QM-10 [Lecture Notes] Introduction to representations

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$\newcommand{\innerproduct}[2]{\langle#1\vert#2\rangle}$

 In the following sections we will introduce the coordinate and momentum representations and discuss the relationship between the two epresentations. Here we briefly recall the main results for representations in a finite dimensional vector spaces. Let $\Big\{    \ket{e_1}, \ket{e_2}, \ldots, \ket{e_N}\Big\}$ be an o.n. 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}

 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 and an $n\times n$ matrix $\underline{\sf T}$ for every operator $T$. 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}

\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}

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{m}{v} = \matrixelement{m}{T}{v} \\ &\Rightarrow&  \innerproduct{m}{v} = \matrixelement{m}{T\Big(\sum_n \ket{n}\bra{n}\Big)}{v} \\ &\Rightarrow&  \innerproduct{m}{v} =\sum_n \matrixelement{m}{T}{n} \innerproduct{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 \( \underline{\sf A} \underline{\sf B} = \underline{\sf C}\).

 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 we now come to the most commonly used representation called the coordinate representation.

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