Bohmian particles

Equation (17), together with equations (10) and (15) provide the working equations of the MQCB method. According to their structure, the position of the classical degrees of freedom, its momentum, the position of the Bohmian particle and the wavefunction in the quantum subspace need to be propagated simultaneously. For clarity, these quantities shall be combined as... 

Quantum-classical coupling through Bohmian particles... 

Here we focus on yet another implementation, the single-particle hydrodynamic approach or QFD-DFT, which provides a natural link between DFT and Bohmian trajectories. The corresponding derivation is based on the realization that the density, p(r, t), and the current density, j(r, t) satisfy a coupled set of classical fluid, Navier-Stokes equations ... 

These new trajectories are the so-called reduced quantum trajectories [30], which are only explicitly related to the system reduced density matrix. The dynamics described by Equation 8.42 leads to the correct intensity (time evolution of which is described by Equation 8.40) when the statistics of a large number of particles are considered. Moreover, Equation 8.42 reduces to the well-known expression for the velocity held in Bohmian mechanics, when there is no interaction with the environment. 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

The interacting waves from myriads of charge centres constitute the electromagnetic radiation field. In particle physics the field connection between balanced charge centres is called a virtual photon. This equilibrium is equivalent to the postulated balance between classical and quantum potentials in Bohmian mechanics, which extends holistically over all space. 

Thus, within the Bohmian formulation of quantum mechanics, quantum trajectories move according to the usual Hamilton s equations, subject to the additional quantum potential defined in equation (3). An ensemble of quantum particles at positions (x(t),X(t)) distributed initially according to... 

The contentious issue of quantum-particle trajectories is put into perspective by the Bohmian model. One interpretation is that the quantum electron has an unspecified diffuse structure, which contracts into a classical point-like object when confined under external influences. The observed trajectory, as in a cloud chamber, may be considered to follow the centre of gravity. 

It is not implausible that there are worlds whose ultimate constituents are Newtonian particles conforming to Newtonian-like laws, worlds whose ultimate constituents are fluids obeying classical fluid mechanics, worlds whose ontology and laws are those of Bohmian quantum mechanics, all of which contain configurations that realize the nomological/causal specifications associated with at least some mental properties. 

The Bohmian quantum-classical and QMF approaches [21,22] have been applied to a model intended as a simplified representation of gaseous oxygen interacting with a platinum surface, Ref. [13,91]. The model consists of a light particle q with mass m colliding with a heavier particle Q with mass M. The heavy particle is bound to an immobile surface. The total Hamiltonian for the system is given by... 

In this respect, the more deep approaches of quantum description of the chemical bonding advocates in making the required steps toward assessing the quantum particle of the chemical bond as based on the derived chemical field released at its turn by the fundamental electronic equations of motion either within Bohmian non-relativistic (Schrodinger) and to explore the first consequences. If successful, the present endeavor will contribute to celebrate the dream in unveiling the true particle-wave nature of the chemical bond. 

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