How Did the (2π)^2 Term Arise in the Casimir Force Derivation?

[kenkhoo]

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>,

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?

 

[kenkhoo]

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

Thanks anyway

 

[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.

 

[kenkhoo]

Ah. Yeah I've forgot about the DOS.
Thanks for the detailed explanation!