1.14: Parseval’s Theorem
- Page ID
- 50115
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is often convenient to normalize a wavepacket in k space. To do so, we can apply Parseval's theorem.
Let's consider the bracket of two functions, f(x) and g(x) with Fourier transform pairs F(k) and G(k), respectively..
\[ \langle f|g\rangle = \int^{\infty}_{-\infty} f(x)^{*}g(x)dx \nonumber \]
Now, replacing the functions by their Fourier transforms yields
\[ \int^{\infty}_{-\infty} f(x)^{*}g(x)dx=\int^{\infty}_{-\infty}[\frac{1}{2\pi}\int^{\infty}_{-\infty}F(k’)e^{-ik’x}dk’]^{*}[\frac{1}{2\pi}\int^{\infty}_{-\infty}G(k)e^{-ikx}dk]dx \nonumber \]
Rearranging the order of integration gives
\[ \int^{\infty}_{-\infty}[\frac{1}{2\pi}\int^{\infty}_{-\infty}F(k’)e^{-ik’x}dk’]^{*}[\frac{1}{2\pi}\int^{\infty}_{-\infty}G(k)e^{-ikx}dk]dx \nonumber \]
\( =\frac{1}{2\pi}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}F(k’)*G(k)\frac{1}{2\pi}e^{-i(k-k’)x}dxdk’dk \)
From Equation (1.9.6) the integration over the complex exponential yields a delta function
\[ \frac{1}{2\pi}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}F(k’)*G(k)\frac{1}{2\pi}e^{-i(k-k’)x}dxdk’dk = \frac{1}{2\pi}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}F(k’)^{*}G(k)\delta (k-k’)dk’dk \nonumber \]
Thus,
\[ \int^{\infty}_{-\infty}f(x)^{*}g(x)dx=\frac{1}{2\pi}\int^{\infty}_{-\infty}F(k)^{*}G(k)dk \nonumber \]
It follows that if a wavefunction is normalized in real space, it is also normalized in k-space, i.e.,
\[ \langle \psi|\psi \rangle = \langle A|A \rangle \nonumber \]
where
\[ \langle A|A \rangle = \frac{1}{2\pi} \int^{\infty}_{-\infty}A(k)^{*}A(k)dk \nonumber \]