15.1: Casimir effect - forces from nothing

For many quantum systems, such as the harmonic oscillator, there is still some energy associated with the lowest quantum state. This “zero-point” energy is real, and can be measured in the ‘Casimir effect’. There is a force between two metallic plates in a vacuum, because moving them would change the wavelength/energy of the zero-point quantised electromagnetic waves between them: this change in energy in response to a move equates to a force.

The wavefunction for transverse standing electromagnetic waves between plates of area A separated by a in the $z$-direction is:

$$\Phi_n = \text{ exp}[i({\bf k.r} − \omega_n t)] \sin (k_nz) \nonumber$$

where ${\bf k}$ lies in the $xy$ plane and $k_n = n\pi /a$. The energy is $E_n = \hbar \omega_n = hc/\lambda = \hbar c \sqrt{{\bf k}^2 + k^2_n}$

and the force per unit area is $F = − \frac{dE}{da} = \frac{d}{da} \left( \hbar \int \sum^{\infty}_{n=1} \omega_n \right) dk_xdk_y/(2\pi )^2 = − \frac{\hbar c\pi^2}{240 a^4}$

Solving this involves a trick of multiplying each term by $|\omega_n|^{−s}$, then taking the limit of $s = 0$. This tiny attractive force has now been measured (Bressi, Phys.Rev Letters, 2002)