Bohmian mechanics

Bohmian mechanics refers to the ontological interpretation of quantum theory pioneered by Bohm [4]. The mathematical structure of the theory is not affected by the different interpretation and the same formalism adopted before [7] will be used here. 

The connection between wave mechanics and hydrodynamics, expressed by equations (4.20) and (4.22), was developed in more detail by Madelung, writing the time dependence of T as an action function, T = which seperates (4.18) into a coupled pair that resembles the field equations of hydrodynamics  

An attractive feature of the hydrodynamic model is that it obviates the statistical interpretation of quantum theory, by eliminating the need of a point particle. It is worth noting that the assumption of a point electron derives from the observation that it responds as a rmit to an electromagnetic signal, which must therefore propagate instantaneously through the interior of the electron, thought to be at variance with the theory of relativity. However, by now it is known from experiment that non-local (instantaneous) response 

On reinterpretation it was pointed out by David Bohm that equation (4.25) differed from the classical Hamilton-Jacobi equation only in the term 

The quantity Vq, called quantum potential vanishes for classical systems as h/m 0. A gradual transition from classical to quantum behaviour is inferred to occur for systems of low mass, such as sub-atomic species. All dynamic properties of classical systems should therefore be dehned equally well for quantum systems, although the relevant parameters are hidden (Bohm, 1952). 

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. 

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The theories that should feature prominently in the understanding of chemical effects have been summarized in this volume, without demonstrating their application. The way forward has been indicated by Primas [67] and in the second volume of this work the practical use of modern concepts such as spontaneous symmetry breaking, non-local interaction, bohmian mechanics, number theories and space-time topology, to elucidate chemical effects will be explored. The aim is to stimulate renewed theoretical interest in chemistry. 

Bohmian Mechanics A Trajectory Picture of Quantum Mechanics. 112... 

Although trajectories are not computed in QFD-DFT, it is clear that there is a strong connection between this approach and the trajectory or hydrodynamical picture of quantum mechanics [20], independently developed by Madelung [21], de Broglie [22], and Bohm [23], which is also known as Bohmian mechanics. From the same hydrodynamical equations, information not only about the system... 

The purpose of this chapter is to show and discuss the connection between TD-DFT and Bohmian mechanics, as well as the sources of lack of accuracy in DFT, in general, regarding the problem of correlations within the Bohmian framework or, in other words, of entanglement. In order to be self-contained, a brief account of how DFT tackles the many-body problem with spin is given in Section 8.2. A short and simple introduction to TD-DFT and its quantum hydrodynamical version (QFD-DFT) is presented in Section 8.3. The problem of the many-body wave function in Bohmian mechanics, as well as the fundamental grounds of this theory, are described and discussed in Section 8.4. This chapter is concluded with a short final discussion in Section 8.5. 

BOHMIAN MECHANICS A TRAJECTORY PICTURE OF QUANTUM MECHANICS... 

Apart from the operational, wave or action-based pictures of quantum mechanics provided by Heisenberg, Schrodinger, or Feynman, respectively, there is an additional, fully trajectory-based picture Bohmian mechanics [20,23]. Within this picture, the standard quantum formalism is understood in terms of trajectories defined... 

An alternative way to obtain the quantum trajectories is by formulating the Bohmian mechanics as a Newtonian-like theory. Then, Equation 8.29 gives rise to a generalized Newton s second law ... 

In the case of a many-body problem, the Bohmian mechanics for an A-body dynamics follows from the one for a single system, but replacing Equation 8.25 by... 

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. 

Some of the chemical concepts with little or no quantum-mechanical meaning outside the Bohmian formulation but, well explained in terms of the new interpretation, include electronegativity, the valence state, chemical potential, metallization, chemical bonding, isomerism, chemical equilibrium, orbital angular momentum, bond strength, molecular shape, phase transformation, chirality and barriers to rotation. In addition, atomic stability is explained in terms of a simple physical model. The central new concepts in Bohmian mechanics are quantum potential and quantum torque. 

The quantum potential can now be identified as a surface effect that exists close to any interface, in this case the vacuum interface. The non-local effects associated with the quantum potential also acquire a physical basis in the form of the vacuum interface, now recognized as the agent responsible for mediating the holistic entanglement of the universe. The causal interpretation of Bohmian mechanics finds immediate support in the postulate of a vacuum interface. There is no difference between classical and quantum entities, apart from size. Logically therefore, the quantum limit depends on... 

The chemistry community, understandably, failed to respond at all, even though Bohmian mechanics probably holds the key to the development of a theory of chemistry, soundly based on quantum theory and relativity. The problem with molecular structure is resolved by the ontological interpretation of quantum-mechanical orbital angular momentum and the key to chemical reactivity and reaction mechanism is provided by the quantum potential function. 

The theories of quantum mechanics and relativity and the concepts underlying these theories are unfamiliar territory for many chemists. To prepare the ground for a reassessment of the chemical importance of these theories against the background of Bohmian mechanics, the relevant concepts have recently been presented from a chemical perspective [7]. Constant reference to this earlier work will be made here. Important equations, discussed and... 

The new theories that spring from the application of Bohmian mechanics to chemical problems reveal a close connection between chemical phenomena and the attributes of space-time. The most fundamental principle of chemistry is the periodic classification of the elements in terms of natural numbers. Examined against the background of number theory a deeper level of periodicity that embraces all nuclides is revealed and found to relate on a cosmic scale to an involution in space-time structure. 

Virtually no aspect of chemistry is left untouched by the new insight gained through consideration of Bohmian mechanics, number theory and symmetry. A preliminary analysis of some of these aspects are presented here in an effort to stimulate further research, that may happen once familiarity with the new approach overcomes traditional scepticism. Most of the material in this book has been published in somewhat different form before. Chapters 1, 2, 4, 5, and 6 are elaborations based on recent review articles [8, 9, 10, 11] by the author and included here with permission by the relevant publishers. 

J.T. Cushing, A. Fine and S. Goldstein (eds.) Bohmian Mechanics and Quantum Theory An... 

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