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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:
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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 1017 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.
A New Harmonic Oscillator: The EM Field • In the Coulomb gauge, vector potential satisfies wave equation • Expanding A in plane waves, Coulomb gauge says Ak 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
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!)
INFINITE VACUUM ENERGY • Electromagnetic zero-point energy of vacuum: E = 2Σk 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 L2 • 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/cm2)a-4 (with a in microns) small but not unobservable (Compare: atmospheric pressure ≈ 106 dyne/cm2.) • Sometimes not attractive
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.”
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
