Zero-point oscillations

These fluctuations will affect the motion of charged particles. A major part of the Lamb shift in a hydrogen atom can be understood as the contribution to the energy from the interaction of the electron with these zero point oscillations of the electromagnetic field. The qualitative explanation runs as follows the mean square of the electric and magnetic field intensities in the vacuum state is equal to... 

It was found that normal zero-point oscillations lie on top of large gluon fluctuations - instantons and anti-instantons with random positions and sizes. The left column - action density and the right column - topological charge density. Here instantons are peaks and anti-instantons are holes. 

A more detailed treatment of the model of Craig and others resulted19 in values 4 or 5 times as small as those quoted above for the contribution of the plasmon zero-point oscillations to 7 it is concluded, consequently, that 7 must contain also other terms. 

This formula represents the model in which the absorbed phase has perfect lateral freedom (two-dimensional ideal gas) on the sorbent surface and vibrates harmonically (fundamental frequency v) normal to the surface the sorbent surface is assumed to have a uniform sorbate potential E < 0 (including the zero-point oscillator energy) relative to the gas. By Equation (2.8), the lifetime corresponding to Equation (2.15) in the special case a = 1 and hv/kT 1 is... 

Knowing the interaction.s, one can proceed to calculate any of the other mechanical properties, and one can expect reasonably good results. Kittel (1976) indicates in particular that results obtained for the bulk moduli are quite good, and that they are improved by the inclusion of quantum effects associated with the zero-point oscillations. A study of the full vibrational spectra of Ar and Kr has been made by Grindlay and Howard (1965). 

Modern inelastic neutron scattering technique has made it possible to discover free protons in solids [49] that is, free protons have been found in manganese dioxides, coals, graphite nitric acid intercalation compounds, polypyrolles and polyanilines, and fi-alumina. Perhaps the said compounds may be called protonic conductors as well, though in the solids the density of free protons is very small and the distribution of proton kinetic momentum is hidden by the zero-point oscillations of the host matrix [49]. 

Figure 3. Charge distribution in the phase space of x and p, of the vibrator with zero point oscillation amplitude xo = 0.03...

It is not heretical to consider the electromagnetic vacuum as a physical system. In fact, it manifests some physical properties and is responsible for a number of important effects. For example, the field amplitudes continue to oscillate in the vacuum state. These zero-point oscillations cause the spontaneous emission [1], the natural linebreadth [5], the Lamb shift [6], the Casimir force between conductors [7], and the quantum beats [8]. It is also possible to generate quantum states of electromagnetic field in which the amplitude fluctuations are reduced below the symmetric quantum limit of zero-point oscillations in one quadrature component [9]. 

A more profound difference between the two representations can be traced in the properties of the zero-point oscillations. In fact, the energy operators... 

It is seen from the definition of the mode functions (18) and (19) that, in contrast to (28), the zero-point oscillations of the electric field strength of multipole photons manifest the spatial inhomogeneity. 

Figure 1. Contribution into the zero-point oscillations (29) from the terms with j = 1 in the case of an ideal spherical cavity as a function of x = kr, (b) the level (28) is shown by the straight line.

The case of outgoing and incoming spherical waves can be examined under the standard assumption that the atom located at the origin has a finite size that permits us to avoid the divergence at kr — 0. It is seen that, in some small vicinity of the atom, the zero-point oscillations, corresponding to the multipole field in an infinite space, strongly exceed those in the ideal spherical cavity (see Fig. 2). 

The point is that the zero-point oscillations are responsible for the so-called shot noise [14,15], determining the quantum limit of uncertainty in different optical measurements. The preceding result shows that the presence of an atom causes the increase of shot noise and hence a deterioration of the quantum limit of precision of measurements, at least, in some vicinity of the atom [22,29]. We discuss this effect in more details in Section VI. 

The zero-point oscillations of the energy density of plane waves of photons have the same magnitude everywhere. In contrast, those calculated in the presence of a singular point (source or absorber) manifest spatial inhomogeneity. Precisely, the vacuum noise is concentrated in some vicinity of the singular point. 

In contrast to (143), this is a diagonal matrix independent of the spatial variables. Hence, in exactly the same way as with the zero-point oscillations of energy density, the vacuum fluctuations of polarization in empty space has the global nature, while those in the presence of the singular point manifest certain spatial inhomogeneity. 

The element PT describes the vacuum noise of transversal (with respect to r) circular polarizations with positive and negative helicity, while PL gives the zero-point oscillations of linear polarization in the longitudinal direction (along r). The explicit form of the unitary transformation (147) is... 

The limitation to second derivatives in computing the force constants is called the harmonic approximation. The calculation does not take into account either the zero-point oscillation or the anharmonicity of the potentials. It is, strictly speaking, valid only at T = 0. Thermal expansion, thermal conductivity and other nonlinear effects are thus not contained in this model. 

Van der Waals forces are also electrostatic, and, generally speaking, dipolar they exist in virtue of (o) permanent dipoles in molecules, (6) dipoles induced by other permanent dipoles and, most characteristically, (c) the coupling of the zero-point oscillations of positive and negative charge which must occur even in molecules without an observable moment. Van der Waals forces normally fall off as the inverse seventh power of the distance. 

That the interaction of the induced and the inducing dipole leads to a lowering of energy is to be seen qualitatively in a very simple way. If two pendulums of frequency Vq are coupled, they develop two new frequencies Vq Avq. In the same way the two zero-point oscillations of the electrical systems in the molecules develop by their interaction new frequencies, as a result of which their total energy is lowered in a way to be discussed in more detail presently. This is the origin of the van der Waals attraction. 

Whether the van der Waals forces arise from the zero-point oscillations and their mutual effects, from the orientation of existing permanent moments, or from the induction of new ones, they are all essentially of the nature of dipole attractions—to which multipole interactions of higher order may be added as correction terms. 

The zero-point oscillation of the atom or molecule may be schematized as an elastic vibration of a charge of mass m. The frequency is then given by the formula... 

Fig. 5. Regular temperature behavior for elastic-stiffness constants such as Young s modulus, the shear modulus, and the bulk modulus. The main features are continuous decrease with increasing temperature, zero slope at zero temperature, a i dependence at very low temperatures, relative flatness at low temperatures, and linear slope at high temperatures. A, the zero-temperature deviation from linear behavior, is a quantum effect due to zero-point oscillations.

In the 2D regime achieved by confining the hght-atom motion to zero point oscillations with amplitude Zq, the weakly bound molecular states exist at a negative a satisafying the inequality a Iq. See Petrov, D.S. and Shlyapnikov, G.V., Interatomic coUisions in a tightly confined Bose gas, Phys. Rev. A, 64, 012706, 2001. 

The coordinate Zj is the distance from the dimer center of mass to the center of the ring described by the nucleus of atom i during zero-point oscillations. 

Therefore, if a statistical equilibrium configuration is at all possible for a system of classical charged particles, then at a temperature of absolute zero must exist a zero-point classical electromagnetic radiation as well as a zero-point oscillating motion for the charges. Of course, zero-point field and motion are... 

This stabilization has to do with zero-point oscillator energies, so dispersion energy is a consequence of the uncertainty principle, like all zero-point energy effects (Section 3.1). 

Fourthly, at T = 0 K (v = 0), the oscillations do not come to an end. The so-called zero oscillations are preserved even at absolute zero temperature. The zero point oscillation energy is equal to /iCO/2. 

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