Quantum potentials

This is called primitive in the sense that the short-time approximation, truncated after the first term, is in its crudest form. Nonprimitive schemes would be those that would improve this approximation, for instance by replacing the bare potential V(x) by an effective quantum potential (see [142, 149]). 

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,... 

In TD-DFT, the wave function is antisymmetrized and therefore, nonfactorizable or entangled. However, as said above, it is not entangled from a dynamical point of view because the quantum forces originated from a nonseparable quantum potential as in Equation 8.34 are not taken into account. 

As is well known, de Broglie abandoned his attempts at a realistic account of quantum phenomena for many years until David Bohm s discovery of a solution of Schrodinger s equation that lends itself to an interpretation involving a physical particle traveling under the influence of a so-called quantum potential. 

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, Non-local quantum potential interpretation of relativistic actions at a distance in many-body problems, in G. Tarozzi and A. Van der Merwe (Eds.), Open Questions in Quantum Physics. Invited Papers on the Foundations of Microphysics, in (Bari, Italy, May 1983), ISBN 9-02-771853-9, Reidel, Dordrecht, 1985, pp. 297-332. 

C. Dewdney, A. Garuccio, A. Kyprianidis, and J. P. Vigier, The anomalous photoelectric effect Quantum potential theory versus effective photon hypothesis, Phys. Lett. A 105A(12), 15-18 (1984). 

P. Gueret and J. P. Vigier, Relativistic wave equations with quantum potential nonlinearity, Lett. Nuovo Cimento 38, 125—128 (1983). 

C. Fenech and J. P. Vigier, Analyse du potentiel quantique de De Broglie dans le cadre de 1 interpretation stochastique de la theorie des quanta (Analysis of the de Broglie quantum potential in the frame of the stochastic interpretation of quantum theory), C. R. Acad. Sci. Paris 293, 249 (1981). 

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). 

Figure 2. Undersized waveguide used in microwave tunneling experiments (a) waveguide with central reduced cross section (b) waveguide filled with two dielectrics with different refractive indexes n > (c) the same quantum potential...

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,... 

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... 

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... 

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. 

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