Question

Help with derivation of the Casimir Effect?

Answer 1

I am at the very last part of a relatively long derivation of the Casimir effect, and I just don't understand the final step D:

So far, I have derived the ground state energy to be

⟨0|H^|0⟩=δ(0)∫∞−∞dp12E.⟨0|H^|0⟩=δ(0)∫−∞∞dp12E.

And for a massless field using Plank units and using E=p2+m2−−−−−−−√E=p2+m2, then E=pE=p. Between the two parallel plates, only virtual particles of discrete pp can exist which are p=hλp=hλ and using ℏ=1ℏ=1 and λ=2dnλ=2dn where dd is the distance between the parallel plates. This means that the summation becomes π2d∑n=1∞nπ2d∑n=1∞nwhich I find by assuming

∑n=1∞n=∑n=1∞ne−an=−dda∑n=0∞e−an.∑n=1∞n=∑n=1∞ne−an=−dda∑n=0∞e−an.

Which gave the sum to infinity of ea(ea−1)2ea(ea−1)2

Using the Taylor expansions, the first two terms of this result are 1a2−1121a2−112 and subsequent terms are irrelevant because I take a→0a→0 for the sum to become the sum of all natural numbers. Placing aπdaπd for aa, this gives the sum πd∑n=1∞nπd∑n=1∞n. So when I put this result back in, I get that the energy density of the vacuum (assuming that δ(0)δ(0) corresponds to the volume of space, is

d2πa2−π24d.d2πa2−π24d.

Okay, so this is the part that I don't get. To eliminate the infinity, I considered adding a third plate and moving it away to infinity, so the lengths are as shown:

L (from first to third plate) |___|____| d L-d

Now here I want to find the relative energy density between the plates, take the derivative with respect to d to find the force, and move L away to infinity to remove the infinities. My problem is that, looking back over my notes, I have added the energy densities between the first and second plate and second and third plate to give

E(d)=d2πa2−π24d+L−d2πa2−π24(L−d)E(d)=d2πa2−π24d+L−d2πa2−π24(L−d)

which works out because then using F=−E′(d)F=−E′(d) gives the force being

−(π24d2+π24(L−d))−(π24d2+π24(L−d))

which works out perfectly to give the force being π24d2π24d2 when you move the third plate to infinity away.

Answer 2

To understand the Casimir Effect, one first has to understand something about a vacuum in space as it is viewed in quantum field theory. Far from being empty, modern physics assumes that a vacuum is full of fluctuating electromagnetic waves that can never be completely eliminated, like an ocean with waves that are always present and can never be stopped. These waves come in all possible wavelengths, and their presence implies that empty space contains a certain amount of energy--an energy that we can't tap, but that is always there.

Now, if mirrors are placed facing each other in a vacuum, some of the waves will fit between them, bouncing back and forth, while others will not. As the two mirrors move closer to each other, the longer waves will no longer fit--the result being that the total amount of energy in the vacuum between the plates will be a bit less than the amount elsewhere in the vacuum. Thus, the mirrors will attract each other, just as two objects held together by a stretched spring will move together as the energy stored in the spring decreases.