The Casimir Effect

River Snively

What I’m going to talk about (in reverse) • • • • The Casimir force Quantizing the electromagnetic field A non-Field Theoretic Casimir force Review of zero-point energy in quantum mechanics

Zero-point Energy (Quick Review)

• Quantum harmonic oscillator: = ћω(n + ½)\ • Ground state energy: • Don’t like this? Remove it by shifting H.

Tunable QHO (A thought experiment) • • • Changing frequency changes energy minimum Therefore, turning the knob takes work This would’ve been unnatural if we’d shifted H

Tunable QHO: reality check

How hard is it to turn the knob?

• • • Say ∆ω with one degree turn is 10 17 s -1 Then ∆E with one degree turn is 33 eV So torque is 5·10 -18 Nm/degree • Conclusion: tough to measure ZPE with a single oscillator.

If only…

A New Harmonic Oscillator: The EM Field • • • In the Coulomb gauge, vector potential satisfies wave equation Expanding A in plane waves, Coulomb gauge says A

k

is perpendicular to k The other two components: wave equation requires they be SHOs with ω = kc • Main hypothesis of QED: Those harmonic oscillators are “quantum” For much clearer explanation of all this see Feynman and Hibbs, Quantum Mechanics and Path Integrals, ch. 9

Max Planck

Field Quantization: Consequences

• For each momentum mode k there are two oscillators, each with ω = kc • • Excitations = photons!

Take another look at E = ћω(n + ½) • Zero point energy: Twice ½ћω, summed over all k’s: (infinite!)

• • • Electromagnetic zero-point energy of vacuum: E = 2Σ

k

½ ћc|k| Crisis avoided if we just consider changes in energy Similar thing: the self-energy of the classical electron

• How could we change the vacuum energy?

One thing we could do: • • • (put it in a box) Then, allowed k modes are nπ/L Moving walls changes summed-over frequencies More realistically, could confine between parallel plates…

The Prototypical Casimir Set-up • Ideal conductors, area L 2 • Separation a (a << L) • An attractive Force (Casimir, 1948): • First measurement: Sparnaay 1958, with 100% uncertainty.

What’s so attractive about this force?

• • No α in sight (*) F/A = (.013 dyne/cm 2 )a -4 small but not unobservable (with a in microns) • (Compare: atmospheric pressure ≈ 10 6 dyne/cm 2 .) Sometimes not attractive

(excerpt)

An alternative set-up

• • Sphere-plate Casimir Effect: Mohideen and Roy, 1998 Verified Casimir at .1 to .9 micron separations to 1%

Conclusion: we’ve seen that…

• • • The Casimir effect can be explained by zero point energy The effect is large enough to observe experimentally (nowadays) The Casimir effect is not inherently “quantum field theoretical,” just inherently “quantum.”

Thank you…

References

• • • • • • • • • Hendrik Casimir, On the attraction between two perfectly conducting plates. Proc. Akad. Wet. Amsterdam (1948).

R. L. Jaffe, The Casimir Effect and the Quantum Vacuum, Phys. Rev. D 72, 021301 (2005).

M. J. Sparnaay, Measurements of attractive forces between flat plates, Physica 24, (1958).

S.K. Lamoreaux, Demonstration of the Casimir Force in the .6 to 6 µm Range, Phys. Rev. Lett. 78, 5 (1997).

U. Mohideen & A. Roy, precision Measurement of the Casimir Force from.1 to .9 µm, Phys. Rev. Lett. 81, 21 (1998).

R.P. Feynman & A.R. Hibbs, Quantum Mechanics and Path Integrals A. Zee, Quantum Field Theory in a Nutshell F. S. Levin & D. A. Micha (editors), Long-Range Casimir Forces V.M. Mostepanenko & N.N. Trunov, The Casimir Effect and its Applications