The quantum mechanical state of a system (to be explained in more detail later), belongs to a Hilbert space. The subject of Hilbert spaces is beyond the scope of this lecture so the special case of unitary spaces will be considered instead. Unitary spaces are finite dimensional Hilbert spaces and as such are useful for their understanding. Also, most of the spaces used when considering spin 1/2 systems are finite dimensional, i.e. unitary spaces, so they will suffice for our needs.

*A note on notation:*The Dirac bra-ket notation in which a vector is denoted by \(\ket{a}\) will be used throughout these notes for consistency. As the text progresses it will become more apparent why this notation is so useful. Also, operators will consistently be denoted with a hat over their symbol, \(\hat{A}\) for example.

## Unitary Spaces

A unitary space \(U\) is a*vector space*over the field of

*complex numbers*with an

*inner product*. For completeness and as a reminder to the reader we give the full definition of a complex vector space. For every two vectors \(\ket{u}\) and \(\ket{v}\) in the space \(U\) (\(\ket{u},\ket{v}\in U\)) and every two complex numbers \(c\) and \(d\) (\(c,d\in \cn \)) the following statements have to be satisfied:

- \(\ket{u}+\ket{v}\in U\) (the vector space is closed under vector addition).
- \(c\cdot \ket{u}\in U\) (the vector space is closed under scalar multiplication.
- \(\ket{u}+\ket{v}=\ket{v}+\ket{u}\) and \(\ket{u}+(\ket{v}+\ket{w})=(\ket{u}+\ket{v})+\ket{w}\) (commutativity and associativity).
- there exist a neutral element, the null vector \(\knull\), for vector addition such that \(\knull+\ket{u}=\ket{u}\).
- there exist an additive inverse, \(-\ket{u}\) such that \(\ket{u}+(-\ket{u})=\knull\).
- \(c\cdot (\ket{u}+\ket{v})=c\cdot \ket{u}+c\cdot\ket{v}\) and \(c\cdot (d\cdot \ket{v})=(cd)\cdot \ket{v}\) (distributivity and assoicativity).
- \(1\cdot \ket{v}=\ket{v}\).

*inner product*defined as a map, \(\langle\;|\;\rangle :U\times U\rightarrow \cn\), that maps a pair of vectors into a scalar and satisfies the following conditions for all \(\ket{u},\ket{v},\ket{w} \in U\) and \(c \in \cn\):

- \(\sprod{u}{v}=\sprod{v}{u}^*\) (conjugate symmetry).
- \(\bra{u}(c\ket{v}+\ket{w})=c\sprod{u}{v}+\sprod{u}{w}\) (linearity).
- \(\sprod{u}{u}\in \rn\) and \(\sprod{u}{u}\geq 0\) and \(\sprod{u}{u}=0\) only for \(\ket{u}=\knull\) (positive definitness).

*span*of a set of vectors \(\{\ket{a_i}|i=1,...,M\}\) is the subspace of all their linear combinations \(span \{\ket{a_i}\}=\{\sum_{i=1}^M c^i\ket{a_i}|\forall c^i\in \cn \}\). The span operation constructs subspaces by making all possible linear combinations of a given set of vectors.

## Orthonormal bases of a unitary space

For each unitary space there exists a set of \(n\)*linearly independent*vectors \(\ket{e_i}\), a

*basis*, such that every vector \(\ket{u}\) in that space can be expressed as a linear combination of those vectors \begin{equation} \ket{u}=\sum_{i=1}^n u^i\ket{e_i}. \label{basis} \end{equation} i.e. they span the whole space. Equivalently a basis is a minimal set of vectors that span the space. While a basis is not uniquely defined (there is an infinite number of sets of \(n\) linearly independent vectors that span the whole space) the number of vectors in a basis is the same for all of them and is a characteristic of the space. The number of basis vectors, \(n\), defines the space

*dimension*and further on we will denote \(n\)-dimensional unitary spaces by \(U_n\). One can always add vectors to the basis set. This expanded set of vectors will still span the whole space however the expansion coefficients \(u_i\) will no longer be unique. These sets are called overcomplete. A very important class of bases are orthonormal bases for which \(\sprod{e_i}{e_j}=\delta_{ij}\), where the Kronecker delta is defined as \(\delta_{ij}=0\) for \(i\neq j\) and \(\delta_{ii}=1\). Orthonormal bases are important since they reflect the geometry of the space. Each vector in an orthonormal basis has unit length and is orthogonal to every other vector in the basis. In these lectures we will only use orthonormal bases. The coefficients in the expansion \ref{basis} can be easily calcualted for an orthonormal basis since \begin{equation} \sprod{e_j}{u}=\sum_{i=1}^n u^i\sprod{e_j}{e_i}=\sum_{i=1}^n u^i\delta_i^{\;j}=u^j. \label{represent} \end{equation} The coefficients \(u^j\) are the magnitudes of the projections of the vector onto the basis vectors. Expression \ref{represent} is true

**only for orthogonal bases**. It is very important to keep in mind that the choice of basis is not unique and the same vector will have different coordinates in different bases. The coordinates of a vector expressed in a given basis can be written as a column: \begin{equation} \left( \begin{array}{c} u^1\\ u^2\\ \vdots \\ u^n \end{array} \right). \end{equation} It can be shown that every unitary space of dimension \(n\) is isomorphic (through the above coordinate correspondence) to the space of \(n\)-tuple columns of complex numbers, \(\cn^n\), with the scalar product \(\sprod{u}{v}\) defined as \begin{equation} \sprod{u}{v}=\sum_{i=1}^n (u_i)^*v^i=(u_1^* u_2^* \ldots u_n^*) \mbox{$\left( \begin{array}{c} v^1\\ v^2\\ \vdots \\ v^n \end{array} \right)$} .\label{scalarprod} \end{equation} It is said the the columns form a representation of the abstract unitary space. The representation is basis dependent - if we change the basis vectors the coefficients in the expansion \ref{basis} of the

*same*abstract vector will change. Every orthonormal basis is, by definition, always represented by the so called

*standard basis*in \(\cn^n\) \begin{equation} \mbox{$\left( \begin{array}{c} 1\\ 0\\ \vdots \\ 0 \end{array} \right)$}, \mbox{$\left( \begin{array}{c} 0\\ 1\\ \vdots \\ 0 \end{array} \right)$}, \ldots , \mbox{$\left( \begin{array}{c} 0\\ 0\\ \vdots \\ 1 \end{array} \right)$}. \end{equation} We will often work with the representation of a given unitary space i.e. the columns of expansion coefficients \(u^i\). For unitary spaces it is often hard to keep this distinction in mind that is very important and non-trivial in Hilbert spaces.