1 \part{Mathematical foundations}
3 Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
5 \chapter{Linear algebra}
7 \section{Vectors and bases}
9 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 ${\vec{e}_i}$
11 \vec{a}=\sum_i^N \vec{e}_i a_i
14 i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
15 The scalar product in an $N$-dimensional Euclidean vector space is defined as
17 (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
20 which satisfies the properties of an inner product (see \ref{math_app:product}) and enables to define a norm
22 ||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
24 that just corresponds to the length of vector \vec{a}.
25 Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
27 (\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=
28 \sum_i\sum_j(\vec{e}_i,\vec{e}_j)a_ib_j \text { ,}
30 which is equal to \eqref{eq:vec_sp} only if
32 (\vec{e}_i,\vec{e}_j)=
33 \delta_{ij} = \left\{ \begin{array}{lll}
34 0 & {\rm for} ~i \neq j \\
35 1 & {\rm for} ~i = j \end{array} \right.
36 \text{ (Kronecker delta symbol),}
38 i.e.\ the basis vectors are mutually perpendicular (orthogonal) and have unit length (normalized).
39 Such a basis set is called orthonormal.
40 The component of a vector can be obtained by taking the scalar product with the respective basis vector.
42 (\vec{e}_j,\vec{a})=(\vec{e}_j,\sum_i \vec{e}_ia_i)=
43 \sum_i (\vec{e}_j,\vec{e}_i)a_i=
44 \sum_i\delta_{ij}a_i=a_j
46 Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vector can be written as
49 \vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow
50 \sum_i\vec{e}_i\otimes \vec{e}_i=\vec{1}
52 if the basis is complete.
53 Indeed, the very important identity representation by the outer product ($\otimes$, see \ref{math_app:product}) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.
55 \section{Operators, matrices and determinants}
57 An operator $O$ acts on a vector resulting in another vector
59 O\vec{a}=\vec{b} \text{ ,}
63 O(\lambda\vec{a}+\mu\vec{b})=\lambda O\vec{a} + \mu O\vec{b} \text{ .}
65 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.
66 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
68 O\vec{e}_i=\vec{e}_jO_{ji}
71 with $O_{ji}$ determining the components of the new vector $O\vec{e}_i$ along $\vec{e}_j$.
73 \section{Dirac notation}