Lisa Larrimore
Physics 115 - Final Presentation
Vacuum Fluctuations and the Casimir Force
I mentioned my results to Niels Bohr, during a walk. That is nice, he said, that is something
new. I told him that I was puzzled by the extremely simple form of the expressions for the
interaction at very large distances and he mumbled something about zero-point energy.
That was all, but it put me on a new track.
—H. B. G. Casimir [1]
The Casimir Force was predicted in 1948 by Dutch physicist Hendrick Casimir. Casimir realized
that when calculating the energy between two parallel uncharged conducting plates, only those virtual
photons whose wavelengths fit an integral number of times into the gap should be counted. Each mode
contributes to a pressure on the plates, and the infinite number of modes outside the plates is in some
sense greater than the infinite number inside the plates, resulting in a small force drawing the plates
together. This experimentally-confirmed force is one observable consequence of the existence of the
vacuum electromagnetic field.
Theoretical Background
First, let us review the tools we have developed through Loudon to analyze the quantum vacuum. In
§4.4, when he quantized the electromagnetic field, Loudon expressed the radiation Hamiltonian as
ĤR =
∑
k
∑
λ
~ωk
(
â†kλâkλ +
1
2
)
.
(1)
Then in §6.2, he expressed this in continuous variables,
ĤR =
∫ ∞
0
~ωâ†(ω)â(ω)dω + vacuum energy,
(2)
where he ignored the infinite contribution due to the vacuum energy. Up till now, we have been able
to ignore this infinite ground state contribution because we have been interested in measuring the
intensity of a light beam, which means that we were detecting changes above this level [2]. Now we
will consider the vacuum state of the electromagnetic field, in which there are no photons excited in
any mode.
In §4.4, Loudon defined the vacuum state, | {0}〉 , as the the state with no photons in any mode
(nkλ = 0 for all k and λ), which means that the destruction operator gives âkλ | {0}〉 = 0 for all k
and λ. Using