Feynman's Approach Research Articles

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

Exchange–correlation entropy from the generalized thermal adiabatic connection

Warm dense matter is a highly energetic phase characterized by strong correlations, thermal effects, and quantum mechanical electrons. Thermal density functional theory is commonly used in simulations of this challenging phase, driving the development of temperature-dependent approximations to the exchange–correlation free energy. Approaches using the adiabatic connection formula are well known at zero temperature and have been recently leveraged at non-zero temperatures as well. In this work, a generalized thermal adiabatic connection (GTAC) formula is proposed, introducing a fictitious temperature parameter. This allows extraction of the exchange–correlation entropy SXC using simulated interaction strength scaling. This procedure uses a Hellmann–Feynman approach to express the exchange–correlation entropy in terms of a temperature- and interaction strength-dependent exchange–correlation potential energy. In addition, analysis of SXC as a function of interaction strength suggests new forms for approximations, and GTAC itself offers a new framework for exploring both the exact and approximate interplay of temperature, density, and interaction strength across a wide range of conditions.

Motivating the Biot-Savart Law

Introducing students to a new law is pedagogically challenging. One can just declare it as fact, but it is easier on (at least some) students if we can derive it when possible, or, if not, at least motivate it somehow. The Biot-Savart law is particularly challenging in this respect, because it is only derivable from more advanced formulations (such as Maxwell’s equations), which are in their turn merely stipulated (this is the approach of Feynman et al., for example). The other possibility is simply to declare its empirical truth, without details (see, for example, Griffiths).

On the breakdown of the perturbative interaction picture in Big Crunch/Big Bang or the true reason why perturbative string amplitudes on temporal orbifolds diverge

We discuss how the perturbative particle paradigm fails in certain background with space-like singularity but asymptotically flat which should admit a S-matrix. The Feynman approach relies on the interaction picture. This approach means that we can interpret interactions as exchanges of particles. Particles are the modes of the quadratic part of the Lagrangian. In certain backgrounds with space-like singularity the interaction Hamiltonian is well defined but the perturbative expansion of the evolution operator through the singularity and the perturbative S matrix do not exist. On the other hand, relying on minisuperspace approximation we argue that the non perturbative evolution operator does exist. The complete breakdown of the perturbative expansion explains why the perturbative computations in the covariant formalism in string theory in temporal orbifold fail, at least at the tree level.

General memory kernels and further corrections to the variational path integral approach for the Bogoliubov-Fröhlich Hamiltonian

The celebrated variational path integral approach to the polaron problem shows remarkable discrepancies with diagrammatic Monte Carlo for the Bogoliubov-Fr\"{o}hlich Hamiltonian which describes an impurity weakly coupled to a Bose condensed atomic gas. It has been shown both via a renormalization group approach and by the method of correlated Gaussian wavefunctions that the model has a subtle UV divergence caused by quantum fluctuations, which are not captured within Feynman's approach. In this work we address the issues with Feynman's approach and show that by extending the model action to a more general form, and by considering higher order corrections beyond the Jensen-Feynman inequality, a good agreement with diagrammatic Monte Carlo can be obtained.

Study of dynamic and thermodynamic properties of zinc-blend and wurtzite cadmium sulfide (CdS) using density functional theory

Abstract The dynamic and thermodynamic properties of wurtzite (wz) and zinc-blend (zb) CdS are investigated within the density functional theory using different approximation methods such as LDA, PBE, and DFT+U. Hellmann–Feynman approach is implemented for the relaxation of atomic position for both phases. To guarantee the accuracy of calculation, the convergence test of total energy with respect to energy cutoff and k-point sampling is performed. The dynamic properties such as phonon dispersion, phonon density of state, frequency along with high symmetry points, static and dynamic polarizability, and dielectric constants are calculated. The obtained values are compared with previous theoretical results. DFT + U approximation gives a good result that is consistent with the available theory. Moreover, the vibrational energy, vibrational free energy, entropy, electron chemical potential, and constant-volume specific heat are obtained within LDA, PBE, and DFT + U approximations.

Depth of Space and Radius of the Electron

The cosmological model of the expanding balloon in 4D-space (CM) delivers in interaction with a homogeneous vector field exactly Newton’s law of gravitation with its 1/r-shape of the gravitational funnel. So far, the depth of space, W, in the 4-th spatial dimension can only be calculated using the theoretical approach of Feynman’s radius of excess rex=a/3 with Schwarzschild-radius a. With this, the connection to the general theory of relativity (GR) is established, but the situation is unsatisfactory. In the present study, the possibilities of an experimental approach to the calculation of spatial depth, W, are explored. The only experimental approach so far is the bending of light on a central mass. We hypothesize in addition to the main effect φ = -4a/y, i.e., the angle of diffraction of a light beam on a heavy central mass in the distance y and with Schwarzschild-radius a, an additional effect close to the center of the form φC ~ -1/y4. This additional effect has on the edge of the central mass about 1/3 of the strength of the main effect. However, its influence disappears very quickly with increasing distance. For this reason the sun cannot be used as the central mass. The bright corona and the strong magnetosphere do not allow measurements close to the sun. However, ESA’s GAIA mission puts the planet Jupiter at the center of interest. This spacecraft measures with extremely high precision the positions of billions of stars. Results of first data analyses have already been published. As a side effect - the application of the CM to small particles provides an indication that the radius of the electron could be in the order of 10-23 m.

Electronic, structural and optical properties of zincblend and wurtizite cadmium selenide (CdSe) using density functional theory and hubbard correction

Zinc blend (zb) and wurtizite (wz) structure of cadmium selenide (CdSe) is determined using density-functional theory within local density approximation (LDA), generalized gradient approximation (GGA), Hubbard-correction (GGA+U) and Hybrid functional approximation (PBE0 or HSE06). The first principle pseudopotential plane wave is used and the relaxed atomic position for the CdSe in zb and wz structure was obtained by using total energy and force minimization method following the Hellmann Feynman approach. The convergence test of total energy with respect to cutoff energy and k-point sampling is performed . The equilibrium lattice constant and unit cell volume of CdSe in both phases are calculated and the obtained value is compared` with experimental values. In addition the band gap of CdSe is analyzed using DFT within LDA, GGA, DFT+U and PBE0 to approximate the unknown exchange correlation functional. The band gap values obtained using LDA and GGA are severally under estimated due to their poor approximation of exchange-correlation potential. This problem was improved by using projector augmented-wave pseudopotential within Hubbard-correction (GGA+U) and the hybrid functional approximation. Optical properties: complex and real parts of dielectric function, energy loss spectrum and absorption coefficient of CdSe in both ZB and WZ phase were studied.

Renormalization of higher-dimensional operators from on-shell amplitudes

On-shell amplitude methods allow to derive one-loop renormalization effects from just tree-level amplitudes, with no need of loop calculations. We derive a simple formula to obtain the anomalous dimensions of higher-dimensional operators from a product of tree-level amplitudes. We show how this works for dimension-6 operators of the Standard Model, providing explicit examples of the simplicity, elegance and efficiency of the method. Many anomalous dimensions can be calculated from the same Standard Model tree-level amplitude, displaying the attractive recycling aspect of the on-shell method. With this method, it is possible to relate anomalous dimensions that in the Feynman approach arise from very different diagrams, and obtain non-trivial checks of their relative coefficients. We compare our results to those in the literature, where ordinary methods have been applied.

Electronic, Structural, and Optical Properties of Zinc Blende and Wurtzite Cadmium Sulfide (CdS) Using Density Functional Theory

Zinc blende (zb) and wurtzite (wz) structure of cadmium sulfide (CdS) are analyzed using density functional theory within local density approximation (LDA), generalized gradient approximation (GGA), Hubbard correction (GGA + U), and hybrid functional approximation (PBE0 or HSE06). To assure the accuracy of calculation, the convergence test of total energy with respect to energy cutoff and k-point sampling is performed. The relaxed atomic position for the CdS in zb and wz structure is obtained by using total energy and force minimization method following the Hellmann–Feynman approach. The structural optimization and electronic band structure properties of CdS are investigated. Analysis of the results shows that LDA and GGA underestimate the bandgap due to their poor approximation of exchange-correlation functional. However, the Hubbard correction to GGA and the hybrid functional approximation give a good bandgap value which is comparable to the experimental result. Moreover, the optical properties such as real and imaginary parts of the dielectric function, the absorption coefficient, and the energy loss function of CdS are determined.

Iterative formulation for the skin effect on cylindrical conductors based on the Feynman's approach

AbstractOnly qualitative aspects on the skin effect in conductors are properly described in the technical literature on power systems applications. Although the qualitative analysis is an important issue, it is not sufficient for a full approach and proper understanding of this electromagnetic phenomenon in power systems, more specifically in power transmission systems. The present work intends to provide a quantitative analysis of the skin effect on conductors based exclusively on the Maxwell Equations, whereby both physical and mathematical treatments of the phenomenon are properly analysed. This approach contrasts with the traditional one that uses a complex formulation based on both the differential Maxwell equations and potential concepts. Finally, a discussion on the application of the Poynting theorem on the external surface of the cylindrical conductor is presented in order to evaluate not only the resistance, but also the inductance as a function of the frequency, which is not properly discussed in the technical literature on power systems.

Perturbative four-point functions from the analytic conformal bootstrap

We apply the analytic conformal bootstrap method to study weakly coupled conformal gauge theories in four dimensions. We employ twist conformal blocks to find the most general form of the one-loop four-point correlation function of identical scalar operators, without any reference to Feynman calculations. The method relies only on symmetries of the model. In particular, it does not require introducing any regularisation and it is free from the redundancies usually associated with the Feynman approach. By supplementing the general solution with known data for a small number of operators, we recover explicit forms of one-loop correlation functions of four Konishi operators as well as of four half-BPS operators {{mathcal{O}}_{20}}_{prime } in mathcal{N} = 4 super Yang-Mills.

One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint Φ3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach.

Physical reality of electromagnetic potentials and the classical limit of the Aharonov–Bohm effect

Recent literature on the Aharonov-Bohm effect has raised fundamental questions on the classical correspondence of this effect and the physical reality of the electromagnetic potentials in quantum mechanics. Reappraisal on Feynman's approach to the classical limit of AB effect is presented. The critique throws light on the significance of quantum interference and quantum phase shifts in any such classical correspondence. Detailed analysis shows that Feynman arguments are untenable on physical grounds and the claim made in the original AB paper that this effect had no classical analog seems valid. The importance of nonintegrable phase factor distinct from the AB phase factor, here termed as Fock-London-Weyl phase factor for the historical reasons, is underlined in connection with the classical aspects/limits. A topological approach incorporating the physical significance of the interaction field momentum is proposed. A new idea emerges from this approach that attributes the origin of the AB effect to the exchange of modular angular momentum.

A sum-over-paths approach to one-dimensional time-independent quantum systems

We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.

Tunable Polarons of Slow-Light Polaritons in a Two-Dimensional Bose-Einstein Condensate

When an impurity interacts with a bath of phonons it forms a polaron. For increasing interaction strengths the mass of the polaron increases and it can become self-trapped. For impurity atoms inside an atomic Bose-Einstein condensate (BEC) the nature of this transition is not understood. While Feynman's variational approach to the Fröhlich model predicts a sharp transition for light impurities, renormalization group studies always predict an extended intermediate-coupling region characterized by large phonon correlations. To investigate this intricate regime and to test polaron physics beyond the validity of the Fröhlich model we suggest a versatile experimental setup that allows us to tune both the mass of the impurity and its interactions with the BEC. The impurity is realized as a dark-state polariton (DSP) inside a quasi-two-dimensional BEC. We show that its interactions with the Bogoliubov phonons lead to photonic polarons, described by the Bogoliubov-Fröhlich Hamiltonian, and make theoretical predictions using an extension of a recently introduced renormalization group approach to Fröhlich polarons.

Occupation times of the Ornstein–Uhlenbeck process: Functional PCA and evidence from electricity prices

We discuss the functional principal component analysis (FPCA) of the occupation times of the Ornstein–Uhlenbeck process. For the eigenvalue problem of the covariance operator of the occupation times we derive the corresponding integral equation in the large time limit and we solve numerically for the principal components. The formulation applies the path-integral approach of Feynman and Kac. The principal components are compared with those from empirical electricity price processes on energy markets. The results indicate that FPCA of the occupation times is a suitable tool in stochastic energy modeling to generate moderately-sized scenario trees.

The Lagrangian and Hamiltonian Aspects of the Electrodynamic Vacuum-Field Theory Models

The important classical Ampere’s and Lorentz laws derivations are revisited and their relationships with the modern vacuum field theory approach to modern relativistic electrodynamics are demonstrated. The relativistic models of the vacuum field medium and charged point particle dynamics as well as related classical electrodynamics problems jointly with the fundamental principles, characterizing the electrodynamical vacuum-field structure, based on the developed field theory concepts are reviewed and analyzed detail. There is also described a new approach to the classical Maxwell theory based on the derived and newly interpreted basic equations making use of the vacuum field theory approach. There are obtained the main classical special relativity theory relationships and their new explanations. The well known Feynman approach to Maxwell electromagnetic equations and the Lorentz type force derivation is also. A related charged point particle dynamics and a hadronic string model analysis is also presented. We also revisited and reanalyzed the classical Lorentz force expression in arbitrary non-inertial reference frames and present some new interpretations of the relations between special relativity theory and its quantum mechanical aspects. Some results related with the charge particle radiation problem and the magnetic potential topological aspects are discussed. The electromagnetic Dirac-Fock-Podolsky problem of the Maxwell and Yang-Mills type dynamical systems is analyzed within the classical Dirac-Marsden-Weinstein symplectic reduction theory. Based on the Gelfand-Vilenkin representation theory of infinite dimensional groups and the Goldin-Menikoff-Sharp theory of generating Bogolubov type functionals the problem of constructing Fock type representations and retrieving their creation-annihilation operator structure is analyzed. An application of the suitable current algebra representation to describing the non-relativistic Aharonov-Bohm paradox is demonstrated. The current algebra coherent functional representations are constructed and their importance subject to the linearization problem of nonlinear dynamical systems in Hilbert spaces is also presented.

On the foundations of quantum physics

Some aspects of the interpretation of microscopic physics in terms of quantum theory are discussed. It is first emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutation relations between ''canonically conjugated'' coordinate and momentum operators leads to a wrong version of quantum mechanics. In this connection the Feynman integral formalism is also discussed. In this formalism the measure is not well defined, and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one. In this respect, the Feynman approach consists of a mnemonic rule to generate perturbation series from an undefined zero-order term. The origin of time in the quantum framework is then analyzed in detail and illustrated by the example of atomic collisions. It is shown that the time-dependent Schr ¨ odinger equation for the closed three-body (two nuclei + electron) system has no physical meaning because in the high-impact energy limit it transforms into an equation with two independent time-like variables; time naturally appears in the stationary Schr ¨ odinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Finally, following the well-known Einstein-Rosen-Podolsky experiment and Bell's inequality, we reiterate that the wave function must be interpreted as an actual field of information, in a form as elementary as the usual material particles and electromagnetic fields. In fact, experimental measurements transfer this quantum information field into the classical world, which is directly discernable. In my conclusion, the relation between physical reality and its mathematical formulation is discussed. 2012 Physics Essays Publication. (DOI: 10.4006/0836-1398-25.1.27) R ´

Electrodynamics onκ-Minkowski space-time

In this paper, we derive Lorentz force and Maxwell's equations on kappa-Minkowski space-time up to the first order in the deformation parameter. This is done by elevating the principle of minimal coupling to non-commutative space-time. We also show the equivalence of minimal coupling prescription and Feynman's approach. It is shown that the motion in kappa space-time can be interpreted as motion in a background gravitational field, which is induced by this non-commutativity. In the static limit, the effect of kappa deformation is to scale the electric charge. We also show that the laws of electrodynamics depend on the mass of the charged particle, in kappa space-time.