Normalization of eigenfunction to Dirac-delta function

Question

In the first chapter of Principles of Quantum Mechanics by R. Shankar, he describes finding the eigenvalues and eigenfunctions of the operator $K=-iD=-i\frac{d}{dx}$. For context, he does this:

What I don't understand is how he arrived at $A=1/\sqrt{2\pi}$. It seems to be because (since this is an infinite-dimensional space) we want to normalize to the Dirac delta function, but I don't understand why $$\frac1{2\pi}\int_{-\infty}^\infty e^{-i(k-k')x}dx=\delta(k-k').\tag{*}$$ He doesn't really explain this. How does he normalize it?

Answer 1

For a function $f(x)$ the Fourier transform is defined as:
\begin{equation} \overset{\boldsymbol{\sim}}{f}\left(k\right)=\dfrac{1}{\sqrt{2\pi}}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!f\left(x\right)e^{ikx}\mathrm dx \tag{01}\label{01} \end{equation} This transformation is invertible, that is: \begin{equation} f\left(x\right)=\dfrac{1}{\sqrt{2\pi}}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!\overset{\boldsymbol{\sim}}{f}\left(k\right)e^{\boldsymbol{-}ixk}\mathrm dk \tag{02}\label{02} \end{equation}

With $f\left(x\right)=\delta\left(x\right)$, equation \eqref{01} yields: \begin{equation} \overset{\boldsymbol{\sim}}{\delta}\left(k\right)=\dfrac{1}{\sqrt{2\pi}}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!\delta\left(x\right)e^{ikx}\mathrm dx =\dfrac{1}{\sqrt{2\pi}} \tag{03}\label{03} \end{equation} That is, the Fourier transform of the $\delta$-function is the constant $1/\sqrt{2\pi}$. Equation \eqref{02} gives: \begin{equation} \delta\left(x\right)=\dfrac{1}{\sqrt{2\pi}}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!\dfrac{1}{\sqrt{2\pi}}e^{\boldsymbol{-}ixk}\mathrm dk =\dfrac{1}{2\pi}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!e^{\boldsymbol{-}ixk}\mathrm dk \tag{04}\label{04} \end{equation}

This is sometimes called the integral definition of the $\delta$-function.

Exchanging the roles of $k$ and $x$ in equation \eqref{04}, the definition becomes: \begin{equation} \delta\left(k\right)=\dfrac{1}{2\pi}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!e^{\boldsymbol{-}ikx}\mathrm dx \tag{05}\label{05} \end{equation}

Replacing $k$ in equation \eqref{05} with $k-k'$ we arrive at: \begin{equation} \delta\left(k-k'\right)=\dfrac{1}{2\pi}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty}\!\!\!e^{\boldsymbol{-}i(k-k')x}\mathrm dx \tag{06}\label{06} \end{equation} which is the equation that Shankar used.