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The metric space approach to quantum mechanics

Paul M. Sharp

Published Jul 31, 2015• 0 citations
OpenAlex

Abstract

The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws. We develop this approach to show that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law. All of these metric spaces have an "onion-shell" geometry consisting of concentric spheres, with functions conserved to the same value lying on the same sphere. We apply this approach to generate metrics for wavefunctions, particle densities, paramagnetic current densities, and external scalar potentials. In addition, we demonstrate the extensions to our approach that ensure that the metrics for wavefunctions and paramagnetic current densities are gauge invariant. \n \nWe use our metric space approach to explore the unique relationship between ground-state wavefunctions, particle densities and paramagnetic current densities in Current Density Functional Theory (CDFT). We study how this relationship is affected by variations in the external scalar potential, pairwise electronic interaction strength, and magnetic field strength. We find that all of the metric spaces exhibit a "band structure", consisting of "bands" of points characterised by the value of the angular momentum quantum number, m. These "bands" were found to either be separated by "gaps" of forbidden distances, or be "overlapping". We also extend this analysis beyond CDFT to explore excited states. \n \nWe apply our metrics in order to gain new insight into the Hohenberg-Kohn theorem and the Kohn-Sham scheme of Density Functional Theory. For the Hohenberg-Kohn theorem, we find that the relationship between potential and wavefunction metrics, and between potential and density metrics, is monotonic and includes a linear region. Comparing Kohn-Sham quantities to many-body quantities, we find that the distance between them increases as the electron interaction dominates over the external potential.

Related papers

Metric-space approach to potentials and its relevance to density-functional theory

Paul M. Sharp, Irene D’Amico

External potentials play a crucial role in modeling quantum systems, since, for a given interparticle interaction, they define the system Hamiltonian. We use the metric-space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wave functions and densities from I. D'Amico et al. [Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of density-functional theory. We explore these by analyzing two many-body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground-state wave functions, and ground-state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wave functions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analyzed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wave function with the Kohn-Sham wave function lies under a threshold of 10%.

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Metric space formulation of quantum mechanical conservation laws

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We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an ``onion-shell'' geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a ``band structure'' in metric space directly arising from the conservation of the $z$ component of the angular momentum.

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Quantum Mechanics in Metric Space: Wave Functions and Their Densities

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Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.

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We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.

•Dec 1, 2018•7 citations

Testing density-functional approximations on a lattice and the applicability of the related Hohenberg-Kohn-like theorem

Vivian V. França, Jeremy P. Coe, Irene D’Amico

We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.

arXiv:1708.08952•Scientific Reports•Jan 8, 2018•13 citations
The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws. We develop this approach to show that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law. All of these metric spaces have an "onion-shell" geometry consisting of concentric spheres, with functions conserved to the same value lying on the same sphere. We apply this approach to generate metrics for wavefunctions, particle densities, paramagnetic current densities, and external scalar potentials. In addition, we demonstrate the extensions to our approach that ensure that the metrics for wavefunctions and paramagnetic current densities are gauge invariant. \n \nWe use our metric space approach to explore the unique relationship between ground-state wavefunctions, particle densities and paramagnetic current densities in Current Density Functional Theory (CDFT). We study how this relationship is affected by variations in the external scalar potential, pairwise electronic interaction strength, and magnetic field strength. We find that all of the metric spaces exhibit a "band structure", consisting of "bands" of points characterised by the value of the angular momentum quantum number, m. These "bands" were found to either be separated by "gaps" of forbidden distances, or be "overlapping". We also extend this analysis beyond CDFT to explore excited states. \n \nWe apply our metrics in order to gain new insight into the Hohenberg-Kohn theorem and the Kohn-Sham scheme of Density Functional Theory. For the Hohenberg-Kohn theorem, we find that the relationship between potential and wavefunction metrics, and between potential and density metrics, is monotonic and includes a linear region. Comparing Kohn-Sham quantities to many-body quantities, we find that the distance between them increases as the electron interaction dominates over the external potential.
External potentials play a crucial role in modeling quantum systems, since, for a given interparticle interaction, they define the system Hamiltonian. We use the metric-space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wave functions and densities from I. D'Amico et al. [Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of density-functional theory. We explore these by analyzing two many-body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground-state wave functions, and ground-state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wave functions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analyzed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wave function with the Kohn-Sham wave function lies under a threshold of 10%.
We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an ``onion-shell'' geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a ``band structure'' in metric space directly arising from the conservation of the zzz component of the angular momentum.
Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density-functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric-space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density-functional theory, which relates the many-body wave function, particle density, and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers mmm. Additionally, varying the magnetic field uncovers a richer complexity for the ``band structure'' present in ground-state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric-space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground-state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.
Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.
We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.
We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.