# Modern Physics: Chapter 5 – Fourier Transform I

XKCD – Fourier

To create a truly localized wavepacket we need to superpose not just two but an infinite amount of sinusoidal waves whose wavelengths and amplitudes vary in a continuous spectrum. To do that, we need to learn about what’s called the Fourier integral:

• $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} a(k) e^{ikx} dk$

where

• $a(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$

Why an exponential function instead of sine and cosine? This is because we just used Euler’s formula:

• $e^{ix} = cos(x) + isin(x)$

and it’s derivations:

• $cos(x) = \frac{e^{ix} + e^{-ix}}{2}$
• $sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$

to replace the sines and cosines with the exponential function.

Essentially, this gives us a framework for expressing practically any function as a superposition of harmonic waves (just plug in the function $f(x)$ in the expression for $a(k)$ and then plug in $a(k)$ into the expression for the Fourier integral). This is known as the Fourier transform.