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.
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.
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%.
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 m. 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.
This paper presents a comprehensive perspective of the metric of quantum states with a focus on geometry in the background independent quantum mechanics. We also explore the possibilities of geometrical formulations of quantum mechanics beyond the quantum state space and Kahler & & manifold. The metric of quantum states in the classical configuration space with the pseudo-Riemannian signature and its possible applications are explored. On contrary to the common perception that a metric for quantum state can yield a natural metric in the configuration space when the limit h → 0, we obtain the metric of quantum states in the configuration space without imposing the limiting condition h → 0. Here Planck’s constant h is absorbed in the quantity like Bohr radii 1 ~ a0. While exploring the metric structures associated with Hydrogen like atom,
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.
Abstract Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.