Quantum Mechanical Operators Research Articles

A Dynamical Density Field That Shows the Localizability of Electrons: The Exchange-Correlation Ehrenfest Force.

A gradual but steady tide in theoretical chemistry is favoring the exploration of atomic and molecular interactions through the dynamical forces perceived and exerted by the particles of a system. By integrating the quantum mechanical force operator over all the spin and all but one of the spatial coordinates of the electrons, the Ehrenfest force density field reveals these forces directly and is separable into a classical term, related to the electric field, and a quantum mechanical correction, which we introduce and analyze for various atoms and molecules in this work. This exchange-correlation Ehrenfest force density field, Fxc(r), excludes the dominant nuclear components that shape the full Ehrenfest field, revealing information about electron sharing, pairing, and delocalization. In a manner similar, though not equal, to the electron localization function, Fxc(r) unveils covalent and core basins. Its divergence, ∇·Fxc(r), indicates the presence of electron shells in atoms and recovers the positions of lone pairs and the shell structure of ionic, polar, and covalent interactions in molecules. It also exhibits a semiquantitative match with the Laplacian of the electron density that we also explore. In alignment with the established role of exchange-correlation as nature's glue, we demonstrate that a significant number of fundamental concepts in chemical bonding can be derived from the Fxc(r) dynamical field.

More on spin-2 operators in holographic quantum mechanics

We study the spectrum, unitarity bound and holographic central charge of spin-2 operators in a warped AdS2 × S2 × T4 × Iψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{I}}_{\\psi } $$\\end{document} × Iρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{I}}_{\\rho } $$\\end{document} background in the Type IIB theory. We were able to identify a class of solutions that is completely independent of the functions that define the background solution. We comment on the relation of our results to the previous ones in the literature.

Quantum mechanical operator Touchard polynomials studied by virtue of operators’ normal ordering and Weyl ordering

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is [Formula: see text] That is replacing [Formula: see text] by [Formula: see text] where [Formula: see text] is the number operator, and using the method of integration within ordered product we find that [Formula: see text] is just the normal ordering form [Formula: see text]. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is [Formula: see text] With use of the Weyl ordering form of [Formula: see text] we prove [Formula: see text] where [Formula: see text] denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory.

Prediction of First-Order Phase Transition with Electron-Phonon Interaction.

Phase transitions in solids occur due to the shifting balance between the binding energies and entropic contributions of different crystal structures, even though the underlying Hamiltonian remains the same. This work demonstrates that incorporating electron-phonon interactions in the Hamiltonian results in distinct free energies at different temperatures, thus leading to a first-order phase transition. Contrary to prior investigations, taking into consideration the quantum mechanical kinetic energy operator of the nucleus by employing Bogoliubov's inequality yields a first-order phase transition. An equation is implicitly derived to determine the critical temperature of the first-order phase transition. Furthermore, an estimation is made to evaluate the latent heat and the resulting positional displacement of the nucleus. Comparing the present findings with previous ones allows setting parameter boundaries for both first- and second-order phase transitions.

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

This paper is concerned with robust performance analysis of linear stochastic systems acting as an integral operator on a standard Wiener process at the input and producing a stationary Gaussian random process at the output. We propose a performance criterion which uses the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the “cost-shaping” analytic function, the covariance-analytic performance criterion depends on higher-order Hardy-Schatten norms of the system transfer function. We discuss the links of these norms with the asymptotic behaviour of cumulants of finite-horizon quadratic functionals of the system output and their variational properties pertaining to system robustness to statistically uncertain inputs which differ from the standard Wiener process. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of mixed products of noncommuting annihilation and creation operators in quantum mechanics. This computational procedure, which is one of the main results of the paper, involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H2-norm of an auxiliary cascaded system. These results are also compared with an alternative approach, which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations, and illustrated by a numerical example.

Quantum field theory for coherent photons: isomorphism between Stokes parameters and spin expectation values

Stokes parameters (S) on the Poincaré sphere are very useful values to describe the polarisation state of photons. However, the fundamental principle on the nature of polarisation is not completely understood, yet, because we have no concrete consensus on how to describe spin of photons, quantum-mechanically. Here, we have considered a monochromatic coherent ray of photons, described by a many-body coherent state, and established a fundamental basis to describe the spin state of photons, in connection with a classical description based on Stokes parameters. We show that a spinor description of the coherent state is equivalent to Jones vector for polarisation states, and obtain the spin operators (Ŝ) of all components based on rotators in an SU(2) group theory. Polarisation controllers such as phase-shifters and rotators are also obtained as quantum-mechanical field operators to change the phase of the wavefunction for polarisation states. We show that the Stokes parameters are quantum-mechanical average of the spin operators, S=⟨Ŝ⟩.

On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics

On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics

A Quantum-Like Tensor Compression Sentence Representation Based on Constraint Functions for Semantics Analysis

To emphasize the semantic impact of local semantic and grammatical information among adjacent words in the input text, we establish a constraint functions-based quantum-like tensor compression sentence representation model by integrating the concept of extending the pure state-based density matrix to the mixed-state projection operator in quantum mechanics. The provided model highlights the semantic significance of mixed word associations in the input text, simultaneously reducing the reliance on information derived solely from dictionary statistics. We combine the correlation coefficient with the attention mechanism to establish the correlation coefficient between words. The quantum-like sentence representation based on pure state density matrix is extended to the projection operator of mixed states. Combining the acquisition of maximum in convex optimization, a constraint functions-based quantum-like text representation pruning model is established to reduce redundant information caused by dimensional expansion of tensor operations. The experimental results on SICK-2014, STS-benchmark, and STS-companion show that the provided model is more effective than the mainstream models in mining semantic information, especially more sensitive to the negative semantics of sentences.

Algebraic Solution of the Quantum Harmonic Oscillator

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.

Efficient and Parallel Implementation of Real and Complex Response Functions Employing the Second-Order Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator.

We present the implementation of an efficient matrix-folded formalism for the evaluation of complex response functions and the calculation of transition properties at the level of the second-order algebraic-diagrammatic construction (ADC(2)) scheme. The underlying algorithms, in combination with the adopted hybrid MPI/OpenMP parallelization strategy, enabled calculations of the UV/vis spectra of a guanine oligomer series ranging up to 1032 contracted basis functions, thereby utilizing vast computational resources from up to 32,768 CPU cores. Further analysis of the convergence behavior of the involved iterative subspace algorithms revealed the superiority of a frequency-separated treatment of response equations even for a large spectral window, including 101 frequencies. We demonstrate the applicability to general quantum mechanical operators by the first reported electronic circular dichroism spectrum calculated with a complex polarization propagator approach at the ADC(2) level of theory.

Sharp focusing of partially coherent Bessel-correlated beams by a graded-index lens.

The nonparaxial focusing of partially coherent Bessel-correlated beams carrying vortices by a graded-index lens is investigated using the decomposition of the incident field into coherent modes and the quantum mechanical operator method. The influence of the coherence state and the incident beam aperture on tight focusing is analyzed. Our results show that a partially coherent Bessel-correlated beam can be focused into a spot of smaller size than coherent light.

Physics analysis with Leibniz’s differential operators dn

We introduce a systematic approach to represent Leibniz’s nth-order differential operator d^n as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator d^n is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.

Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg algebras

We consider 3d \mathcal{N}=4𝒩=4 theories on the geometry \Sigma \times \mathbb{R}Σ×ℝ, where \SigmaΣ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on \mathbb{R}ℝ. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminiscent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.

An approach to the Gaussian RBF kernels via Fock spaces

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers, the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics, and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform. The core function of the coordinate–momentum exchange operators in the addition law of fractional Fourier transform was analyzed too. In this paper, the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators (IWOP) are used to establish the entanglement fractional Fourier transform theory to the extent of quantum. A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

Thermal properties of relativistic Dunkl oscillators

For a better understanding of the physical systems, even at the quantum level, the thermal quantities can be investigated. Recently, we realized that the parity of the system can also be examined simultaneously, by substituting the Dunkl operator with the ordinary differential operator in quantum mechanics. In this manuscript, we consider two relativistic Dunkl-oscillators and investigate their thermal quantities with well-known statistical methods. Besides, we establish a relationship between the Dunkl-Dirac oscillator and the Dunkl-Anti-Jayne-Cummings model by defining the Dunk creation and annihilation operators. Therefore, we conclude that our model can be regarded as an appropriate scenario for the theory of an open quantum system coupled to a thermal bath.

A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

The Golden–Thompson trace inequality, which states that Tr eH+K ≤ Tr eHeK, has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or H=−−Δ+m and K = potential, Tr eH+(1−u)KeuK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr XsYtX1−sY1−t ≤ Tr XY for positive operators X, Y and for 12≤s,t≤1, and s+t≤32. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the 32,1 range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem.

Quantum Spectral Problems and Isomonodromic Deformations

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2times 2 linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé mathrm {III}_3 equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (epsilon _1+epsilon _2=0). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (epsilon _2=0). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the epsilon _2rightarrow 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Studying the Two New Convolutions of Fractional Fourier Transform by Using Dirac Notation

Based on quantum mechanical representation and operator theory, this paper restates the two new convolutions of fractional Fourier transform (FrFT) by making full use of the conversion relationship between two mutual conjugates: coordinate representation and momentum representation. This paper gives full play to the efficiency of Dirac notation and proves the convolutions of fractional Fourier transform from the perspective of quantum optics, a field that has been developing rapidly. These two new convolution methods have potential value in signal processing.

Flow of time during energy measurements and the resulting time-energy uncertainty relations

Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.