Energy Expectation Value Research Articles

Supersymmetric Yang-Mills, spherical branes, and precision holography

Using supersymmetric localization we compute the free energy and BPS Wilson loop vacuum expectation values for planar maximally supersymmetric Yang-Mills theory on Sd in the strong coupling limit for 2 ≤ d < 6. The same calculation can also be performed in supergravity using the recently found spherical brane solutions. We find excellent agreement between the two sets of results. This constitutes a non-trivial precision test of holography in a non-conformal setting. The free energy of maximal SYM on S6 diverges in the strong coupling limit which might signify the onset of little string theory. We show how this divergence can be regularized both in QFT and in supergravity. We also consider d = 7 with a small negative ’t Hooft coupling and show that the free energy and Wilson loop vacuum expectation value agree with the results from supergravity after addressing some subtleties.

The backreaction of massive scalar field on FLRW and de Sitter spaces

In this paper, we obtain the effect of backreaction on the scale factor of the Friedmann–Lemaître–Robertson–Walker (FLRW) and de Sitter spaces. We consider a non-minimally coupled massive scalar field to the curvature scalar. For our purpose, we use the results of vacuum expectation values of energy–momentum tensor, which have been obtained previously. By substituting the quantum energy density into the Friedmann equation, we obtain the linear order perturbation of the scale factor. So, the effect of backreaction leads to the new scale factor.

Fermionic Model with a Non-Hermitian Hamiltonian

This paper deals with the mathematical spectral analysis and physical interpretation of a fermionic system described by a non-Hermitian Hamiltonian possessing real eigenvalues. A statistical thermodynamical description of such a system is considered. Approximate expressions for the energy expectation value and the number operator expectation value, in terms of the absolute temperature $T$ and of the chemical potential $\mu$, are obtained, based on the Euler-Maclaurin formula.

A gauge theory of quantum gravity: quantization of the non-interacting field

The non-interacting field belonging to a new SO(1,3) gauge field theory equivalent to general relativity is canonically quantized in the Lorentz gauge and the physical Fock space for non-interacting gauge particles is constructed. To assure both Lorentz covariance and positivity of the norm and energy expectation value for physical states restrictions needed in the construction of the physical Fock space are put consequentially on state vectors, and not on the algebra of creation and annihilation operators as usual—alltogether providing the second step in consistently quantizing gravitation

Nonequilibrium path-ensemble averages for symmetric protocols.

According to the nonequilibrium work relations, path-ensembles generated by irreversible processes in which a system is driven out of equilibrium according to a predetermined protocol may be used to compute equilibrium free energy differences and expectation values. Estimation has previously been improved by considering data collected from the reverse process, which starts in equilibrium in the final thermodynamic state of the forward process and is driven according to the time-reversed protocol. Here, we develop a theoretically rigorous statistical estimator for nonequilibrium path-ensemble averages specialized for symmetric protocols, in which forward and reverse processes are identical. The estimator is tested with a number of model systems: a symmetric 1D potential, an asymmetric 1D potential, the unfolding of deca-alanine, separating a host-guest system, and translocating a potassium ion through a gramicidin A ion channel. When reconstructing free energies using data from symmetric protocols, the new estimator outperforms existing rigorous unidirectional and bidirectional estimators, converging more quickly and resulting in a smaller error. However, in most cases, using the bidirectional estimator with data from a forward and reverse pair of asymmetric protocols outperforms the corresponding symmetric protocol and estimator with the same amount of simulation time. Hence, the new estimator is only recommended when the bidirectional estimator is not feasible or is expected to perform poorly. The symmetric estimator shows similar performance to a unidirectional protocol of half the length and twice the number of trajectories.

Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines.

The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.

Enhancement of the metrological sensitivity limit through knowledge of the average energy

We consider the problem of quantum phase estimation with access to arbitrary measurements in a single suboptimal basis. The achievable sensitivity limit in this case is determined by the classical Cram\'{e}r-Rao bound with respect to the fixed basis. Here we show that the sensitivity can be enhanced beyond this limit if knowledge about the energy expectation value is available. The combined information is shown to be equivalent to a direct measurement of an optimal linear combination of the basis projectors and the phase-imprinting Hamiltonian. Application to an atomic clock with oversqueezed spin states yields a sensitivity gain that scales linearly with the number of atoms. Our analysis further reveals that small modifications of the observable can have a strong impact on the sensitivity.

Breakdown of the equivalence between gravitational mass and energy due to quantum effects

We review our recent theoretical results about inequivalence between passive and active gravitational masses and energy in the semiclassical variant of general relativity, where the gravitational field is not quantized but matter is quantized. To this end, we consider the simplest quantum body with internal degrees of freedom — a hydrogen atom. We concentrate our attention on the following physical effects, related to electron mass. The first one is the inequivalence between passive gravitational mass and energy at the microscopic level. Indeed, the quantum measurement of gravitational mass can give a result which is different from the expected one, [Formula: see text], where the electron is initially in its ground state; [Formula: see text] is the bare electron mass. The second effect is that the expectation values of both the passive and active gravitational masses of stationary quantum states are equivalent to the expectation value of the energy. The most spectacular effects are the inequivalence of the passive and active gravitational masses and the energy at the macroscopic level for an ensemble of coherent superpositions of stationary quantum states. We show that, for such superpositions, the expectation values of passive and active gravitational masses are not related to the expectation value of energy by Einstein’s famous equation, [Formula: see text]. In this paper, we also improve several drawbacks of the original pioneering works.

The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Abstract Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

Variational relativistic correction to the Thomas-Fermi model of atoms

A relativistic correction to the Thomas-Fermi model of atoms is obtained by utilizing the variational solutions proposed by Csavinszky [Phys. Rev. 166 (1968) 53]. The relativistic effect appears important for heavier elements, and our correction improves the ionization energies more noticeably with an increasing atomic number. The radial expectation values are calculated and compared with Hartree-Fock values. The non-relativistic results for 〈r2〉 are slightly better, and our relativistic correction enhanced the expectation value of the potential energy 〈1/r〉.

Micro-scale classification of offshore wind energy resource ——A case study of the New Zealand

The classification of wind energy resources can provide reference for assessing optimal turbine locations. The traditional wind energy classification scheme is usually a single consideration of partial wind energy factors. This study aims to construct a micro-scale classification scheme of wind energy resource that incorporates a comprehensive consideration of wind energy factors, environment risk factor and cost factors. The wind energy classification of the New Zealand waters is presented as a case study by calculating the expectation values of wind energy. Results show that the expectation values of wind energy in most of the New Zealand are above 0.6 (belongs to rich area). Only a small range coast area is relatively low in expectation values of wind energy, but still above 0.5 (belongs to subrich area). The North Cape waters, the East Cape waters, the South Taranaki Bight and its northwest waters are relative large center of expectation values of wind energy. Comparing with the traditional wind energy classification schemes, our scheme can exhibit the regional difference of wind energy classification more obviously. However, the weight coefficient of the factors considered in this study is not fixed. Reasonably adjusting, adding or deleting the weight coefficients of the related factors according to different focus, the wind energy classification scheme proposed by this study can be widely used to provide scientific reference for the site selection of wind energy development under different actual demands.

Benchmark study of an auxiliary-field quantum Monte Carlo technique for the Hubbard model with shifted-discrete Hubbard-Stratonovich transformations

Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are suitable also for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered as a generalization of the discrete HSTs by Hirsch [J. E. Hirsch, Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by Rom et al. [N. Rom et al., Chem. Phys. Lett. 270, 382 (1997)]. An improvement of the acceptance ratio is found for a real auxiliary field, while an improvement of the average sign is found for a pure-imaginary auxiliary field. Efficiencies of the different HSTs are tested in the single-band Hubbard model at and away from half filling by studying the staggered magnetization and energy expectation values, respectively.

Determination of energy levels, probabilities, and expectation values of particles in the three-dimensional box at quantum numbers ≤5

Static potential boxes represent an area that is limited by an infinite potential wall, where there is no external potential in it. The solution of the Schrodinger equation gives a wave function that can explain the energy level and determine the probability of finding a particle at any point. The main focus of the research in this paperwas to solve the solution to the energy levels, probabilities, and expectation values of a particle in a three-dimensional box, where control limits are given for the potential box width, namely and L for each situation. In this research, the results showed that probability meetings depend on changes in box width and quantum number of particles, while particle energy levels depend on the length of the box and the quantum number of particles. Based on the relation between probabilities and expectation values, the most stable size of the box to find the particles in ground state to the fourth excitation state is at the width of and L.

Chromo-Electric Field Energy in Quark Models

We study the contribution of the chromo-electric field to the total energy of the hadrons. We analyze the case of quarks considered as extended sources of the field. Specific forms of the interaction potential of the quarks are obtained. Moreover, the color charge distribution of these sources allows to determine the zero point energy that is commonly introduced, as a free (independent) parameter, in the total quark model Hamiltonian of the hadronic systems. The ground states of charmonium ($q \bar q$) and nucleon ($ q q q$) are studied in more detail: their mass is calculated within a relativized dynamical model and the expectation value of the chromo-electric field energy is consistently obtained. \end{abstract}

Quantum dynamics and spectroscopy of dihalogens in solid matrices. I. Efficient simulation of the photodynamics of the embedded I2Kr18 cluster using the G-MCTDH method.

The molecular dynamics following the electronic BΠu30+⟵XΣg+1 photoexcitation of the iodine molecule embedded in solid krypton are studied quantum mechanically using the Gaussian variant of the multiconfigurational time-dependent Hartree method (G-MCTDH). The accuracy of the Gaussian wave packet approximation is validated against numerically exact MCTDH simulations for a fully anharmonic seven-dimensional model of the I2Kr18 cluster in a crystal Kr cage. The linear absorption spectrum, time-evolving vibrational probability densities, and I2 energy expectation value are accurately reproduced by the numerically efficient G-MCTDH approach. The reduced density matrix of the chromophore is analyzed in the coordinate, Wigner and energy representations, so as to obtain a multifaceted dynamical view of the guest-host interactions. Vibrational coherences extending over the bond distance range 2.7 Å < RI-I < 4.0 Å are found to survive for several vibrational periods, despite extensive dissipation. The present results prepare the ground for the simulation of time-resolved coherent Raman spectroscopy of the I2-krypton system addressed in Paper II.

Theoretical ranging performance model that considers laser pointing errors for laser ranging systems in detecting diffusely reflecting targets.

A photon-counting lidar ranging system equipped with a single-photon detector and high-power laser sources for tracking low earth orbit (LEO) diffusely reflecting targets has attracted a large amount of attention, and the ranging performance is the key indicator of a laser ranging system. The fluctuation of the pointing error has a considerable effect on the ranging performance, and a theoretical model can be used to optimize the systematic parameters when the ranging system is designed. In this paper, the joint probability distribution function of the laser pointing error is first derived, from which an improved lidar equation that considers the pointing error is derived. The energy expectation value of laser illuminating LEO uncooperative targets is calculated and analyzed by this analytical expression. A Monte Carlo method is used to verify the energy expectation value, and the simulation results match the theoretical results well. The average signal photon number (calculated by the improved lidar equation) also matches well with the experiment conducted by the Shanghai Astronomical Observatory station. Next, the range performance model consists of the range walk error, and the ranging precision is further developed; the results indicate that the range walk error and ranging precision caused by the pointing error are increased to 31.75cm and 2.5cm, respectively, when the laser source is powerful (∼2 J) and the pointing error is approximately 1''. The value of the pointing error should be constrained to be less than 20% of the laser divergence angle to avoid a significant influence on the ranging performance.

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t(r) expressed in terms of its quantal source, the single‐particle density matrix: t A(r), the integrand of the kinetic energy expectation value; t B(r), the trace of the kinetic energy tensor; t C(r), a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information‐entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.

Viability of phantom dark energy as a quantum field in first-order perturbation theory of FLRW spacetime

In the standard cosmological framework of the 0th-order FLRW metric and the use of perfect fluids in the stress-energy tensor, dark energy with an equation-of-state parameter $w < -1$ (known as phantom dark energy) implies negative kinetic energy and vacuum instability when modeled as a scalar field. However, the accepted values for present-day $w$ from Planck and WMAP9 include a significant range of values less than $-1$. A flip of the sign in front of the kinetic energy term in the Lagrangian remedies the negative kinetic energy but introduces ghostlike instabilities, which perhaps may be rendered unobservable, but certainly not without great cost to the theory. Staying within the confines of observational constraints and general relativity, we treat dark energy as a quantum scalar field in the background of this 1st-order FLRW space-time, find an approximation for the Green's function, and calculate the expectation value of the field's kinetic energy for $w<-1$ using adiabatic expansion to renormalize and obtain a finite value. We find that the kinetic energy is positive for values of $w$ less than $-1$ in 0th- or 1st-order FLRW space, thus giving more theoretical credence to observational values of $w<-1$ and demonstrating that phantom dark energy does not categorically have negative kinetic energy. For a nonminimal coupling parameter $\xi=0$, kinetic energy is positive for $w \gtrsim -1.22$, which includes virtually all values of constant $w$ allowed by cosmological data constraints, and more negative values of $w$ give positive kinetic energy for non-zero values of $\xi$. Also, our results are generally applicable for a massive free field or a field with a small potential in a 0th- or 1st-order FLRW background dominated by a fluid with a constant $w$. [abridged]

Nodal variational principle for excited states

It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.

Spectroscopy and decay properties of charmonium

The mass spectra of charmonium are investigated using a Coulomb plus linear (Cornell) potential. Gaussian wave functions in position space as well as in momentum space are employed to calculate the expectation values of potential and kinetic energy respectively. Various experimental states (X(4660)(53S1), X(3872)(23P1), X(3900)(21P1), X(3915)(23P0) and X(4274)(33P1) etc.) are assigned as charmonium states. We also study the Regge trajectories, pseudoscalar and vector decay constants, electric and magnetic dipole transition rates, and annihilation decay widths for charmonium states.