Zero point energy

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In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system. The quantum mechanical system that encapsulates this energy is the zero-point field. The concept was first proposed by Albert Einstein and Otto Stern in 1913. The term "zero-point energy" is a calque of the German Nullpunktenergie. All quantum mechanical systems have a zero point energy. The term arises commonly in reference to the ground state of the quantum harmonic oscillator and its null oscillations.

In quantum field theory, it is a synonym for the vacuum energy, an amount of energy associated with the vacuum of empty space. In cosmology, the vacuum energy is taken to be the origin of the cosmological constant1 which is thought by many to produce dark energy.2 Experimentally, the zero-point energy of the vacuum leads directly to the Casimir effect, and is directly observable in nanoscale devices.

Because zero point energy is the lowest possible energy a system can have, this energy cannot be removed from the system. A related term is zero-point field, which is the lowest energy state of a field; i.e. its ground state, which is non-zero.3

Contents

History

In 1900, Max Planck derived the formula for the energy of a single "energy radiator", i.e. a vibrating atomic unit, as:

\epsilon = \frac{h\nu}{ e^{\frac{h\nu}{kT}}-1}

Here, h is Planck's constant, ν is the frequency, k is Boltzmann's constant, and T is the absolute temperature.

In 1913, using this formula as a basis, Albert Einstein and Otto Stern published a paper of great significance in which they suggested for the first time the existence of a residual energy that all oscillators have at absolute zero. They called this "residual energy" and then Nullpunktsenergie (in German), which later became translated as zero-point energy. They carried out an analysis of the specific heat of hydrogen gas at low temperature, and concluded that the data are best represented if the vibrational energy is taken to have the form:4

\epsilon = \frac{h\nu}{ e^{\frac{h\nu}{kT}}-1} + \frac{h\nu}{2}

Thus, according to this expression, even at absolute zero the energy of an atomic system has the value ½hν.5

Foundational physics

In classical physics, the energy of a system is relative, and is defined only in relation to some given state (often called reference state). Typically, one might associate a motionless system with zero energy, although doing so is purely arbitrary.

In quantum physics, it is natural to associate the energy with the expectation value of a certain operator, the Hamiltonian of the system. For almost all quantum-mechanical systems, the lowest possible expectation value that this operator can obtain is not zero; this lowest possible value is called the zero-point energy. (Caveat: If we add an arbitrary constant to the Hamiltonian, we get another theory which is physically equivalent to the previous Hamiltonian. Because of this, only relative energy is observable, not the absolute energy. This does not change the fact that the minimum momentum is non-zero, however.)

The origin of a minimal energy that isn't zero can be intuitively understood in terms of the Heisenberg uncertainty principle. This principle states that the position and the momentum of a quantum mechanical particle cannot both be known simultaneously, with arbitrary accuracy. If the particle is confined to a potential well, then its position is at least partly known: it must be within the well. Thus, one may deduce that within the well, the particle cannot have zero momentum, as otherwise the uncertainty principle would be violated. Because the kinetic energy of a moving particle is proportional to the square of its velocity, it cannot be zero either. This example, however, is not applicable to a free particle—the kinetic energy of which can be zero.

In thermodynamics, since temperature is defined as the average translational kinetic energy of a moving particle, the existence of non-zero minimal energy of the particle implies that it is impossible to achieve the temperature of absolute zero.

Varieties of zero-point energy

The idea of zero-point energy occurs in a number of situations, and it is important to distinguish these, and note that there are many closely related concepts.

In ordinary quantum mechanics, the zero-point energy is the energy associated with the ground state of the system. The most famous such example is the energy E={\hbar\omega\over 2} associated with the ground state of the quantum harmonic oscillator. More precisely, the zero-point energy is the expectation value of the Hamiltonian of the system.

In quantum field theory, the fabric of space is visualized as consisting of fields, with the field at every point in space and time being a quantized simple harmonic oscillator, with neighboring oscillators interacting. In this case, one has a contribution of E={\hbar\omega\over 2} from every point in space, resulting in a technically infinite zero-point energy. The zero-point energy is again the expectation value of the Hamiltonian; here, however, the phrase vacuum expectation value is more commonly used, and the energy is called the vacuum energy.

In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of vacuum fluctuations or the zero-point energy to the particle masses.

Experimental evidence

A phenomenon that is commonly and erroneously presented as evidence for the existence of zero-point energy in quantum field theory is the Casimir effect. This effect was proposed in 1948 by Dutch physicist Hendrik B. G. Casimir (Philips Research), who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. A small force can be measured between the plates, which can be regarded as the change of the zero-point energy of the electromagnetic field between the plates. However, there is still some debate on this issue, since the Casimir effect can be shown to be equally well described by a different theory involving charge-current interactions (the radiation-reaction picture), as argued by Robert Jaffe of MIT 6. Other such vacuum-induced phenomena include the spontaneous emissions of light (photons) by atoms and nuclei, the observed Lamb shift of the energy levels of atoms, and the anomalous value of electron's gyromagnetic ratio, to name a few.

Gravitation and cosmology

Unsolved problems in physics: Why doesn't the zero-point energy of vacuum cause a large cosmological constant? What cancels it out?

In cosmology, the zero-point energy offers an intriguing possibility for explaining the speculative positive values of the proposed cosmological constant. In brief, if the energy is "really there", then it should exert a gravitational force. In general relativity, mass and energy are equivalent; both produce a gravitational field.

One obvious difficulty with this association is that the zero-point energy of the vacuum is absurdly large. Naively, it is infinite, but one must argue that new physics takes over at the Planck scale, and so its growth is cut off at that point. Even so, what remains is so large that it would visibly bend space, and thus, there seems to be a contradiction. There is no easy way out, and reconciling the seemingly huge zero-point energy of space with the observed zero or small cosmological constant has become one of the important problems in theoretical physics, and has become a criterion by which to judge a candidate Theory of Everything.

"Free energy" devices

As a scientific concept, the existence of zero point energy is not controversial although it may be debated. But perpetual motion machines and other power generating devices based on zero point energy are highly controversial. Descriptions of practical zero point energy devices have thus far lacked cogency. Experimental demonstrations of zero point energy devices have thus far lacked credibility. For reasons such as these, claims to zero point energy devices and great prospects for zero point energy are deemed pseudoscience.

The discovery of zero point energy does not improve the world's prospects for perpetual motion machines. Much attention has been given to reputable science suggesting that zero point energy is infinite. But zero point energy is a minimum energy below which a thermodynamic system can never go, thus none of this energy can be withdrawn without altering the system to a different form in which the system has a lower zero point energy. The calculation that underlies the Casimir experiment, a calculation based on the formula predicting infinite vacuum energy, shows the zero point energy of a system consisting of a vacuum between two plates will decrease at a finite rate as the two plates are drawn together. The vacuum energies are predicted to be infinite, but the changes are predicted to be finite. Casimir combinined the projected rate of change in zero point energy with the principle of conservation of energy to predict a force on the plates. The predicted force, which is very small and was experimentally measured to be within 5% of its predicted value, is finite.7 Even though the zero point energy might be infinite, there is no theoretical basis or practical evidence to suggest that infinite amounts of zero point energy are available for use, that zero point energy can be withdrawn for free, or that zero point energy can be used in violation of conservation of energy.

In principle, there remains the prospect of finding something that can be irreversibly altered or consumed to draw a net positive amount of energy through a zero point energy effect. Enthusiasm should be tempered by the realization that the Casimir effect produces tiny amounts of energy and those only in a non-renewable fashion.

References

  1. ^ http://users.physik.tu-muenchen.de/sfb375/Server/ringberg_2003/bauer.pdf
  2. ^ http://arxiv.org/ftp/physics/papers/0506/0506017.pdf
  3. ^ Gribbin, John (1998). Q is for Quantum - An Encyclopedia of Particle Physics. Touchstone Books. ISBN 0-684-86315-4. 
  4. ^ Laidler, Keith, J. (2001). The World of Physical Chemistry. Oxford University Press. ISBN 0198559194. 
  5. ^ Introduction to Zero-Point Energy - Calphysics Institute
  6. ^ Jaffe, R. L., Physical Review D. 72, 021301(R) (2005)
  7. ^ http://math.ucr.edu/home/baez/physics/Quantum/casimir.html - The article refers to an "implied force" from the change in energy, which is the force required by conservation of energy.

Further reading

External links

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  • This page was last modified on 14 November 2008, at 03:17.

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