Quantum potential energy

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,... 

Without this term the quantum equation becomes identical with the classical expression. The only factor which can cause Vq to vanish is the mass. Not surprisingly, massive macroscopic objects have Vq —> 0 and are predicted to behave classically whereas sub-atomic entities with appreciable quantum potential energy Vq > 0, are known to exhibit quantum behaviour. The clear implication that there is no sharply defined classical/quantum limit, but... 

In a region of constant quantum potential energy, the expression... 

It will now be shown that the existence of quantum potential energy eliminates the need to allow for repulsion between sub-electronic charge elements in an extended electron fluid. An electron, whatever its size or shape is described by a single wave function that fixes the electron density at any point... 

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. 

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. 

Once decoupled from the core the energy of the valence electron, now in the valence state, is entirely in the form of quantum potential, energy. 

The observation [66] that the energy, Eg = h2/8mrl, of a valence electron, decoupled from the core, but confined to the ionization sphere, consists entirely of quantum potential energy, has been interpreted to represent the electronegativity of an atom, also defined as the chemical potential of the... 

In terms of the causal model the kinetic energy in every stationary state with me = 0 is zero hence the total calculated energy is pure quantum potential energy. To confirm this, recall that Vq must be constant for the confined particle, i.e. 

It is assumed that the tendency of a molecular mixture to interact can be analyzed as a function of the chemical (quantum) potential energy field and some action variable that reflects mass ratios or amounts of substance. Spontaneous chemical change occurs as the chemical potential of a system decreases, i.e. while Ap < 0, and ceases when Ap = 0, at equilibrium. The quantity here denoted by Ap, also known as the affinity, a of the system, is the sum over all molecules, reactants and products... 

There is no way to understand the energy difference between the S and A states (figure 5) in terms of pre-assigned interaction of the particles constituting the molecule. As the wave function is real, the momenta of all particles are zero and the energy is just the sum of classical and quantum potential energies... 

To see how importance sampling is applied in the quantum case, we consider the evaluation of the average quantum potential energy using Eqs. (4.11) and (4.15). Since the kinetic energy contains some special difficulties, we shall examine it separately. The potential energy is diagonal in coordinate representation so that we can write... 

As for the classical potential, the gradient of quantum potential energy defines a quantum force. A quantum object therefore has an equation of motion, m x= —VH — VV. For an object in uniform motion (constant potential) the quantum force must vanish, which requires = 0 or a constant, —k say. 

A quantum object confined to an impenetrable spherical enclosme of radius r has quantum-potential energy... 

When the valence electron reaches the ionization limit, its potential energy with respect to the nucleus goes to zero, and in uniform distribution, it has no kinetic energy. The calculated confinement energy Eg) can therefore only represent quantum potential energy, defined as [16]... 

Ionization radii are of fundamental importance in chemistry. By definition, they represent the volume to which activated valence electrons are confined, and hence the quantum-potential energy of the valence state. This quantity is the same as the classical concept of electronegativity [25]. Not only is the entire theory of chemical reactivity entangled with electronegativity, but the ionization sphere also features directly in the simulation of interatomic interactions. Previous efforts to model ionization radii theoretically invariably involved some unsubstantiated assumptions. The present calculation proceeds without such assumptions, from derived extranuclear electronic arrangements. 

The formation of a diatomic molecule involves the interaction between two activated valence electrons. In the case of heteropolar interaction, the difference in quantum potential energy (electronegativity) of these two electrons results in a skewed charge distribution, which may be expressed as a difference 8Q in charge, measured at the... 

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