[NOTES/QM-10001] Representations in an Inner Product Space

 We present a general discussion of representations in vector space.We will assume a finite dimensional vector spaces. We will give the final results about representations without a detailed discussion.

Choose an orthornormal basis

We work with an orthonormal (o.n.) basis. Here it is most convenient to use Dirac bra-ket notation. Use of bra-ket notation facilitates remembering different results and formula.Let $\big\{    \ket{e_1}, \ket{e_2}, \ldots,\ket{e_N}\big\}$ be an ortho normal  basis in a finite dimensional vector space. The orthogonality property is\begin{equation}     \innerproduct{e_m}{e_n}= \delta_{mn}.  \label{EQ01}\end{equation}The completeness property of the basis vectors states that the sum of all  projection operators $\ket{e_n}\bra{e_n}$ is the identity operator.\begin{equation}  \sum_{n=1}^N \ket{e_n}\bra{e_n}= \hat{I}. \label{EQ02}.\end{equation}

Representing a ket vector

A representation, with respect to a chosen basis, is constructed by forming  a $n$- component column vector $\underline{\sf f}$ for every vector $\ket{f}$ in the vector space  The rules for writing these representatives are
\begin{equation}  \ket{f} \to {\underline{\sf f}}  = \left(  \begin{array}{c}\innerproduct{e_1} {f} \\[2mm]                  \innerproduct{e_2}{f} \\[2mm]                  ..\\[2mm]                \innerproduct{e_n}{f}  \end{array}  \right)\label{EQ03}\end{equation}

Representing operators as matrices

An operator \(\hat T\) is represented by  $n\times n$ matrix which will be denoted as $\underline{\sf T}$. The matrix representation of an operator w.r.t. an o.n. basis  is easy to remember. The \(m\,n\) matrix element of the matrix representing an operator \(\hat T\) is just \(\matrixelement{m}{\hat T}{n}\). So we write\begin{equation}  \hat{T} \to {\underline{\sf T}} =\left(  \begin{array}{cccc}  \matrixelement{e_1}{T}{e_1} & \matrixelement{e_1}{T}{e_2}& ... &               \matrixelement{e_1}{T}{e_n} \\[2mm]   \matrixelement{e_2}{T}{e_1} & \matrixelement{e_2}{T}{e_2}& ... &               \matrixelement{e_2}{T}{e_n} \\[2mm]    ... & ... & ... &  ...\\[2mm]    \matrixelement{e_n}{T}{e_1} & \matrixelement{e_n}{T}{e_2}& ... &               \matrixelement{e_n}{T}{e_n} \\[2mm]  \end{array}   \right)\label{EQ04}\end{equation}

Action of an operator on vector

Having introduced a representation, all equations involving vectors and operators can be written as matrix equations. This is most conveniently achieved by making use of completeness relation \eqref{EQ02}, for example, the relation $ \ket{v}= T \ket{u}$ gets translated into a matrix relation as follows.

 \begin{eqnarray}   \ket{v}= T \ket{u}   &\Rightarrow&  \innerproduct{e_m}{v} = \matrixelement{m}{T}{v} \\   &\Rightarrow&  \innerproduct{e_m}{v} =   \matrixelement{e_m}{T\Big(\sum_n \ket{e_n}\bra{e_n}\Big)}{v} \\   &\Rightarrow&  \innerproduct{e_m}{v} =\sum_n    \matrixelement{e_m}{T}{e_n} \innerproduct{e_n}{v}\end{eqnarray}

 The last equation is just the matrix equation $\underline{\sf v} =\underline{\sf T} \underline{\sf u}$ written in terms of representatives of the abstract vectors $\ket{u}, \ket{v}$. Similarly an operator equation  \(AB=C\) becomes a matrix equation \( \underline{\sf A} \underline{\sf B} =\underline{\sf C}\).

Change of basis

Let \(\{\ket{u_n}, n=1,\ldots,n\}\) and \(\{\ket{v_n}, n=1,\ldots,n\}\) be two sets of o.n. bases. The components of a vector \(\ket{f}\) w.r.t. to the two bases will be \(\innerproduct{f}{u_n}\) and \(\innerproduct{f}{v_n}\).

 The formulae for change of basis can be easily written down using the completeness relation. Thus we have the relation\begin{eqnarray} \innerproduct{v_j}{f}  &=& \bra{v_j}\Big(\sum_k \ket{u_k}\bra{u_k}\Big)\ket{f}\\ &=& \sum_k \innerproduct{v_j}{u_k} \innerproduct{u_k}{f}\label{EQ09}\end{eqnarray}between the components of a vector \(\ket{f}\) w.r.t. the chosen bases. In a matrix notation, let \({\underline f}\) and  \(\underline{\underline{f}}\) denote the column vectors representing the vector \(\ket{f}\) w.r.t. the two bases. Then the above relation, \eqref{EQ09}, can be written as\begin{equation}\underline{\underline{\sf f}} = \underline {\sf X}\,  \underline{\sf f},  \end{equation}where \(\sf \underline{X} \) is a matrix with elements \({\sf X}_{jk}=\innerproduct {v_j}{u_k}.\) We use the notation \(\underline{\sf T}\) and \(\underline{\underline{T}}\) to denote the matrices representing the operator \(\hat T\) w.r.t. the two bases.  The relation between the matrix representations of an operator \(\hat T\) w.r.t. the two bases takes the form \begin{equation}  \underline{\underline{\sf T}} = \underline{\sf X}\,\underline{\sf T}\, \underline{\sf X}^\dagger\end{equation} Using the o.n. property of the two bases, it may be checked that the  matrix \(\sf X\), giving the transformation from one basis to another, is a unitary matrix.

Possible choices for o.n. bases

Our discussion raises an issue of existence and choice of orthonormal basis for purpose of constructing representation. Very often most computations and derivation of results can be carried out without choosing a basis in a manner. However for many practical applications a choice provided by eigenvectors of hermitian operators which form {\tt complete a set of commuting operators.}. For a mechanical system of point particles one such system is the set of coordinates of all the particles. Another example of set of a complete commuting operators is the set of all components of momentum operators of the particles in the system.

A note on use of representations in QM

In quantum mechanics, most of the time, one has to work with infinite dimensional vector space. The number of components of vectors become infinite and so do the number of rows and columns of a matrix representing an operator. Two of the most useful representations,  are the coordinate and the momentum representations. In these representations the discrete indices get replaced by a continuous index, such as $x$, or $p$. These indices can take all real values instead of positive integral values, as is the case for a finite dimensional vector space. In these situations we cannot display corresponding column vectors or matrices, but all rules of matrix multiplication will apply; the summation over an index $n$ is replaced by an integration over a continuous index such as $x$ or $p$. Keeping this in mind one can discuss the most commonly used representation called the coordinate representation.A mathematical discussion of infinite dimensional spaces, the Hilbert space, requires a long and hard mathematical preparation and is beyond the  scope of present discussion.