1.14: Parseval’s Theorem
- Page ID
- 50115
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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
\[ \langle A|A \rangle = \frac{1}{2\pi} \int^{\infty}_{-\infty}A(k)^{*}A(k)dk \nonumber \]