From: hackbard <hackbard@hackdaworld.org>
Date: Tue, 7 Feb 2012 22:01:36 +0000 (+0100)
Subject: more math, started operators
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=1b61eac526bb34de8c8060551f70c6e7cd437d89;p=lectures%2Flatex.git

more math, started operators
---

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index 4c8e6f9..c182a96 100644
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -6,11 +6,12 @@ Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
 
 \section{Vectors and bases}
 
-A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$.
+A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$
 \begin{equation}
 \vec{a}=\sum_i^N \vec{e}_i a_i
 \label{eq:vec_sum}
 \end{equation}
+i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
 The scalar product for an $N$-dimensional real vector space is defined as
 \begin{equation}
 (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
@@ -51,3 +52,23 @@ Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vecto
 if the basis is complete.
 Indeed, the very important identity representation by the outer product ($\cdot$) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.
 
+\section{Operators, matrices and determinants}
+
+An operator $O$ acts on a vector resulting in another vector
+\begin{equation}
+O\vec{a}=\vec{b} \text{ ,}
+\end{equation}
+which is linear if
+\begin{equation}
+O(\lambda\vec{a}+\mu\vec{b})=\lambda O\vec{a} + \mu O\vec{b} \text{ .}
+\end{equation}
+Thus, for a linear operator, it is sufficient to describe the effect on the complete set of basis vectors, which enables to describe the effect of the operator on any vector.
+Since the result of an operator acting on a basis vector is a vector itself, it can be expressed by a linear combination of the basis vectors
+\begin{equation}
+O\vec{e}_i=\vec{e}_jO_{ji}
+\text{ ,}
+\end{equation}
+with $O_{ji}$ determining the components of the new vector $O\vec{e}_i$ along $\vec{e}_j$.
+
+\section{Dirac notation}
+