What is the Bessel-Parseval relation and how does it work?

[Domnu]

The following link: http://electron6.phys.utk.edu/QM1/modules/m1/free_particle.htm mentions something about the Bessel-Parseval relation... could someone explain what this is exactly and how it works?

 

[Zacku]

The following link: http://electron6.phys.utk.edu/QM1/modules/m1/free_particle.htm mentions something about the Bessel-Parseval relation... could someone explain what this is exactly and how it works?

If you have an L2(R) (complex) function $$f$$ whose Fourier transform writes
$$\tilde{f}(q) = \int_{-\infty}^{+ \infty} dq f(x) e^{-2 \pi i qx }$$
then the Bessel-Parseval theorem states that
$$\int_{-\infty}^{+\infty} \left| f(x) \right|^2 dx = \int_{-\infty}^{+\infty} \left| \tilde{f}(q) \right|^2 dq$$

This theorem also works (and is simpler to understand) in the discret case i.e. considering the Fourier series of $$f$$ as a specific case of the general Pythagore theorem.

 

[Domnu]

Wowww... are you serious? The theorem must be ridiculously helpful then...