Homework Help: Simple derivation of Casimir Force

1. May 23, 2012

kenkhoo

1. The problem statement, all variables and given/known data
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>,

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?

2. May 23, 2012

kenkhoo

oh this question is moot. It's basically multiplication of the DOS.

Thanks anyway

3. May 23, 2012

Dickfore

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

In an interval $(k_i, k_i + dk_i)$ of the ith component, there are
$$dn_i = \frac{L}{2\pi} \, dk_i$$
To find the total number of states within an infinitesimal volume of k space
$$dn = \mathrm{\Pi}_{i = 1}^{d}{dn_{i}} = \frac{L^{d}}{(2\pi)^{d}) \, d^{d}k$$
So, the famous factor $L^{d}/(2\pi)^{d}$ gives the density of states in k space.

4. May 24, 2012

kenkhoo

Ah. Yeah Ive forgot about the DOS.
Thanks for the detailed explaination!