-m+1)} \over
{(2l-1)(2l+3)}}}\;Y_{l+1}^m\end{equation}

\begin{eqnarray}
\lefteqn{-\sin(\theta) e^{\pm i\varphi} Y_l^m = } \nonumber \\
\lefteqn{\mp \sqrt{{{(l\mp m)(l\mp m-1)} \over {(2l-1)(2l+1)}}}
 Y_{l-1}^{m\pm 1} \pm \sqrt{{{(l\pm m+2)(l\pm m+1)} \over
{(2l+1)(2l+3)}}} Y_{l+1}^{m\pm 1}}
\end{eqnarray}

\section {Fouriereihen}

\begin{equation}
\int_a^b \sin\left({n \pi x \over a} \right) \sin\left({m \pi x \over
a} \right) dx = {a \over 2} \delta_{nm}
\end{equation}

\begin{equation}
\sum_{n=1}^\infty \cos(nx)=-{1 \over 2} + \pi
\sum_{n=-\infty}^{+\infty} \delta(x-2\pi n)\end{equation}

\begin{equation}
\sum_{n=1}^\infty \sin(nx)=-\pi \sum_{n=-\infty}^{-\infty}
\delta'(x-2\pi n)\end{equation}

\begin{equation}\sum_{n=-\infty}^{+\infty} e^{i n x} = 2\pi
\sum_{n=-\infty}^{+\infty} \delta(x-2\pi n)\end{equation}

\begin{equation}
\int Y_l^m d\Omega = {1 \over \sqrt{4 \pi}} 4\pi \delta_{l0} \delta_{m0}
\end{equation}

\chapter {Operatoralgebra}

\begin{eqnarray}
[A,B]     & = & AB-BA \\
\{A,B\}   & = & AB+BA \\
\left[A,B\right]^\dagger & = & \left[B^\dagger, A^\dagger\right] \\
\left[A,B\right]
          & = & - \left[B,A\right]\\
\left[AB,C\right]
          & = & A[B,C]+[A,C]B\\
\left[A,BC\right]
          & = & B[A,C]+[A,B]C\\
\left[\left[A,B\right],C\right]
          & = & \{A, \{B,C\}\} - \{B, \{A,C\}\}\\
\left[AB,CD\right]
          & = & AC[B,D]+A[B,C]D+C[A,D]B+[AC]DB\\
\left[AB,CD\right]
          & = & A \{A,C\}D - AC \{B,D\} - C\{A,D\}B + \{C,A\}DB\\
e^{At}    & = & \sum_{n=0}^\infty {(At)^n \over n!}\\
e^A e^B   & = & e^{A+B} e^{\einhalb [A,B]}\\
\end{eqnarray}

%Baker-Haussdorff-Formel:
%\begin{equation}
%
%\end{equation}

\chapter{Quantenmechanik}

\section{Normierte Radialwellenfunktionen $R_{n,l}(r)$}

\begin{tabular}{|c|c|c|c|l|}
\hline
n &  l & Schale & Orbital & Funktion \\
\hline
1 & 0 & K & s &
$\displaystyle
R_{10}(r) = 2 \cdot \left( {Z \over a}\right)^{3 \over 2}
\exp\left(-{Z r \over a } \right)
$\\
2 & 0 & L & s &
$\displaystyle
R_{20}(r) = 2 \cdot \left( {Z \over 2a}\right)^{3 \over 2}
\left( 1 - {Z r \over 2a}\right)
\exp\left(-{Z r \over 2a } \right)
$\\
2 & 1 & L & p &
$\displaystyle
R_{21}(r) = {1 \over \sqrt{3}} \left( {Z \over 2a}\right)^{3 \over 2}
\left({Z r \over a}\right)
\exp\left(-{Z r \over 2a } \right)
$\\
3 & 0 & M & s &
$\displaystyle
R_{30}(r) = 2 \cdot \left( {Z \over 3a}\right)^{3 \over 2}
\left(1- {{2 Z r}\over {3 a}}+ {{2 (Z r)^2}\over {27 a^2}}\right)
\exp\left(-{Z r \over 3 a } \right)
$\\
3 & 1 & M & p &
$\displaystyle
R_{31}(r) = {{4 \sqrt{2}}\over 3}\left( {Z \over 3a}\right)^{3 \over
2} {{Z r}\over a} \left( 1 - { Zr \over 6 a}\right) \exp\left(-{Z r
\over 3 a } \right)
$\\
\hline
\end{tabular}

\section{Winkelanteil der Wellenfunktionen
(= Kugelfl�chenfunktionen) \label{sec_Ylm}}

\begin{center}
\begin{tabular}{|c|c|l|}
\hline
l & m & Funktion \\
\hline
0 & 0 &
$\displaystyle Y_0^0={1 \over {\sqrt{4\pi}}} $ \\
\hline
1 & +1 &
$\displaystyle Y_1^1=-\sqrt{{3 \over {8\pi}}} \sin(\theta) e^{i \varphi} $ \\
1 & 0 &
$\displaystyle Y_1^0=\sqrt{{3 \over {4\pi}}} \cos(\theta) $ \\
1 & -1 &
$\displaystyle Y_1^{-1}=\sqrt{{3 \over {8\pi}}}\sin(\theta) e^{-i\varphi} $ \\
\hline
2 & +2 &
$\displaystyle Y_2^2=\sqrt{{15\over{32\pi}}}\sin^2(\theta) e^{2 i\varphi}$ \\
2 & +1 &
$\displaystyle Y_2^1=-\sqrt{{15\over {8\pi}}}\cos(\theta) \sin(\theta) e^{i
\varphi}$ \\
2 &  0 &
$\displaystyle Y_2^0=\sqrt{{5\over{16\pi}}}[3\cos^2(\theta)-1]$\\
2 & -1 &
$\displaystyle Y_2^{-1}=\sqrt{{15\over {8\pi}}} \cos(\theta) \sin(\theta) e^{-i
\varphi}$ \\
2 & -2 &
$\displaystyle Y_2^{-2}=\sqrt{{15 \over {32\pi}}} \sin^2(\theta) e^{-2i\varphi} $\\
\hline
3 & +3 &
$\displaystyle
Y_3^3 = - \sqrt{{35 \over 64 \pi}} \sin^3(\theta) e^{3 i \phi}
$\\
3 & +2 &
$\displaystyle
Y_3^2 = \sqrt{{105 \over 32 \pi}} \sin^2(\theta) e^{2 i \phi}
$\\
3 & +1 &
$\displaystyle
Y_3^1 = - \sqrt{{21 \over 64 \pi}} \sin(\theta) (5 \cos^2(\theta)-1)
e^{i \phi}
$\\
3 & 0  &
$\displaystyle
Y_3^0 = \sqrt{{7 \over 16 \pi}} (5 \cos^3(\theta)- 3 \cos(\theta))
$\\
3 & -1 &
$\displaystyle
Y_3^{-1} = \sqrt{{21 \over 64 \pi}} \sin(\theta) (5
\cos^2(\theta)-1) e^{-i \phi}
$\\
3 & -2 &
$\displaystyle
Y_3^{-2} = \sqrt{{105 \over 32 \pi}} \sin^2(\theta) e^{-2 i \phi}
$\\
3 & -3 &
$\displaystyle
Y_3^{-3} = + \sqrt{{35 \over 64 \pi}} \sin^3(\theta) e^{-3 i \phi}
$\\
\hline
\end{tabular}
\end{center}


\section{Normierte Eigenfunktionen $\psi_{n,l,m}(r,\theta,\phi)$}

\begin{tabular}{|l|l|}
\hline
$\displaystyle \psi_{100} ={1 \over {\sqrt{\pi}}} \left( {Z \over {a_0}}\right)^{3
\over 2 } \exp(-{{Zr}\over {a_0}})
$ &
$\displaystyle
 = R_{10} \cdot Y_0^0
$\\
$\displaystyle \psi_{200} = {1\over {4 \sqrt{2 \pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left( 2-{{Zr}\over {a_0}}\right)
\exp(-{{Zr}\over {2 a_0}})
$&
$\displaystyle
 = R_{20} \cdot Y_0^0
$\\
$\displaystyle \psi_{210} =  {1\over {4 \sqrt{2 \pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left( {{Zr}\over {a_0}}\right)
\exp(-{{Zr}\over {2 a_0}}) \cos(\theta)
$&
$\displaystyle
 = R_{21} \cdot Y_1^0
$\\
$\displaystyle \psi_{21\pm 1} = {1\over {8 \sqrt{\pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left( {{Zr}\over {a_0}}\right)
\exp(-{{Zr}\over {2 a_0}}) \sin(\theta) e^{(\pm i \phi)}
$&
$\displaystyle
 = R_{21} \cdot Y_1^{\pm1}
$\\
$\displaystyle \psi_{300} = {1\over {81 \sqrt{3 \pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left(27 - 18 {{Zr}\over {a_0}} + 2
\left({{Zr}\over {a_0}} \right)^2\right) \exp(-{{Zr}\over {3 a_0}})
$&
$\displaystyle
 = R_{30} \cdot Y_0^0
$\\
$\displaystyle \psi_{310} = {{\sqrt{2}}\over {81 \sqrt{\pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left(6 - {{Zr}\over {a_0}} \right)
{{Zr}\over {a_0}} \exp(-{{Zr}\over {3 a_0}}) \cos(\theta)
$&
$\displaystyle
 = R_{31} \cdot Y_1^0
$\\
$\displaystyle \psi_{31\pm 1} ={1 \over {81 \sqrt{\pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left(6 - {{Zr}\over {a_0}} \right)
{{Zr}\over {a_0}} \exp(-{{Zr}\over {3 a_0}}) \sin(\theta) e^{\pm i
\phi}
$&
$\displaystyle
 = R_{31} \cdot Y_1^{\pm 1}
$\\
$\displaystyle \psi_{320} = {1 \over {81 \sqrt{6 \pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left({{Zr}\over {a_0}} \right)^2
\exp(-{{Zr}\over {3 a_0}}) (3 \cos^2(\theta)-1)
$&
$\displaystyle
 = R_{32} \cdot Y_2^0
$\\
$\displaystyle \psi_{32\pm 1} = {1 \over {81 \sqrt{\pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left({{Zr}\over {a_0}} \right)^2
\exp(-{{Zr}\over {3 a_0}}) \sin(\theta) \cos(\theta)-1) e^{\pm i \phi}
$&
$\displaystyle
 = R_{32} \cdot Y_2^{\pm 1}
$\\
$\displaystyle \psi_{32\pm 2} = {1 \over {81 \sqrt{\pi}}} \left( {Z \over
{a_0}}\right)^{3 \over 2 } \left({{Zr}\over {a_0}} \right)^2
\exp(-{{Zr}\over {3 a_0}}) \sin^2(\theta) e^{\pm 2 i \phi}
$&
$\displaystyle
 = R_{32} \cdot Y_2^{\pm 2}
$\\
\hline
\end{tabular}

\chapter{Verteilungsfunktionen}

Im folgenden die in der Physik wichtigsten Verteilungfunktionen.

Maxwell-Boltzmann-Verteilung

\[
f(\vec{v}) = n \left( {m \over 2\pi k_B t}\right)^{3\over 2}
\exp{\left(-{m v^2 \over 2 k_B T}\right)}
\]

Fermi-Dirac-Verteilung

\[
f_F(\vec{v}) = {1\over 4} \left( {m \over \hbar \pi} \right)^3
{1 \over {\exp{\left({{1\over 2} v^2 - k_B T_0}\over k_B T\right)}+
1}} \]

Bose-Einstein-Verteilung

\[
f_B(\vec{v}) = {1\over 4} \left( {m \over \hbar \pi} \right)^3
{1 \over {\exp{\left({{1\over 2} v^2 - k_B T_0}\over k_B T\right)}-
1}} \]


\chapter{Elektrodynamik}

\section{SI und cgs}

\begin{tabular} {|l|l|}
\hline
SI (MKSA) & Gau�'sche Einheiten (cgs) \\
\hline\hline
$\displaystyle  \nabla \cdot \vec{E} = {\rho \over {\epsilon_0}} $ &
$\displaystyle  \nabla \cdot \vec{E} = 4 \pi \rho $\\
$\displaystyle \vec{E} = {1 \over {4 \pi \epsilon_0}} \int_V {{\rho \vec{r}} \over
{r^3}}  dV$ &
$\displaystyle \vec{E} = \int_V {{\rho \vec{r}} \over {r^3}}  dV $\\
$\displaystyle \nabla \times \vec{E} = -{{\partial \vec{B}}\over {\partial t}}$ &
$\displaystyle c \nabla \times \vec{E} = -{{\partial \vec{B}}\over {\partial t}}$ \\
$\displaystyle \vec{E} = -\nabla \phi - {{\partial \vec{A}}\over {\partial t}}$ &
$\displaystyle \vec{E} = -\nabla \phi - {1 \over c}{{\partial \vec{A}}\over
{\partial t}}$ \\
\hline
$\displaystyle \nabla \times \vec{B} = \mu_0 \left(j + \epsilon_0 {{\partial E}\over
{\partial t}} \right)$ &
$\displaystyle \nabla \times \vec{B} = 4 \pi j + {1 \over c} {{\partial E}\over
{\partial t}}$ \\
$\displaystyle \vec{B} = {\mu_0 \over {4 \pi}} \int {{\left( j + \epsilon_0
{{\partial E}\over {\partial t}}\right) \times \vec{r}}\over {r^3}}
dV$ &
$\displaystyle \vec{B} =  \int {{\left( j + {1 \over {4 \pi c}}
{{\partial E}\over {\partial t}}\right) \times \vec{r}}\over {r^3}}
dV$ \\
$\displaystyle \nabla \cdot \vec{B} = 0$ & $\displaystyle \nabla \cdot \vec{B} = 0$ \\
$\displaystyle \vec{B} = \nabla \times \vec{A}$ & $\displaystyle \vec{B} = \nabla \times \vec{A}$
\\
\hline
$\displaystyle \nabla \cdot \vec{j} + {{\partial \rho}\over {\partial t}} = 0$ &
$\displaystyle \nabla \cdot \vec{j} + {1\over c}{{\partial \rho}\over {\partial t}}
= 0$ \\
\hline
$\displaystyle \nabla \cdot \vec{A} + {1 \over c^2} {{\partial \phi}\over {\partial
t}} = 0$ &
$\displaystyle \nabla \cdot \vec{A} + {1 \over c} {{\partial \phi}\over {\partial
t}}$ \\
\hline
$\displaystyle \vec{F} = e (\vec{E}+ \vec{u}\times \vec{B})$ &
$\displaystyle \vec{F} = e (\vec{E}+{1 \over c} \vec{u}\times \vec{B})$ \\
$\displaystyle {{\partial F^{ij}}\over {\partial x^i}} = {{j^j}\over {\epsilon_0}}$
& $\displaystyle {{\partial F^{ij}}\over {\partial x^i}} = 4 \pi j^j$ \\
wobei & wobei \\
$F^{ij}=\left(
\begin{array}{cccc}
0 & -c B_z & c B_y & +E_x \\
c B_x & 0 & -c B_z &  +E_y \\
-c b_y & c B_x & 0 & + E_z \\
-E_x & -E_y & -E_z & 0\\
\end{array}
\right)$ &
$F^{ij}=\left(
\begin{array}{cccc}
0 & - B_z &  B_y & +E_x \\
 B_x & 0 & -B_z &  +E_y \\
- b_y &  B_x & 0 & + E_z \\
-E_x & -E_y & -E_z & 0\\
\end{array}
\right)$ \\
\hline
\end{tabular}

\section{Relativit�tstheorie}

Kontravarianter 4-Vektor:
\begin{equation}
A^{\alpha} = (A^0, A^1, A^2, A^3) = (A^0, \vec{A})
\end{equation}

Kovarianter 4-Vektor
\begin{equation}
A_{\alpha} = (A_0, A_1, A_2, A_3) = (A_0, \vec{A})
\end{equation}

\subsection{Transformationseigenschaften}

\paragraph{Kontravarianter 4-Vektor}
\begin{equation}
A'^{\alpha} = { \partial x'^{\alpha} \over \partial x^{\beta}}
 A^{\beta} =
{ \partial x'^{\alpha} \over \partial x^{0}} A^{0} +
{ \partial x'^{\alpha} \over \partial x^{1}} A^{1} +
{ \partial x'^{\alpha} \over \partial x^{2}} A^{2} +
{ \partial x'^{\alpha} \over \partial x^{3}} A^{3}
\end{equation}

\paragraph{Kovarianter 4-Vektor}
\begin{equation}
A'_{\alpha} = { \partial x'^{\beta} \over \partial x^{\alpha}}
A_{\beta} =
{ \partial x^{0} \over \partial x'^{\alpha}} A^{0} +
{ \partial x^{1} \over \partial x'^{\alpha}} A^{1} +
{ \partial x^{2} \over \partial x'^{\alpha}} A^{2} +
{ \partial x^{3} \over \partial x'^{\alpha}} A^{3}
\end{equation}

\paragraph{Kontravarianter Tensor}
\begin{equation}
F'^{\alpha \beta} = { \partial x'^{\alpha} \over \partial x^{\gamma}}
{ \partial x'^{\beta} \over \partial x^{\delta}} F^{\gamma \delta }
\end{equation}

\paragraph{Kovarianter Tensor}
\begin{equation}
F'_{\alpha \beta} = { \partial x^{\gamma} \over \partial x'^{\alpha}}
{ \partial x^{\delta} \over \partial x'^{\beta}} F_{\gamma \delta}
\end{equation}

\paragraph{Gemischter Tensor}
\begin{equation}
F'^{\alpha .}_{. \beta} = { \partial x'^{\alpha} \over \partial
x^{\gamma}}
{ \partial x^{\delta} \over \partial x'^{\beta}} F^{\gamma
.}_{.\delta}
\end{equation}

\subsection{Vektorraumeigenschaften}

Linienelement
\begin{equation}
ds^2 = (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 = g_{\alpha \beta}
dx^{\alpha} dx^{\beta}
\end{equation}

Mit dem {\em metrischen Tensor} $g_{\alpha \beta}$
\begin{equation}
g_{\alpha \beta} = \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{array}
\right)
\end{equation}

Es gilt:

\begin{eqnarray}
&g_{\alpha \beta} = g^{\alpha \beta} \\
&g_{\alpha \beta} g^{\alpha \beta} = \delta^{.\beta}_{\alpha .} \\
&x_{\alpha} = g_{\alpha \beta} x^{\beta} \\
&x^{\alpha} =  g_{\alpha \beta} x_{\beta} \\
& A^{\alpha} = (A^0, A^1, A^2, A^3) \Rightarrow A_{\alpha} =(A_0,
-A_1, -A_2, -A_3) \\
\end{eqnarray}

Skalarprodukt
\begin{equation}
A \cdot  B = B_{\alpha} A^{\alpha} = B^0 A^0 - \vec{A} \cdot  \vec{B}
\end{equation}

\chapter{Quantenfeldtheorie}

\section{$\gamma$-Matrizen}

\begin{eqnarray}
  \Tr(1) & = & 4 \\
  \Tr(\mbox{ungerade Zahl von $\gamma$'s}) & = & 0 \\
  \Tr(\gamma^\mu \gamma^\nu) & = & 4 g^{\mu\nu}\\
  \Tr(gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma) & = &
     4 (g^{\mu \nu} g^{\rho \sigma} - g^{\mu \rho} g^{\nu \sigma}
        + g^{\mu \sigma} g^{\nu\rho})\\
  \Tr(\gamma^5) & = & 0 \\
  \Tr(\gamma^\mu \gamma^\nu \gamma^5) & = & 0\\
  \Tr(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^5)
     & = & -4 i \varepsilon^{\mu \nu \rho \sigma}\\
  \Tr(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \cdots)
     & = & \Tr(\cdots \gamma^\sigma \gamma^\rho \gamma^\mu \gamma^\nu \cdots) \\
  \gamma^\mu \gamma_\nu & = & 4 \\
  \gamma^\mu \gamma^\nu \gamma_\mu & = & - 2 \gamma^\nu \\
  \gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu & = & 4 g^{\nu \rho} \\
  \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu & = &
    -2 \gamma^\rho \gamma^\sigma \gamma^\nu \\
\end{eqnarray}

\section{4-Dimensionaler $\varepsilon$-Tensor}

\begin{eqnarray}
  \varepsilon^{\alpha \beta \gamma \delta}
  \varepsilon_ {\alpha \beta \gamma \delta} & = & 24 \\
  \varepsilon^{\alpha \beta \mu \nu}
  \varepsilon_{\alpha \beta \mu \nu}  & = &
    -6 \delta^{\mu \cdot}_{\cdot \nu }\\
  \varepsilon^{\alpha \beta \mu \nu}
  \varepsilon_{\alpha \beta \rho \sigma}  & = &
   -2\left( \delta{^mu\cdot}_{\cdot \nu}
   \delta{^\nu\cdot}_{\cdot \sigma} - \delta^{\mu\cdot}_{\nu \cdot}
   \delta^{\mu\cdot}_{\cdot \rho} \right)
\end{eqnarray}


\tableofcontents
\end{document}
