Examples of Fourier Transforms

Skip to main content

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf{A}} \newcommand{\BB}{\vf{B}} \newcommand{\CCv}{\vf{C}} \newcommand{\EE}{\vf{E}} \newcommand{\FF}{\vf{F}} \newcommand{\GG}{\vf{G}} \newcommand{\HHv}{\vf{H}} \newcommand{\II}{\vf{I}} \newcommand{\JJ}{\vf{J}} \newcommand{\KKv}{\vf{Kv}} \renewcommand{\SS}{\vf{S}} \renewcommand{\aa}{\VF{a}} \newcommand{\bb}{\VF{b}} \newcommand{\ee}{\VF{e}} \newcommand{\gv}{\VF{g}} \newcommand{\iv}{\vf{imath}} \newcommand{\rr}{\VF{r}} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF{u}} \newcommand{\vv}{\VF{v}} \newcommand{\ww}{\VF{w}} \newcommand{\grad}{\vf{\nabla}} \newcommand{\zero}{\vf{0}} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\LLv}{\vf{L}} \newcommand{\OOb}{\boldsymbol{O}} \newcommand{\PPv}{\vf{P}_\text{cm}} \newcommand{\RRv}{\vf{R}_\text{cm}} \newcommand{\ff}{\vf{f}} \newcommand{\pp}{\vf{p}} \newcommand{\tauv}{\vf{\tau}} \newcommand{\Lap}{\nabla^2} \newcommand{\Hop}{H_\text{op}} \newcommand{\Lop}{L_\text{op}} \newcommand{\Hhat}{\hat{H}} \newcommand{\Lhat}{\hat{L}} \newcommand{\defeq}{\overset{\rm def}{=}} \newcommand{\absm}{\vert m\vert} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 21.7 Examples of Fourier Transforms

This section asks you to find the Fourier transform of a cosine function and a Gaussian. Hints and answers are provided, but the details are left for the reader.

Activity 21.7. The Fourier Transform of the Cosine.

Find the Fourier transform of the cosine function \(f(x)=\cos kx\text{.}\)

Hint.

Hint: You must integrate a combination of an exponential and a cosine. It is ALWAYS easier to integrate exponentials, so use the exponential form of the cosine function, (2.6.3).

Answer.

\begin{align} {\cal{F}}(\cos kx)\amp =\tilde{f}(k^{\prime})\tag{21.7.1}\\ \amp = \pi\, \left(\delta(k-k^{\prime})+\delta(k+k^{\prime})\right)\tag{21.7.2} \end{align}

Activity 21.8. The Fourier Transform of a Gaussian.

Find the Fourier transform of the quantum-normalized Gaussian (21.2.12)

\begin{equation} f(x) =\frac{1}{(\sqrt{\pi}\,\sigma)^{1/2}}\, e^{-\frac{(x-x_0)^2}{2\sigma^2}}\tag{21.7.3} \end{equation}

Hint.

Complete the square in the exponential, see Section A.1, and use the formula for the integral of a Gaussian, see (21.2.1).

Answer.

\begin{align} \tilde{f}(k)\amp ={\cal{F}}\left( \frac{1}{(\sqrt{\pi}\,\sigma)^{1/2}}\, e^{-\frac{(x-x_0)^2}{2\sigma^2}}\right)\tag{21.7.4}\\ \amp = \left(\frac{\sigma}{\sqrt{\pi}}\right)^{1/2}\, e^{-\frac{k^2\sigma^2}{2}}\, e^{-ikx_0}\tag{21.7.5} \end{align}

Notice that the Fourier transform of a Gaussian is also a Gaussian, but now the factors of \(\sigma\) are in the numerators of the exponential and the normalization constant, instead of the denominators. What do the positions of the factors of \(\sigma\) tell you about how the shapes of the two Gaussian’s are related to each other? Also, notice that there is an overall phase factor in the Fourier transform of the Gaussian that depends on the position \(x_0\) of the peak of the original Gaussian.