The Quantum Potential

Beck, T. L. Marchioro, T. L., The quantum potential distribution theorem, in Path Integrals from meV to MeV Tutzing 1992, Grabert, H. Inomata, A. Schulman, L. Weiss, U., Eds., World Scientific Singapore, 1993, pp. 238-243... 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

Since TD-DFT is applied to scattering problems in its QFD version, two important consequences of the nonlocal nature of the quantum potential are worth stressing in this regard. First, relevant quantum effects can be observed in regions where the classical interaction potential V becomes negligible, and more important, where p(r, t) 0. This happens because quantum particles respond to the shape of K, but not to its intensity, p(r, t). Notice that Q is scale-invariant under the multiplication of p(r, t) by a real constant. Second, quantum-mechanically the concept of asymptotic or free motion only holds locally. Following the classical definition for this motional regime,... 

P. R. Holland and J. P. Vigier, The quantum potential and signaling in the Einstein-Podolsky-Rosen experiment, Found. Phys. 18(7), 741-750 (1988). 

J. P. Vigier, Superluminal propagation of the quantum potential in the causal interpretation of quantum mechanics, Lett. Nuovo Cimento 24(8) (Ser. 2), 258-264 (1979). 

In 1952 David Bohm rediscovered aspects of earlier proposals by de Broglie and Madelung, which had been rejected years before, and established the concept of non-local interaction via the quantum potential. It appears to provide fundamental answers for the understanding of chemistry, but remains on the fringes, while awaiting recognition by the establishment. 

David Bohm gave new direction to Madelung s proposal by using the decomposition of the wave equation for a radically new interpretation of quantum theory. He emphasized the similarity between the Madelung and Hamilton-Jacobi equations of motion, the only difference between them being the quantum potential energy term,... 

Bohm s failure to give an adequate explanation to support the pilot-wave proposal does not diminish the importance of the quantum-potential concept. In all forms of quantum theory it is the appearance of Planck s constant that signals non-classical behaviour, hence the common, but physically meaningless, proposition that the classical/quantum limit appears as h —> 0. The actual limiting condition is Vq —> 0, which turns the quantum-mechanical... 

The quantum potential energy however, depends on the wave function over the entire space occupied by the electron, i.e. 

A system of this type is not holistic, but partially holistic, which means that pairwise interaction occurs between the holistic units. The distinction drawn here between holistic and partially holistic systems is not in line with the terminology used in general philosophic discourse and in order to avoid any confusion it is preferable to distinguish between systems that interact either continuously, or discontinuously, with the quantum potential field. Quantum potential, like the gravitational potential, occurs in the vacuum, presumably with constant intensity. The quantum potential energy of a quantum object therefore only depends on the wave function of the object. 

The archetype of quantum objects is the photon. It is massless, has unit spin, carries no charge, and responds to the quantum potential field. By comparison, an electron is a massive fermion with half-spin and unit negative charge. It responds to both classical and quantum potentials. The only property that these two entities have in common is their wave nature,... 

Gauge fields (M) that restore local phase invariance are evidently closely related also to the quantum-potential field. The wave function of a free electron, with temporal and spatial aspects of the phase factor separated, may be written as... 

The total energy of an electron in the potential field V of an atomic core tends to zero as V —> 0 on compression to ro- The calculated energy of the decoupled valence electron can therefore only arise from the quantum potential and it will be argued that this energy represents the concept, intuitively defined before as the electronegativity of an atom. The wave function (3.36) of the valence electron will be argued to determine the chemical interaction of an atom with its environment. 

To build a theory on these axioms it is necessary to have a clear understanding of the assumed nature of the electron and the conditions under which electron exchange between atoms becomes possible. These conditions will be taken to define an atomic valence state. The electronic configuration that dictates the mode of interaction between atoms of different elements will be interpreted to define the quantum potential energy of a valence electron in the valence state of an atom. This quantity will be shown to correspond to what has traditionally been defined empirically as the electronegativity of an atom. 

The problem has been resolved [69] by redefining electronegativity as the chemical potential of the valence state, calculated as the quantum potential of the valence electron, confined to its ionization sphere, i.e. x2 = h2/8mrl, expressed in eV. Whereas x corresponds to Pauling electronegativities, subject to simple periodic scaling, x2 corresponds to the Mulliken scale by the same type of operation. All of the many electronegativity scales in existence are simply related to the ionization radii, from which they ultimately derive. 

The only difference between the classical and quantum formulations resides in the additional potential-energy term h2V2A/2mA, known as the quantum potential, Vq. In the classical case Vq 0. A quantum-mechanically stationary state occurs when Vq = k, a constant independent of x, i.e. 

The importance of the quantum potential lies therein that it defines the classical limit with Vq —> 0, or more realistically where the quantity h/m —> 0, which implies h/p = A —> 0. It means that quantum effects diminish in importance for systems with increasing mass. Massless photons and electrons (with small mass) behave non-classically, and atoms less so. Small molecules are at the borderline, and macro molecules approach classical behaviour. When the system is in an eigenstate (or stationary state) of energy E, the kinetic energy E — V = k) is by definition equal to zero. 

Entropy production during chemical change has been interpreted [7] as the result of resistance, experienced by electrons, accelerated in the vacuum. The concept is illustrated by the initiation of chemical interaction in a sample of identical atoms subject to uniform compression. Reaction commences when the atoms, compacted into a symmetrical array, are further activated into the valence state as each atom releases an electron. The quantum potentials of individual atoms coalesce spontaneously into a common potential field of non-local intramolecular interaction. The redistribution of valence electrons from an atomic to a metallic stationary state lowers the potential energy, apparently without loss. However, the release of excess energy, amounting to Au = fivai — fimet per atom, into the environment, requires the acceleration of electronic charge from a state of rest, and is subject to radiation damping [99],... 

Molecular structure and shape are related to orbital angular momentum and chemical change is shown to be dictated by the quantum potential. The empirical parameters used in computer simulations such as molecular mechanics and dynamics are shown to derive in a fundamental way from the relationship between covalence and the golden ratio. 

It is noted that a system of particles reaches equilibrium when the resulting forces on them are zero, and hence the quantum force on a free particle must be perceived to vanish. This requires the quantum potential to be either zero or a constant, independent of position. The first condition relates to a classical particle, whereas the second condition implies... 

In terms of the quantum-potential formulation particle trajectories can be associated with the quantum HJ equation (6) in exactly the same way as in the classical case [34, 35]. As before, particle trajectories associated with the phase S may be obtained by constructing the normals to S, each one distinguished by its initial coordinates. By this procedure Bohm managed to revive the pilot-wave model of De Broglie. It means that a point particle of mass m on a trajectory x = x(t), is now associated with the physical... 

The intensity of a wave is proportional to the square of the amplitude, i.e. I = R2. Multiplication of the amplitude by a real constant therefore scales the intensity, but the quantum potential stays the same,... 

The effect of Vq is seen to be independent of the intensity of the quantum field and to depend only on its form. This is in sharp contrast to the effect of classical waves. The effect of the quantum potential on a quantum particle has been likened to a ship on automatic pilot being guided by radio waves. Here, too, the effect of the radio waves is independent of their intensity and... 

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential... 

The quantum potential therefore arises in the effects of an internal stress in the fluid and depends on derivatives of the fluid density rather than on external factors. 

The idea of action at a distance was resisted both by Newton, and by Einstein [42] who called it "spooky", but it has now been demonstrated experimentally [43, 44] that local realistic theory cannot account for correlations between measurements performed at well separated sites. The conclusion is that quantum theory permits hidden variables and is non-local. This conclusion is at variance with relativity, but, as pointed out by Bohm [34], the nonlocality of quantum theory only applies to complex wave functions and does not imply that the quantum potential can be used to transmit signals faster than light. 

In this case the two systems evidently behave independently. Situations like this are fairly common in chemistry, generally associated with an approach to the classical limit in which the quantum potential becomes negligible and non-local interactions insignificant. Although the basic law therefore refers inseparably to the whole universe, it tends to fragment into numerous independent parts, each constituted of further sub-units that are non-locally connected internally. The key to this fragmentation is the lack (or nature) of chemical interaction between sub-units, which can be treated in the traditional way. 

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