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# Zero-point energy

In a quantum mechanical system such as the particle in a box or the quantum harmonic oscillator, the lowest possible energy is called the zero-point energy. According to classical physics, the kinetic energy of a particle in a box or the kinetic energy of the harmonic oscillator may be zero if the velocity is zero. Quantum mechanics with its uncertainty principle implies that if the velocity is measured with certainty to be exactly zero, the uncertainty of the position must be infinite. This either violates the condition that the particle remain in the box, or it brings a new potential energy in the case of the harmonic oscillator. To avoid this paradox, quantum mechanics dictates that the minimal velocity is never equal to zero, and hence the minimal energy is never equal to zero.

 Contents

## A few formulae

A particle in a box is defined by the potential energy

V(x) = 0 for $-l/2 \leq x \leq l/2$

which is defined to be infinite for $\left|x\right| > l/2$. The wave function with the minimal energy eigenvalue is then

ψ(x) = Csin(πx / l)

where C is an important normalization constant. The (zero-point) energy of this wave function is pure kinetic and equal to

$E_0 = \frac{\hbar^2\pi^2}{2ml^2}$

which is non-zero. Similarly, the zero-point energy of the quantum harmonic oscillator with the frequency ω is equal to

$E_0 = \frac{1}{2} \hbar\omega.$

Both of these simplest cases have a useful generalization to the case of quantum field theory. Quantum field theory - such as Quantum Electrodynamics - may be regarded as a collection of infinitely many harmonic oscillators, and quantum mechanics therefore predicts a nonzero vacuum energy. Although the absolute value of the vacuum energy is partly a matter of convention, the difference between the vacuum energy of various configurations has a physical meaning.

## Existence

Does electromagnetic zero-point energy exist, and if so, are there any practical applications and does it have any connection with dark energy? The theoretical basis for electromagnetic zero-point energy is clear. According to Sciama (1991):

"Even in its ground state, a quantum system possesses fluctuations and an associated zero-point energy, since otherwise the uncertainty principle would be violated. In particular the vacuum state of a quantum field has these properties. For example, the electric and magnetic fields in the electromagnetic vacuum are fluctuating quantities."

The Casimir effect is an example of a one-loop effect in quantum electrodynamics that can be simply explained by the zero-point energy.

## History

The concept of zero-point energy originated with Max Planck in 1911. The average energy of a harmonic oscillator in this hypothesis is (where h is Planck's constant and ν is frequency):

$E= \frac{h\nu}{2} + \frac{h\nu}{e^{h\nu/kT}-1}$

At the same time Einstein and Hopf (1910) and Einstein and Stern (1913) were also studying the properties of zero-point energy. Shortly thereafter Nernst (1916) proposed that empty space was filled with zero-point electromagnetic radiation. Then in 1925 the existence of zero-point energy was shown to be “required by quantum mechanics, as a direct consequence of Heisenberg's uncertainty principle” (Sciama 1991). As any textbook on quantum optics will show (e.g. Loudon 1983), the way to quantize the electromagnetic field is to associate each mode of the field with a harmonic oscillator with the result that the minimum energy per mode of the electromagnetic quantum vacuum is hν / 2 .

## Problems and answers

Zero-point energy shares a problem with the Dirac sea: both are potentially infinite. In the case of zero-point energy, there are reasons for believing that a cutoff does exist in the zero-point spectrum corresponding to the Planck scale. Even this results in an enormous amount of zero-point energy whose existence is assumed to be negated (in spite of the unmistakable mandate of the Heisenberg uncertainty principle) by the claim that the mass equivalent of the energy should gravitate, resulting in an absurdly large cosmological constant, contrary to observations. Matters are not quite so straightforward.

In response to the question “Do Zero-Point Fluctuations Produce a Gravitational Field?” Sciama (1991) writes:

"We now wish to comment on the unsolved problem of the relation between zero-point fluctuations and gravitation. If we ascribe an energy hν / 2 to each mode of the vacuum radiation field, then the total energy of the vacuum is infinite. It would clearly be inconsistent with the original assumption of a background Minkowski space-time to suppose that this energy produces gravitation in a manner controlled by Einstein’s field equations of general relativity. It is also clear that the space-time of the real world approximates closely to the Minkowski state, at least on macroscopic scales. It thus appears that we must regularize the zero-point energy of the vacuum by subtracting it out according to some systematic prescription. At the same time, we would expect zero-point energy differences to gravitate. For example, the (negative) Casimir energy between two plane-parallel perfect conductors would be expected to gravitate; otherwise, the relativistic relation between a measured energy and gravitation would be lost."

It is precisely localizable differences in the zero-point energy that may prove to be of some practical use and that may be the basis of dark energy phenomena. Moreover it has also been found that asymmetries in the zero-point field that appear upon acceleration may be associated with certain properties of inertia, gravitation and the principle of equivalence (Haisch, Rueda and Puthoff 1994; Rueda and Haisch 1998; Rueda, Haisch and Tung 2001).

## Properties

Lastly, insights may be offered on certain quantum properties (Compton wavelength, de Broglie wavelength, spin) and on mass-energy equivalence (E=mc2) if it proves to be the case that zero-point fluctuations interact with matter in a phenomenon identified by Schrödinger known as zitterbewegung (Haisch and Rueda 2000; Haisch, Rueda and Dobyns 2001; Nickisch and Mollere 2002).

As intriguing as these latter possibilities are, the first order of business is to unambiguously detect and measure zero-point energy. While a Casimir experiment such as that of Forward (1984) can in principle measure energy that may be attributed to the existence of real zero-point energy, there are alternative explanations involving source-source quantum interactions in place of real zero-point energy (see Milonni 1994). To move beyond this ambiguity of interpretation experiments that will test for the reality of measurable zero-point energy will need to be devised.

## Pop Culture References

In the movie The Incredibles, the villain Syndrome uses a ray that can immobilize an opponent, suspending him in mid-air. Director Brad Bird, speaking in a DVD commentary, says that in searching for a name for the device (or at least a better one than "the Immobi-ray"), he came across and used a reference to "zero-point energy", which Syndrome himself uses to describe his weapon (Of course, this is simply a cool name rather than a practical application at this time!). In Half-Life 2, one of the weapons at your disposal is the "Zero Point Energy Field Manipulator", better known by its nickname the "Gravity Gun". It allows you to pick up and launch any medium-sized objects, and was used to market the game's detailed physics engine. The television show Stargate Atlantis also makes references to zero-point energy in the form of Zero-Point Modules or ZPMs. These ZPMs are used to power the technology of The Ancients, such as the Galactic Stargate which allows the exploration team to travel to the Pegasus Galaxy and the energy shield which protects the city of Atlantis.

## References

1. Einstein, A. and Hopf, L., Ann. Phys., 33, 1096 (1910a); Ann. Phys., 33, 1105 (1910b).
2. Einstein, A. and Stern, O., Ann. Phys., 40, 551 (1913).
3. Forward, R., Phys. Rev. Phys. Rev. B, 30, 1700 (1984). http://www.calphysics.org/articles/Forward1984.pdf
4. Haisch, B. and Rueda, A., Phys. Lett. A, 268, 224 (2000). http://xxx.arxiv.org/abs/gr-qc/9906084
5. Haisch, B., Rueda, A., and Dobyns, Y., Ann. Phys., 10, No. 5, 393 (2001). http://xxx.arxiv.org/abs/gr-qc/0002069
6. Haisch, B., Rueda, A. and Puthoff, H.E. 1994, Phys. Rev. A., 69, 678. http://www.calphysics.org/articles/PRA94.pdf
7. Loudon, R., The Quantum Theory of Light, (Oxford: Clarendon Press) (1983).
8. Milonni, P., The Quantum Vacuum: an Introduction to Quantum Electrodynamics (New York: Academic) (1994).
9. Nernst, W., Verh. Dtsch. Phys. Ges., 18, 83 (1916).
10. Nickisch, L. J. and Mollere, J., physics/0205086 (2002). http://www.arxiv.org/abs/physics/0205086
11. Rueda, A. and Haisch, B., Found. Phys., 28, No. 7, 1057 (1998a) http://xxx.arxiv.org/abs/physics/9802030; Phys. Lett. A, 240, 115 (1998b). http://xxx.arxiv.org/abs/physics/9802031
12. Rueda, A., Haisch, B. and Tung, R., preprint, gr-qc/0108026 (2001). http://xxx.arxiv.org/abs/gr-qc/0108026
13. Sciama, D. W. in “The Philosophy of Vacuum” (S. Saunders and H. R. Brown, eds.), (Oxford: Clarendon Press) (1991).

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