Photons and Fourier Series
Date: Summer 2013
I read that wave packets can be written as sums of monochromatic waves. Similarly, I know that Fourier series is a means of writing any periodic function as an infinite sum of sine and cosine waves. The definition of "monochromatic" that I got on the Internet said that a monochromatic wave was a wave with a single frequency. Sine and cosine waves have single frequencies. Does this mean that the phases of wave packets such as photons can be written as a mere Fourier series?
Fourier decomposition is a common means to solve problems involving waves. In classical physics, i.e., simply using Maxwell?s equations for describing light propagation, the field is called ?Fourier Optics.? In fact, you can explain such phenomena as diffraction in light by summing up the propagating light as a series of Fourier waves. In Quantum Mechanics, a similar process is used to describe the wave nature of light and matter. It turns out to be quite a useful method for looking at (and solving) certain problems in physics and engineering (e.g., the wave nature of light, the vibrations of the face of a drum, the behavior of electrical circuits to name a few).
Kyle J. Bunch, PhD, PE
Your insight exceeds your age (That is a complement). Without an explanation which would be too long for this format, try reading ?The Quantum Universe? by Brian Cox and Jeff Forshaw?. The book is not easy reading but I think it is within your understanding. The only comment I would make is do not just say ?mere? Fourier series. Fourier analysis is a powerful and flexible tool.
Thanks for the question. Wave packets in quantum mechanics are usually written not just as discrete sum on integer frequencies, such as sin(x) + sin(2x) + sin(3x) +...., but as a continuous sum such as sin(1x) + sin(1.0000001x) + sin(1.0000002x) + sin(1.0000003x) + .... Because we take all frequencies into account (and not just integer frequencies), we use an integral to represent all frequencies. This integral representation is called a Fourier transform.
The phase of a wavepacket is the relative phase between two waves in a wavepacket. It is a difficult concept to picture, and is best reserved for advanced courses on quantum scattering, sources, and sinks.
I hope this helps.
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