Quantum force/velocity

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. 

Here the velocity V is a function (not an operator), often referred to in the literature as the quantum force, and is given by... 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

In Equation 4.56, the real quantities p, v, and j are the charge density, velocity field, and current density, respectively. The above equations provide the basis for the fluid dynamical approach to quantum mechanics. In this approach, the time evolution of a quantum system in any state can be completely interpreted in terms of a continuous, flowing fluid of charge density p(r,t) and the current density j(r,t), subjected to forces arising from not only the classical potential V(r, t) but also from an additional potential VqU(r, t), called the quantum or Bohm potential the latter arises from the kinetic energy and depends on the density as well as its gradients. The current... 

The internal (vibrational and rotational) motion of molecule A is the same as that of the pseudo-molecule, while the center of mass of molecule B is at the fixed center of force. The force from the center of force on the pseudo-molecule is determined as the force between A and B, with A at the position of the pseudo-molecule and B at the position of the center of force. The scattering geometry is illustrated in Fig. 4.1.1. The pseudo-molecule moves with a velocity v = va — vB relative to the fixed center of force. We have drawn a line through the force center parallel to v that will be convenient to use as a reference in the specification of the scattering geometry. In addition to the internal quantum states of the pseudo-molecule and velocity v, the impact parameter b and angle are used to specify the motion of the molecule. 

To calculate the quantum potential it is first necessary to find R, which by equation 9 is clearly a function of all particle coordinates, because of nonlocal interaction in terms of the molecular quantum entanglement inferred before. Even when all particle coordinates and velocities (v = 0) are identical in the two states, the state S must therefore still be at a lower energy. The difference can only be attributed to the quantum potential which differs for the two states. The force that holds the molecule together is therefore seen to be a special case of non-local interaction within the molecule. 

In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

Computer simulation of molecular dynamics is concerned with solving numerically the simultaneous equations of motion for a few hundred atoms or molecules that interact via specified potentials. One thus obtains the coordinates and velocities of the ensemble as a function of time that describe the structure and correlations of the sample. If a model of the induced polarizabilities is adopted, the spectral lineshapes can be obtained, often with certain quantum corrections [425,426]. One primary concern is, of course, to account as accurately as possible for the pairwise interactions so that by carefully comparing the calculated with the measured band shapes, new information concerning the effects of irreducible contributions of inter-molecular potential and cluster polarizabilities can be identified eventually. Pioneering work has pointed out significant effects of irreducible long-range forces of the Axilrod-Teller triple-dipole type [10]. Very recently, on the basis of combined computer simulation and experimental CILS studies, claims have been made that irreducible three-body contributions are observable, for example, in dense krypton [221]. 

Various quantum effects arising due to the motion of dielectric boundaries, including the modification of the Casimir force and creation of photons, in both one and three dimensions, and for different orientations of the velocity vector with respect to the surface, have been studied in detail in the series of papers by Barton and his collaborators [142-148]. In one paper [142] the term mirror-induced radiation (MIR) was introduced. 

The question posed in the title may be short, but it does not have a snappy answer. If one asks "What is the force between planets ", everybody knows that it is given with great accuracy by Newton s law of universal gravi-tation. But for electrons, the story is much more complicated and indeed the question itself must be recast. There are at least three reasons for this (a) physical systems involving electrons must be described by quantum mechanics (QM) (b) in many such systems, the relative velocities are either not very small compared to c and/or high accuracy is of interest, so special relativity must be used, and (c) electrons have spin. 

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