Quicky on the Casimir Effect

[spock0149]

Hey folk,

I'm not sure if this would be more approproate in the Calculus section because I think the anser probably involves Fourier transforms.


My understanding of the Casimir effect is that only standing waves can form between the two plates, and thus the energy looks something like:

$$\sum_{n=0}^\infty 2\pi \hbar \frac{n}{L}$$

My question is the following:

All other possible frequencies CANCEL via destructive interference. Can anyone explain to me how to show this mathematically??

I've always 'known' this, but want to see it in the math.

Any help appreciated!

Spock

 

[dlgoff]

"All other possible frequencies CANCEL via destructive interference."

I don't think this correct. The other waves\frequencies exit outside of the the volume between the plates. These are the ones causing the force.

Someone correct me if this is not a way of thinking of it.

 

[denian]

,

The Casimir effect is a fascinating phenomenon that occurs when two uncharged parallel plates are placed in a vacuum. It is caused by the presence of virtual particles in the vacuum, which create a pressure between the plates that pushes them together. This effect has been experimentally observed and has important implications in various fields of physics, including quantum field theory and cosmology.

In regards to your question about the frequencies canceling out via destructive interference, this can indeed be shown mathematically. The key lies in understanding the boundary conditions imposed by the two plates. These boundary conditions restrict the possible frequencies of the standing waves that can exist between the plates. As you correctly mentioned, only standing waves can form between the plates, and these standing waves must satisfy the boundary conditions.

Using Fourier transforms, we can express the energy of the system as a summation of all possible frequencies that satisfy the boundary conditions. However, since the boundary conditions only allow for certain frequencies, this summation will only include those frequencies and all others will cancel out via destructive interference. This can be seen by considering the phase differences between the waves at the boundaries of the plates, which must be equal for the waves to constructively interfere.

I hope this helps to clarify the mathematical understanding of the Casimir effect. It is a complex phenomenon that requires a deep understanding of quantum mechanics and field theory, but with the right tools and equations, we can gain a better understanding of its underlying principles. Keep exploring and asking questions, Spock. That is the true spirit of science.