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International Journal of Quantum Chemistry

On the relationship between the one‐electron and Bohm's quantum potential

Miroslav Kohout

Published Nov 13, 2001• 16 citations
Publisher

Abstract

Abstract The one‐electron potential, derived from the electron density, is a three‐dimensional function, whereas Bohm's quantum potential depends on the spatial coordinates of all involved electrons. To analyze the relationship between the two potentials in the many‐electron case, first the dimensionality of the quantum potential needs to be reduced to match that of the one‐electron potential. A possible approach for such dimensionality reduction by calculating the expectation value of the quantum potential over all but one electron, using the factorization of the real part of the wave function into a marginal and conditional function is presented, and a relation between such three‐dimensional local quantum potential and the one‐electron potential is given. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

DOI
10.1002/qua.10050

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Abstract The one‐electron potential, derived from the electron density, is a three‐dimensional function, whereas Bohm's quantum potential depends on the spatial coordinates of all involved electrons. To analyze the relationship between the two potentials in the many‐electron case, first the dimensionality of the quantum potential needs to be reduced to match that of the one‐electron potential. A possible approach for such dimensionality reduction by calculating the expectation value of the quantum potential over all but one electron, using the factorization of the real part of the wave function into a marginal and conditional function is presented, and a relation between such three‐dimensional local quantum potential and the one‐electron potential is given. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
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We propose a new approach to the Rayleigh–Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piecewise constant zero-order potential. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 235–242, 2000
There is a consensus today that the the main lesson of the Aharonov-Bohm effect is that a picture of electromagnetism based on the local action of the field strengths is not possible in quantum mechanics. Contrary to this statement, it is argued here that when the source of the electromagnetic potential is treated in the framework of quantum theory, the Aharonov-Bohm effect can be explained without the notion of potentials. It is explained by local action of the field of the electron on the source of the potential. The core of the Aharonov-Bohm effect is the same as the core of quantum entanglement: the quantum wave function describes all systems together.
The causal interpretation of quantum mechanics, as originally stated by deBroglie and Bohm, had several attractive features. Among these is the possibility that it could address some of the most fundamental questions on quantum phenomena. However, subsequent theoretical conjectures, which have now been included in the orthodox view of the deBroglie Bohm theory, are unphysical and have done much to undermine the original theory's appeal. We, therefore, return to the original theory as our starting point and address one of its perplexing areas: the quantum potential. By avoiding the unphysical conjectures we are led to an understanding of the quantum potential which is distinctly different from that of the orthodox deBroglie Bohm view.