15.1: Casimir effect - forces from nothing

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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)