Photons - Wavelength and Wavepackets

[Usaf Moji]

Hi, forgive me if this sound noobish...

For photons, I've been struggling with the interrelationship between Maxwell waves and Schrodinger waves, and further, as to the relationship between the wavelength of light and the "wavelength" of the Gaussian envelopes of wave packets.

(I guess this is only for those who accept that photons can be described using Schrodinger's equation(s) and the idea of wave packets - yeah, I saw that wikipedia note that says otherwise.)

In textbooks, a light wave is often depicted as having little wiggles within a broader Gaussian envelope - like the way they look in this guy's notes:

http://www.phys.unsw.edu.au/~sjc/physics1/summer/q25.jpg

I understand that when we speak of the "wavelength" of light, we're referring to the wavelength of the smaller sinusoidal parts within the Gaussian envelope. And I understand that the Gaussian envelope results from summing several of these smaller sinusoidal waves of varying wavelengths (varying due to the uncertainty principle).

So my question is, is the wavelength of the larger Gaussian envelope proportional to the (mean) wavelength of the sinusoids within it? In other words, if all else were equal, would red light have longer envelopes than say blue light?

All responses appreciated.

 

[Meir Achuz]

So my question is, is the wavelength of the larger Gaussian envelope proportional to the (mean) wavelength of the sinusoids within it? In other words, if all else were equal, would red light have longer envelopes than say blue light?

The length of the envelope is unrelated to the wave length of the light, unless you go to the extreme where the length of the envelope gets close to the wave length.

 

[denian]

I can understand your confusion and appreciate your curiosity about the relationship between Maxwell waves and Schrodinger waves, and how they relate to the wavelength of light and wave packets. Let me try to clarify these concepts for you.

Firstly, Maxwell's equations describe the behavior of electromagnetic waves, including light. These waves are continuous and can be described by a sinusoidal function, with a specific wavelength and frequency. On the other hand, Schrodinger's equation is a quantum mechanical equation that describes the behavior of subatomic particles, including photons. This equation describes the probability of finding a particle in a certain location, rather than its exact position.

Now, when we talk about the "wavelength" of light, we are referring to the distance between two consecutive peaks or troughs of the sinusoidal wave. This is a property of the electromagnetic wave itself, not the particle it is associated with. On the other hand, the "wavelength" of a wave packet refers to the distance between the peaks or troughs of the Gaussian envelope, which is a representation of the particle's probability distribution. This envelope is formed by the superposition of multiple sinusoidal waves with different wavelengths, due to the uncertainty principle.

So, to answer your question, the wavelength of the larger Gaussian envelope is not directly proportional to the mean wavelength of the sinusoids within it. It is a result of the superposition of these waves and can vary depending on the specific combination of sinusoids.

In terms of the color of light, the wavelength of the electromagnetic wave determines its color, with shorter wavelengths corresponding to blue light and longer wavelengths corresponding to red light. However, the wavelength of the wave packet does not determine the color of the light, as it is a property of the particle rather than the wave.

I hope this helps to clarify the relationship between photons, wavelength, and wave packets. Keep exploring and asking questions, as it is through curiosity and inquiry that we advance our understanding of the world around us.