Simple derivation of Casimir Force

In summary, the conversation discussed the derivation of the Casimir Force on each plate in a two parallel plate system and how (2∏)^2 appeared in the integral for <E>. It was concluded that it is a multiplication of the density of states, which gives the total number of states within an infinitesimal volume of k space. This factor is important in understanding the behavior of the Casimir effect.
  • #1
kenkhoo
8
0

Homework Statement


Derive the Casimir Force on each plate, for a two parallel plate system (L x L), separated at a distance of 'a' apart.

The solution was found in en.wikipedia.org/wiki/Casimir_effect#Derivation_of_Casimir_effect_assuming_zeta-regularization. (sorry I couldn't include link yet). Now my question is how did the (2∏)^2 came out in the integral for <E>,

ffb0365edd056bc1efcbf0a26f82c31f.png


I would think it as the constant from Fourier transform but I was unable to prove that. Any idea how did that thing pop up of nowhere?
 
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  • #2
oh this question is moot. It's basically multiplication of the DOS.

Thanks anyway
 
  • #3
Assume a large hypercubic box in d dimensions of length L. Impose periodic boundary conditions (PBCs) on any function:
[tex]
\psi(x_1 + L, x_2, \ldots, x_d) = \psi(x_1, x_2 + L, \ldots, x_d) = \ldots = \psi(x_1, x_2, \ldots, x_d + L)
[/tex]
Then, we can expand the function in multidimensional Fourier series:
[tex]
\psi(\mathbf{x}) = \sum_{\mathbf{k}}{c_{\mathbf{k}} \, e^{i \mathbf{k} \cdot \mathbf{x}}}
[/tex]
where
[tex]
\mathbf{k} = \frac{2\pi}{L} \langle n_1, n_2, \ldots, n_d \rangle
[/tex]
is a multidimensional wave vector that can take on discrete values.

In an interval [itex](k_i, k_i + dk_i)[/itex] of the ith component, there are
[tex]
dn_i = \frac{L}{2\pi} \, dk_i
[/tex]
To find the total number of states within an infinitesimal volume of k space
[tex]
dn = \mathrm{\Pi}_{i = 1}^{d}{dn_{i}} = \frac{L^{d}}{(2\pi)^{d}) \, d^{d}k
[/tex]
So, the famous factor [itex]L^{d}/(2\pi)^{d}[/itex] gives the density of states in k space.
 
  • #4
Ah. Yeah I've forgot about the DOS.
Thanks for the detailed explanation!
 
  • #5


The (2π)^2 term in the integral for <E> comes from the fact that we are considering a two-dimensional system, with the plates being parallel to each other. This means that the allowed modes of the electromagnetic field have a two-dimensional momentum space, with each dimension ranging from 0 to π/L. When we integrate over all possible modes, we get (2π)^2 as the factor for each dimension, giving us a total factor of (2π)^2 in the integral. This is a result of the Fourier transform in two dimensions, which relates the spatial coordinates to the momentum coordinates.
 

1. What is the Casimir force?

The Casimir force is a physical phenomenon that arises from the quantum fluctuations of the electromagnetic field between two closely spaced parallel plates. It causes the plates to attract each other due to the difference in energy of the fluctuations inside and outside of the plates.

2. How is the Casimir force derived?

The Casimir force can be derived using quantum field theory, specifically the concept of zero-point energy. This involves calculating the energy of the electromagnetic field between the plates and subtracting the energy outside of the plates. The resulting difference leads to a net attractive force between the plates.

3. Can the Casimir force be observed in everyday life?

The Casimir force is a very small force and can only be observed in extremely small distances, typically on the scale of nanometers. However, it has been observed in experiments using specialized equipment such as atomic force microscopes.

4. What are some real-world applications of the Casimir force?

The Casimir force has potential applications in nanotechnology and microelectromechanical systems (MEMS). It can also be used to create stable levitation and to control the movement of small objects without physical contact.

5. Are there any limitations or challenges in studying the Casimir force?

One limitation of studying the Casimir force is that it is very difficult to isolate and measure due to the presence of other forces and environmental factors. Additionally, the calculations involved in deriving the Casimir force can be complex and require advanced mathematical techniques.

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