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\)
Section 21.4 Parceval’s Theorem for Fourier Transforms
Parceval’s theorem states that integral of the squared norm of a function is equal to the integral of the squared norm of its Fourier transform.
\begin{gather}
\int_{-\infty}^{\infty}
\left(f(x)\right)^*
\left(f(x)\right)\, dx
=
\int_{-\infty}^{\infty}
\left(\tilde f(k)\right)^*
\left(\tilde f(k)\right)\, dk\tag{21.4.1}
\end{gather}
This pure mathematics result is important in quantum mechanics, see
Section 21.9 , where it shows that if a quantum state is normalized in the position representation, it is also normalized in the momentum representation.
Proof (Optional).
The proof is straightforward, requiring only the definition of the inverse Fourier transform
(21.3.2) and its complex conjugate in line
(21.4.3) , a change in the order of integration between lines
(21.4.4) and
(21.4.5) , and the exponential definition of the delta function
(17.11.2) in line
(21.4.6) .
\begin{align}
\int_{-\infty}^{\infty} \amp
\left(f(x)\right)^*
\left(f(x)\right)\, dx\tag{21.4.2}\\
\amp =
\int_{-\infty}^{\infty}
\left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
\tilde{f} (k)\, e^{ikx}\, dk\right)^*\tag{21.4.3}\\
\amp\qquad\qquad
\left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
\tilde{f} (k')\, e^{ik'x}\, dk'\right)\, dx\tag{21.4.4}\\
\amp =
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
\left(\left(\tilde f(k)\right)^*
\left(\tilde f(k')\right)\right.\tag{21.4.5}\\
\amp\qquad\qquad\left.
\cancelto{\delta(k-k')}{
\left(\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-ikx}\, e^{ik'x}\,
dx\right)}
\right) dk\, dk'\tag{21.4.6}\\
\amp =
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
\left(\tilde f(k)\right)^*
\left(\tilde f(k')\right)\delta(k-k')\, dk\, dk'\tag{21.4.7}\\
\amp =
\int_{-\infty}^{\infty}
\left(\tilde f(k)\right)^*
\left(\tilde f(k)\right)\, dk\tag{21.4.8}
\end{align}
