Quantum Mechanical Quantities Research Articles

Active learning for quantum mechanical measurements

The experimental evaluation of many quantum mechanical quantities requires the estimation of several directly measurable observables, such as local observables. Due to the necessity to repeat experiments on individual quantum systems in order to estimate expectation values of observables, the question of how many repetitions to allocate to a given directly measurable observable arises. We show that an active learning scheme can help to improve such allocations, and the resultant decrease in experimental repetitions required to evaluate a quantity with the desired accuracy increases with the size of the underlying quantum mechanical system. Published by the American Physical Society 2024

Development of modulation, pairing mechanism, and slip system with optimum vanadium substitution at Bi-sites in Bi-2212 ceramic structure

Present study focuses extensively on the change in electrical, superconducting and microhardness parameters with partial substitution of trivalent V+3 impurities replacing Bi+3 ions in Bi-2212 ceramic compound with the aid of dc electrical resistivity and microhardness test measurements. Experimental findings, calculation results, and phenomenological discussions provide that the optimum vanadium substitution level is found to be x = 0.01 in the Bi2.0-xVxSr2.0Ca1.1Cu2.0Oy (Bi-2212) ceramic system for the highest conductivity, crystallinity quality, superconducting, and mechanical performance features depending on the decreased microscopic structural problems. All the findings are wholly verified by scanning electron microscopy (SEM) and X-Ray diffraction (XRD) analyses. The dc electrical measurements indicate that the optimum vanadium ions support the pairing mechanism for the formation of new polaronic states in the clusters of microdomains, and hence expand superconducting energy gap due to the enhancement of amplitude part of pair wave function in the spin-density wave systems. The excess vanadium content degrades all the basic thermodynamics and quantum mechanical quantities mentioned due to the stress-induced phase transformation. Numerically, the Bi-2212 advanced ceramic matrix prepared by the optimum vanadium impurity is noticed to present the smallest residual resistivity value of 0.08 mΩ cm, room temperature resistivity value of 8.84 mΩ cm, and broadening degree of 0.36 K. Similarly, the ceramic material is found to possess the highest residual resistivity ratio of 3.05, carrier concentration number of 0.153041, critical transition offset and onset value of 84.66 K and 85.02 K, respectively. Besides, the microhardness findings reveal that the same compound with the least sensitivity to the applied test loads exhibits the largest Hv value of 4.799 GPa, Young's moduli of 393.303 GPa, yield strength of (0.969 GPa), and elastic stiffness coefficient of 15.5574 (GPa)7/4 under the applied test load of 0.245 N. The XRD investigations show that the presence of optimum vanadium impurity supports the formation of a high superconducting phase, c-axis length, and average crystallite size. All the findings are morphologically confirmed by the SEM images. It is found that the crystallographically best crystallinity quality and view of surface morphology is observed for the optimum vanadium substitution level. All in all, new higher properties for the conductivity, crystallinity quality, surface morphology, superconducting, and microhardness parameters based on the optimum vanadium replacement encourage the Bi-2212 crystal system to use in much more application places.

Thermodynamics of the Spin Square

Small spin systems at the interface between analytical studies and experimental application have been intensively studied in recent decades. The spin ring consisting of four spins with uniform antiferromagnetic Heisenberg interaction is an example of a completely integrable system, in the double sense: quantum mechanical and classical. However, this does not automatically imply that the thermodynamic quantities of the classical system can also be calculated explicitly. In this work, we derive analytical expressions for the density of states, the partition function, specific heat, entropy, and susceptibility. These theoretical results are confirmed by numerical tests. This allows us to compare the quantum mechanical quantities for increasing spin quantum numbers s with their classical counterparts in the classical limit srightarrow infty . As expected, a good agreement is obtained, except for the low temperature region. However, this region shrinks with increasing s, so that the classical state variables emerge as envelopes of the quantum mechanical ones.

TBMaLT, a flexible toolkit for combining tight-binding and machine learning.

Tight-binding approaches, especially the Density Functional Tight-Binding (DFTB) and the extended tight-binding schemes, allow for efficient quantum mechanical simulations of large systems and long-time scales. They are derived from ab initio density functional theory using pragmatic approximations and some empirical terms, ensuring a fine balance between speed and accuracy. Their accuracy can be improved by tuning the empirical parameters using machine learning techniques, especially when information about the local environment of the atoms is incorporated. As the significant quantum mechanical contributions are still provided by the tight-binding models, and only short-ranged corrections are fitted, the learning procedure is typically shorter and more transferable as it were with predicting the quantum mechanical properties directly with machine learning without an underlying physically motivated model. As a further advantage, derived quantum mechanical quantities can be calculated based on the tight-binding model without the need for additional learning. We have developed the open-source framework-Tight-Binding Machine Learning Toolkit-which allows the easy implementation of such combined approaches. The toolkit currently contains layers for the DFTB method and an interface to the GFN1-xTB Hamiltonian, but due to its modular structure and its well-defined interfaces, additional atom-based schemes can be implemented easily. We are discussing the general structure of the framework, some essential implementation details, and several proof-of-concept applications demonstrating the perspectives of the combined methods and the functionality of the toolkit.

Deep contextual disambiguation of homonyms and polysemants

AbstractA new metric method for word sense disambiguation is put forward, being formulated within a phenomenological analogy to the wave function of an observable quantity in quantum mechanics where the actual meaning of a multivalued word (a homonym or a polysemant) is determined by its context. The choice of the actualized sense is determined by the minimal semantic distance between the semantics of possible word senses and that of the context, where the meanings of the word in question and the context follow from their semantic fields based on lexicographic hyperchains. In contrast to the common models, our method accounts for semantic polarity. The formulated method showed good results in disentangling polysemy, which was not possible to achieve within the contextualized embedding approacdh based on bidirectional encoder representations from transformers (BERT).

The physical origin of Schrödinger equation

Quantum mechanics is one of the basic theories of modern physics. Here, the famous Schrödinger equation and the differential operators representing mechanical quantities in quantum mechanics are derived, just based on the principle that the translation invariance (symmetry) of a system in Hamiltonian mechanics should be preserved in quantum mechanics. Moreover, according to the form of the differential operators, the commutation relation in quantum mechanics between the generalized coordinate and the generalized momentum can be directly obtained. We believe that the results in this paper are very useful for understanding the physical origin of quantum mechanics.

Quantity in Quantum Mechanics and the Quantity of Quantum Information

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

Controlling the TiN Electrode Work Function at the Atomistic Level: A First Principles Investigation

The paper reports on a theoretical description of work function of TiN, which is one of the most used materials for the realization of electrodes and gates in CMOS devices. Indeed, although the work function is a fundamental quantity in quantum mechanics and also in device physics, as it allows the understanding of band alignment at heterostructures and gap states formation at the metal/semiconductor interface, the role of defects and contaminants is rarely taken into account. Here, by using first principles simulations, we present an extensive study of the work function dependence on nitrogen vacancies and surface oxidation for different TiN surface orientations. The results complement and explain a number of existent experimental data, and provide a useful tool to tailoring transport properties of TiN electrodes in device simulations.

Symmetry and its breaking in a path-integral approach to quantum Brownian motion.

We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.

Many-body multivaluedness of particle-current variance in closed and open cold-atom systems

The quantum variance of an observable is a fundamental quantity in quantum mechanics, and the variance provides additional information other than the average itself. By examining the relation between the particle-current variance $(\delta J)^2$ and the average current $J$ in both closed and open interacting fermionic systems, we show the emergence of a multi-valued Lissajous curve between $\delta J$ and $J$ due to interactions. As a closed system we considered the persistent current in a benzene-like lattice enclosing an effective magnetic flux and solved it by exact diagonalization. For the open system, the steady-state current flowing through a few lattice sites coupled to two particle reservoirs was investigated using a Lindblad equation. In both cases, interactions open a loop and change the topology of the corresponding $\delta J$-$J$ Lissajous curve, showing that this effect is model-independent. We finally discuss how the predicted phenomena can be observed in ultracold atoms, thus offering an alternative way of probing the dynamics of many-body systems.

Dynamical Quantum Phase Transitions: A Geometric Picture.

The Loschmidt echo is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model as an example, we show that this indeed is not the case and that a simple semiclassical approximation to systems well described by mean-field theory is, in fact, in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the postquench dynamics provided in Supplemental Material visualize this new point of view.

Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox “absolute” quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.

Invariant eigen-operator calculated vibration mode of lattice in the case of absorbing an atom onto crystal surface

The influence of diffusion and defects of crystal surface on the crystal vibration mode are an important and basic subject in surface physics research. The frequency of lattice vibration corresponds to the energy band of the system. Since the vibrations of the atoms in the crystal lattice are not isolated from each other, and the crystal lattice is periodic, thereby forming a lattice wave in the crystal. The lattice wave represents that all the atoms in the crystal vibrate at an identical frequency, which is often called a vibration mode. The lattice chain model has been studied as the vibrating mode of phonon and the energy-band in solid state physics. The vibrating modes of the lattice chain model have been analyzed with the Newton equation and the Born-von-Karman boundary condition in the literaure. In general, it is difficult to solve this problem due to the complex nonlinear characteristic of the interactions between the matter particles and the environment. Noting the complicacy in directly diagonalizing quantum Hamiltonian operator of a long chain, we introduce the invariant eigenoperator method (IEO) for deriving the energy gap of a given crystal lattice without solving its eigenstates in the Heisenberg picture. The Heisenberg equation is as important as the Schrödinger equation. However, it has been seldom used for directly deriving the energy-gap in previous studies. Following the Heisenberg's original idea that most observable physical quantity in quantum mechanics is energy spectrum, Hong-yi Fan, one of the authors of the present paper, developed the IEO method. This method provides a natural result of combining both the Schrödinger operator and the Heisenberg equation. Using the IEO method, we study the vibration modes of crystal lattice, which are affected by absorbing an atom with mass m0, which is different from the mass of atom in the crystal. Moreover, the attractive potential constantβ0 of the lattice surface differs from the inner constantβ. With the help of invariant eigen-operator method, we deduce the vibration mode ω=√(2β(1-cosh α))/ħm, where α=ln[-(mβ0+m0(-2β+β0)+√β0√-4 mm0β+(m+m0)2β0)/2m0β]. Our numerical results show that vibration mode ω depends not only on the absorption potential and the mass of the absorbed atom, but also on the mass of the lattice atom and the inner potential. In general, by discussing the vibration modes via some numerical solutions or approximate methods, we show the relations between the system vibration modes with different parameters which describe the environment influences. These results can deepen our understanding of quantum Brownian motion and demonstrate the applicability of the IEO method.

Unveiling the nature of post-linear response Z-vector method for time-dependent density functional theory.

We report a theoretical study on the analysis of the relaxed one-particle difference density matrix characterizing the passage from the ground to the excited state of a molecular system, as obtained from time-dependent density functional theory. In particular, this work aims at using the physics contained in the so-called Z-vector, which differentiates between unrelaxed and relaxed difference density matrices to analyze excited states' nature. For this purpose, we introduce novel quantum-mechanical quantities, based on the detachment/attachment methodology, for analysing the Z-vector transformation for different molecules and density functional theory functionals. A derivation pathway of these novel descriptors is reported, involving a numerical integration to be performed in the Euclidean space on the density functions. This topological analysis is then applied to two sets of chromophores, and the correlation between the level of theory and the behavior of our descriptors is properly rationalized. In particular, the effect of range-separation on the relaxation amplitude is discussed. The relaxation term is finally shown to be system-specific (for a given level of theory) and independent of the number of electrons (i.e., the relaxation amplitude is not simply the result of a collective phenomenon).

Gauge transformations and conserved quantities in classical and quantum mechanics

We are taught that gauge transformations in classical and quantum mechanics do not change the physics of the problem. Nevertheless, here we discuss three broad scenarios where under gauge transformations: (i) conservation laws are not preserved in the usual manner; (ii) non-gauge-invariant quantities can be associated with physical observables; and (iii) there are changes in the physical boundary conditions of the wave function that render it non-single-valued. We give worked examples that illustrate these points, in contrast to general opinions from classic texts. We also give a historical perspective on the development of Abelian gauge theory in relation to our particular points. Our aim is to provide a discussion of these issues at the graduate level.

The gravitational constant as a quantum mechanical expression

A quantitatively verifiable expression for the gravitational constant is derived in terms of quantum mechanical quantities. This derivation appears to be possible by selecting a suitable physical process in which the transformation of the equation of motion into a quantum mechanical wave equation can be obtained by Einstein’s geodesic approach. The selected process is the pi-meson, modeled as the one-body equivalent of a two-body quantum mechanical oscillator in which the vibrating mass is modeled as the result of the two energy fluxes from the quark and the antiquark. The quantum mechanical formula for the gravitational constant appears to show a quantitatively verifiable relationship with the Higgs boson as conceived in the Standard Model.

To Principles of Quantum Theory Construction

New insight to the principles of the quantum theory construction is given. It is based on the symmetry study of main differential equations of mechanics and electrodynamics. It has been shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. The substantial consequence of the given consideration is the proof of the main postulate of quantum mechanics, that is, the proof of the statement, that to any quantum mechanical quantity can be set up into the correspondence the Hermitian matrix. It is shown, that a non-abelian character of the multiplicative group of the quaternion ring leads to the nonapplicability of the quaternion calculus for the construction of new versions of quantum mechanics directly. The given conclusion seems to be actual, since there is a number of modern publications with the development of the quantum mechanics theory using the quaternions with the standard basis {e, i, j, k}. The correct way for the construction of new versions of quantum mechanics on the quaternion base is discussed in the paper presented. It is realized by means of the representation of the quaternions through the basis of the linear space of complex numbers over the field of real numbers, under the multiplicative group of which the equations of the dynamics of mechanical systems are invariant. At the same time the quaternion calculus is applicable in electrodynamics, at that the new versions of quantum electrodynamics can be constructed by an infinite number of the ways corresponding to an infinite number of the matrix representations of the standard quaternion basis {e, i, j, k}. The given conclusion is the consequence of the high symmetry of Maxwell equations.

Conservation of the spin and orbital angular momenta in electromagnetism

We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic field theory still lacks corresponding locally conserved spin and orbital AM currents. In this paper, we construct these missing spin and orbital AM densities and fluxes that satisfy the proper continuity equations. We show that these are physically measurable and conserved quantities. These are, however, not Lorentz-covariant, so only make sense in the single laboratory reference frame of the measurement probe. The fluxes we derive improve the canonical (nonconserved) spin and orbital AM fluxes, and include a ‘spin–orbit’ term that describes the spin–orbit interaction effects observed in nonparaxial optical fields. We also consider both standard and dual-symmetric versions of the electromagnetic field theory. Applying the general theory to nonparaxial optical vortex beams validates our results and allows us to discriminate between earlier approaches to the problem. Our treatment yields the complete and consistent description of the spin and orbital AM of free Maxwell fields in both quantum-mechanical and field-theory approaches.

Quantum harmonic oscillator with position-dependent mass in the displacement operator formalism

The position-dependent effective mass quantum harmonic oscillator problem is considered within the displacement operator framework. Using the analytic and the algebraic approaches, exact expressions for quantum mechanical quantities of the system have been obtained. In the limit of no deformation, results of the constant mass oscillator are recovered.

Position-dependent effective mass system in a variable potential: displacement operator method

Within the framework of the translation operator for a quantum system with position-dependent mass, we examine the quantum state of a position-dependent mass system in a variable potential. By imposing conditions of resolvability, we arrive at a potential with a quartic and a quadratic term. It emerges naturally that the energy eigen states of the system are negative. We have found the quantum mechanical quantities: energy spectrum, eigen functions and uncertainty relation. These quantities depend on the parameters of the potential.