$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
The momentum representation is defined and its connection with the coordinate representations is discussed. The transformation bewteen the two is effected by \(\innerproduct{x}{p}\) which are just the momentum eigenfunctions in the coordinate representation. Delta function normalization and the box normalization is discussed for the momentum eigenfunctions.
A brief account of representations in a finite dimensional vector spaces is presented. The use of an ortho norrnal basis along with Dirac notation makes all frequently used formula very intuitive. The formulas for representing a vector by a column vector and an operator by matrices are given. The results for change of o.n. bases are summarized.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
The choice of orthonormal basis of eigenvectors of position operator gives rise to the coordinate representation. The wave function, being the expansion coefficient of state vector in this basis, gives the probability amplitude for outcomes of position measurements.In the coordinate representation the momentum operator assumes a simple form $\widehat{p} =-i\hbar \dd{x}$.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3] \langle#1|#2|#3\rangle} \newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$A tabular comparison of coordinate and momentum representations is presented.
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