Nonrelativistic Problem Research Articles

Unraveling a Cavity-Induced Molecular Polarization Mechanism from Collective Vibrational Strong Coupling.

We demonstrate that collective vibrational strong coupling of molecules in thermal equilibrium can give rise to significant local electronic polarizations in the thermodynamic limit. We do so by first showing that the full nonrelativistic Pauli-Fierz problem of an ensemble of strongly coupled molecules in the dilute-gas limit reduces in the cavity Born-Oppenheimer approximation to a cavity-Hartree equation for the electronic structure. Consequently, each individual molecule experiences a self-consistent coupling to the dipoles of all other molecules, which amount to non-negligible values in the thermodynamic limit (large ensembles). Thus, collective vibrational strong coupling can alter individual molecules strongly for localized "hotspots" within the ensemble. Moreover, the discovered cavity-induced polarization pattern possesses a zero net polarization, which resembles a continuous form of a spin glass (or better polarization glass). Our findings suggest that the thorough understanding of polaritonic chemistry, requires a self-consistent treatment of dressed electronic structure, which can give rise to numerous, so far overlooked, physical mechanisms.

Nonrelativistic Quantum Mechanical Problem for the Cornell Potential in Lobachevsky Space

In Friedmann–Lobachevsky space-time with a radius of curvature slowly varying over time, we study numerically the problem of motion of a particle moving in the Cornell potential. The mass of the particle is taken to be a reduced mass of the charmonium system. In contrast to the similar problem in flat space, in Lobachevsky space the Cornell potential has a finite depth and, as a consequence, the number of bound states of the system is finite and motion with a continuum energy spectrum is also possible. In this paper, we study the bound states as well as the scattering states of the system.

Квантование энергетических уровней в 1D квантовом колодце в случае мгновенных стационарных состояний при нерелятивистском движении стенки и микрочастицы

The paper considers the problem of finding energy levels in the 1D quantum well in case of its width alteration at the nonrelativistic rate. According to the reviewed literature, the exact solution is known only in the case of nonrelativistic motion of the 1D quantum well wall at the constant rate. It is shown that motion with the constant rate is physically unrealizable. Therefore, it is necessary to find at least small areas of the Schrodinger equation solution for a wider range of nonrelativistic alterations in the 1D quantum well width. Analysis results presented in the study show existence of such areas. The found areas correspond to the instantaneous stationary states satisfying the Bohr quantization condition. In this case, the Dirichlet condition is also satisfied on the moving wall. It means that in this case energy of the level with the n number also becomes a function of the k second quantum number, which takes into account dynamic alteration in the 1D quantum well width. Variants were found of the k second quantum number spectrum and of the quantum level spectrum in various cases of the wall continuous motion with zero initial speed and finite acceleration. Within the framework of the analysis used, formulas were obtained to change the difference between energies of the two arbitrary levels. An analysis was made for the boundaries of the wall speed and the 1D quantum well width in considering the nonrelativistic problem. The obtained results and their possible applications are under discussion, including analysis of the problems related to nanotechnology

New Bound State Solutions of 3-Dimensional Modified Eckart Potential Plus a New Modified Deformed Hylleraas Potential in RNCQM and NRNCQM Symmetries

Abstract: In this study, the solution of the deformed Klein-Gordon and deformed Schrödinger equations (DKGE and DSE for short) for the modified Eckart potential plus a new modified deformed Hylleraas potential (MEPNMDHP, for short) with the improved approximation to deal with the centrifugal term is investigated using Bopp’s shift and standard perturbation theory methods in the symmetries of relativistic noncommutative quantum mechanics RNCQM and nonrelativistic noncommutative quantum mechanics NRNCQM. To the best of our knowledge, this problem is examined in literature in usual RQM and NRQM. The new potential suggested describes some selected diatomic molecules such as the homogeneous diatomic molecules (N2 and H2) and heterogeneous diatomic molecules (HCl, HBr, SO, NO, and HI). The new values that we get appeared sensitive to the quantum numbers (j,s,m) in addition to the usual states’ numbers (n,l), the potential depths of the potential (V_0,V_1,V_2), the range of the potential α and noncommutativity parameters (Θ,σ,χ). We have highlighted three physical phenomena that automatically generate a result of the topological properties of non-commutativity. The first physical phenomenon is the perturbative spin-orbit coupling, the second is the magnetic induction, and the third is the rotational proper phenomenon. In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of quantum field theory and quantum mechanics. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for l=0 and l≠0, so the approximate solutions are available. We have observed that the DKGE under the MEPNMDHP has a physical behavior similar to the Duffin-Kemmer equation that can describe the dynamic state of a particle with spin-1 in the symmetries of RNCQM. Four special cases; i.e., l wave is investigated in the context of DKGE and Schrödinger theories. The new relativistic and nonrelativistic energy for some potentials, such as only modified Eckart potential and only new modified Hylleraas potential, have also been obtained by varying some potential parameters. Keywords: Klein-Gordon equation, Schrödinger equation, Eckart potentials, Modified Hylleraas potential, Diatomic molecules, Noncommutative geometry, Star products, Bopp's shift method. PACs: 03.65. Ge; 03.65. Pm; 03.65.−w, 32.30 – r, 02.30. Gp.

Nonrelativistic and relativistic simple wave problems solved with Smoothed Particle Hydrodynamics

To simulate the expansion of the matter created in relativistic nuclear collisions, codes in 3+1 dimensions are used and we are developing a new one. To benchmark such codes, the Sod’s shock tube is often used. A closely related problem is the one-dimensional expansion of a gas into vacuum. In this paper, we study this problem classically and relativistically with the Smoothed Particle Method and test various techniques to improve the precision and speed of the solution.

Role of the Electromagnetic Vacuum in the Transition from Classical to Quantum Mechanics

We revisit the nonrelativistic problem of a bound, charged particle subject to the random zero-point radiation field (zpf), with the purpose of revealing the mechanism that takes it from the initially classical description to the final quantum-mechanical one. The combined effect of the zpf and the radiation reaction force results, after a characteristic time lapse, in the loss of the initial conditions and the concomitant irreversible transition of the dynamics to a stationary regime controlled by the field. In this regime, the canonical variables x, p become expressed in terms of the dipolar response functions to a set of field modes. A proper ordering of the response coefficients leads to the matrix representation of quantum mechanics, as was proposed in the early days of the theory, and to the basic commutator \(\left[ {\hat{x}},{\hat{p}}\right] =i\hbar \). Further, the connection with the corresponding Fokker–Planck equation valid in the Markov approximation, allows one to obtain the (nonrelativistic) radiative corrections of qed. These results reaffirm the essentially electrodynamic and stochastic nature of the quantum phenomenon, as proposed by stochastic electrodynamics.

A pedagogical approach to relativity effects in quantum mechanics

A simple but precise approach to relativistic quantum mechanics is presented. The approach is based on the use of a Schrödinger-like, little-known but well-studied quantum mechanics wave equation. Such formal similitude allows undergraduate students to quantitatively explore how the results corresponding to a typical non-relativistic quantum problem change when the particle is moving at relativistic speeds. No additional mathematical skills are required. We argue in favor of the academic use of this approach for including the implications of the special theory of relativity in introductory quantum mechanics courses.

Diatomic Molecules with the Improved Deformed Generalized Deng–Fan Potential Plus Deformed Eckart Potential Model through the Solutions of the Modified Klein–Gordon and Schrödinger Equations within NCQM Symmetries

In this study, the deformed Klein–Gordon equation and Schr¨odinger equations were solved with the improved deformed generalized Deng–Fan potential plus the deformed Eckart potential (IDGDFDE-P, in short) model using Bopp’s shift and standard perturbation theory methods in the symmetries of extended quantum mechanics. By employing the improved approximation to the centrifugal term, the relativistic and nonrelativistic bound-state energies are obtained for some selected diatomic molecules such as N2, I2, HCl, CH, LiH, and CO. The relativistic energy shift ΔEtotdfe (n, a, c, b, V0, V1, V2, Θ, σ, χ, j, l, s, m) and the perturbative nonrelativistic corrections ΔEnrdfe (n, α, c, b, V0, V1, V2, Θ, σ, χ, j, l, s, m) appeared as functions of the parameters (α, c, b, V0, V1, V2) and the parameters of noncommutativity (Θ, σ, χ), in addition to the atomic quantum numbers (n, j, l, s, m). In both relativistic and nonrelativistic problems, we show that the corrections to the energy spectrum are smaller than for the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present results under the IDGDFDE-P model is are consistent with what is obtained in the literature. In the new symmetries of noncommutative quantum mechanics (NCQM), it is not possible to get the exact analytical solutions for l = 0 and l ̸ = 0. Only the approximate ones can be obtained. We have clearly shown that the Schr¨odinger and Klein–Gordon equations in the new symmetries can physically describe two Dirac equations and the Duffin–Kemmer equation within the IDGDFDE-P model in the extended symmetries.

Diatomic molecules and fermionic particles with improved Hellmann-generalized Morse potential through the solutions of the deformed Klein-Gordon, Dirac and Schrödinger equations in extended relativistic quantum mechanics and extended nonrelativistic quantum mechanics symmetries

In this paper, we investigate the new approximate bound state solution of deformed Klein--Gordon, Dirac and Schr\"{o}dinger equations in the symmetries of extended relativistic quantum mechanics ERQM and extended nonrelativistic quantum mechanics ENRQM have been obtained with a newly proposed potential called improved Hellmann-generalized Morse potential (IHGMP, for short). To the best of our knowledge, this problem is examined in literature in the usual RQM and NRQM with Hellmann-generalized Morse potential. The potential is a superposition of Hellmann potential, generalized Morse or Deng-Fan potential, and some other exponential terms. By employing the improved approximation to deal with the centrifugal term, Bopp's shift and standard perturbation theory method. The new approximate analytical energy shift and the corrections of bound state energyeigenvalues in ERQM and ENRQM are obtained for some selected diatomic molecules such as (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF). The new values that we get appeared sensitive to the quantum numbers $% \left( j,l,s,m\right) $, the potential depths of the improved Hellmann-generalized Morse potential ($a,b$), the range of the potential $%\alpha $, the dissociation energy $D_{e}$, the equilibrium bond length $%r_{e} $, and noncommutativity parameters$\left( \Theta ,\sigma ,\chi \right)$ . We have highlighted three physical phenomena that automatically generate a result of the topological properties of noncommutativity, the first physical phenomena are the perturbative spin-orbit coupling, the second the magnetic induction while the third corresponds to the rotational proper phenomena. In both relativistic and nonrelativistic problems, we show thatthe corrections on the spectrum energy are smaller than the main energy in the ordinary cases of quantum field theory and quantum mechanics. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for $l=0$ and $l\neq 0$, the approximate solutions are available. Four special cases, i.e., l wave are investigated in the context of deformed Klien-Gordon and Schr\"{o}dinger theories. The relativistic energy equations and the new nonrelativistic energy for some potentials such as improved Hellmann potential and improved generalized Morse potential have also been obtained by varying some potential parameters. We have clearly shown that the Schr\"{o}dinger and Klein Gordon equations in the new symmetries canphysically describe each of the two Dirac equations and the Duffin--Kemmer equation under the effect of IHGMP.

A New Theoretical Investigation of the Modified Equal Scalar and Vector Manning–Rosen Plus Quadratic Yukawa Potential Within the Deformed Klein–Gordon and Schrödinger Equations Using the Improved Approximation of the Centrifugal Term and Bopp’s Shift Method in RNCQM and NRNCQM Symmetries

We employed the elegant tool of Bopp’s shift and standard perturbation theory methods to obtain a new relativistic and nonrelativistic approximate bound state solution of deformed Klein–Gordon DKG and deformed Schrödinger DSE equations using the modified equal vector scalar Manning–Rosen plus quadratic Yukawa potential (MVSMQY-Ps) model. Furthermore, we have employed the improved approximation to the centrifugal term for some selected diatomic molecules such as N2, I2, HCl, CH and LiH in the symmetries of extended quantum mechanics to obtain the approximate solutions. The relativistic shift energy [Formula: see text] and the perturbative nonrelativistic corrections [Formula: see text] appeared as a function of the parameters ([Formula: see text]), the parameters of noncommutativity ([Formula: see text]), in addition to the atomic quantum numbers ([Formula: see text]. In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under MVSMQAY-Ps is consistent with what is obtained in the literature. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for [Formula: see text] and [Formula: see text], and can only be solved approximately. We have observed that the DKGE under the MVSMQY-Ps model has a physical behavior similar to the Duffin–Kemmer equation for meson with spin-1; it can describe a dynamic state of a particle with spin-1 in the symmetries of RNCQM. Moreover, we have treated composite systems such as molecules made of [Formula: see text] particles of masses [Formula: see text] in the frame of noncommutative algebra. The NRNCQM and RNCQM results obtained within Bopp’s shift and standard perturbation theory methods overlap entirely with the results obtained by ordinary NCQM, and it displays that the theoretical investigation in this study is excellent.

Bound States for the Spin-1/2 Aharonov-Bohm Problem in a Rotating Frame

In this paper, we study the effects of rotation in the spin-1/2 non-relativistic Aharonov-Bohm problem for bound states. We use a technique based on the self-adjoint extension method and determine an expression for the energies of the bound states. The inclusion of the spin element in the Hamiltonian guarantees the existence of bound state solutions. We perform a numerical analysis of the energies and verify that both rotation and the spin degree of freedom affect the energies of the particle. The main effect we observe in this analysis is a cutoff value manifested in the Aharonov-Bohm flux parameter that delimits the values for the positive and negative energies.

New relativistic and nonrelativistic model of diatomic molecules and fermionic particles interacting with improved modified Mobius potential in the framework of noncommutative quantum mechanics symmetries

In this study, the deformed Klein–Gordon equation, Dirac equation, and Schrödinger equations were solved with the improved the modified Mobius square potential model (IMSPM, in short) using Bopp’s shift and standard perturbation theory methods in the symmetries of extended quantum mechanics. By employing the improved approximation to the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some selected diatomic molecules (H2, I2, CO, NO, and HCl). The relativistic shift energy ΔEmsptot(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) and the perturbative nonrelativistic corrections ΔEmspnr(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) appeared as a function of the parameters (α,A,B,V0), the parameters of noncommutativity (Θ,σ,χ), in addition to the atomic quantum numbers (n,j,l,s,m). In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under the improved modified Mobius square potential model is consistent with what is obtained in the literature. In the new symmetries of NCQM, is not possible to get the exact analytical solutions for l=0 and l≠0 can only be solved approximately. Through this research, we came to two noteworthy results, the first is related to the deformed Klein-Gordon equation under the influence of the improved modified Mobius square potential model becomes equivalent to the Duffin-Kemmer equation for a meson with spin-s, while the second result concerns the deformed Schrödinger equation, which can now describe the state of high-energy fermionic particles similar to the Dirac equation in the literature.

A new theoretical study of the deformed unequal scalar and vector Hellmann plus modified Kratzer potentials within the deformed Klein–Gordon equation in RNCQM symmetries

In this paper, within the framework of relativistic quantum mechanics and using the improved approximation scheme to the centrifugal term for any [Formula: see text]states via Bopp’s shift method and standard perturbation theory, we have obtained the modified energy eigenvalues of a newly proposed modified unequal vector and scalar Hellmann plus modified Kratzer potentials (DUVSHMK-Ps) for some diatomic N2, I2, CO, NO, O2 and HCl molecules. This study includes corrections of the first-order in noncommutativity parameters [Formula: see text]. This potential is a superposition of the attractive Coulomb Yukawa potential plus the Kratzer potential and new central terms appear as a result of the effects of noncommutativity properties of space–space. The obtained energy eigenvalues appear as a function of noncommutativity parameters, the strength parameters [Formula: see text] and [Formula: see text] of the (scalar vector) Hellmann potential, the screening range parameter [Formula: see text], the dissociation energy of the vector, and scalar potential [Formula: see text], the equilibrium inter-nuclear distance [Formula: see text] in addition to the atomic quantum numbers [Formula: see text]. Furthermore, we obtained the corresponding modified energy of DUVSHMK-Ps in the symmetries of non-relativistic noncommutative quantum mechanics (NRNCQM). In both relativistic and non-relativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM.

A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries

In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.

Two-body bound states through Yukawa forces and perspectives on hydrogen and deuterium

We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb force arises in the limit of massless mediators such as photons. We use the solutions to explain several features of deuterium. We also present heuristic estimates of ground state energies and energy level degeneracy breaking. Studying this Yukawa two-body system provides insights into hydrogen, deuterium, and two-body bound states, generally. We describe our undergraduate-accessible procedure for numerically solving non-relativistic two-body bound state problems with mathematica in the appendix.

Approximate Solutions to the Klein-Fock-Gordon Equation for the Sum of Coulomb and Ring-Shaped-Like Potentials

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass M, described by the Klein-Fock-Gordon equation with equal scalar Sr→ and vector Vr→ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at E<Mc2 and a continuous at E>Mc2 energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group SU1,1 for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞ go over into the corresponding expressions for the nonrelativistic problem.

K-Shell photoionization cross sections of one-electron uranium cation by relativistic complex basis function method

The complex basis function (CBF) method is a square-integrable (L2) function method that represents continuum wave functions, and has been applied successfully to various problems, including the calculation of resonance state energies and photoionization cross sections. The previous applications of this method have been limited to non-relativistic problems, and no serious attempt has been made for the relativistic Hamiltonian. In this study, we calculated the photoionization cross sections from the imaginary part of the frequency-dependent polarizability by applying the CBF method to the inhomogeneous Dirac equation of the one-electron uranium cation. The results obtained by employing a pure Coulomb potential to the K-shell photoionization cross sections were in good agreement with previous theoretical values and experimental values for the neutral uranium atom. Phase shifts were obtained by matching the CBF solutions with the numerical ones, which were further propagated by the fourth-order Runge–Kutta method, and thus, the photoionization differential and total cross sections were successfully calculated. The calculated results were analyzed from various viewpoints, including the multipole effect of electromagnetic fields, momentum conservation of photon and photoelectron, interference of multiple perturbation terms, and difference in the orbital angular momentum of the small component wave functions. These analyses demonstrate the effectiveness of the CBF method for the inhomogeneous Dirac equation and thus its applicability to multielectron systems.

Triply heavy baryons in the constituent quark model * * Supported by China Postdoctoral Science Foundation (2019M650617), National Natural Science Foundation of China (11535005, 11775118), and by Spanish Ministerio de Economía, Industria y Competitividad under contractsno. FPA2017-86380-P and FPA2016-77177-C2-2-P. P.G.O. acknowledges the financial support from Spanish MINECO's Juan de la Cierva-Incorporación programme, Grant Agreement No. IJCI-2016-28525.

The constituent quark model is used to compute the ground and excited state masses of QQQbaryons containing either c or bquarks. The quark model parameters previously used to describe the properties of charmonium and bottomonium states were used in this analysis. The non-relativistic three-body bound state problem is solved by means of the Gaussian expansion method which provides sufficient accuracy and simplifies the subsequent evaluation of the matrix elements. Several low-lying states with quantum numbers and are reported. We compare the results with those obtained by the other theoretical formalisms. There is a general agreement for the mass of the ground state in each sector of triply heavy baryons. However, the situation is more puzzling for the excited states, and appropriate comments about the most relevant features of our comparison are given.

Nonrelativistic potential well problem in GUP formalism: Laplace transform approach

The Schrodinger equation is considered in the minimal length formalism with the potential well problem. The equation, in an approximate scheme, appears in fourth-order form, which has not been extensively discussed in the literature. We transform the problem into Laplace space and the wave functions as well as energy spectra of the modified Schrodinger equation are calculated using the basic properties of this integral transform. The bound and continuous states are discussed in full detail.

Landau-Pomeranchuk-Migdal effect in sequential bremsstrahlung: From large- N QCD to N=3 via the SU(N) analog of Wigner 6‐j symbols

Consider a high-energy parton showering as it traverses a QCD medium such as a quark-gluon plasma. Interference effects between successive splittings in the shower are potentially very important but have so far been calculated (even in idealized theoretical situations) only in soft emission or large-$N_{\rm c}$ limits, where $N_{\rm c}$ is the number of quark colors. In this paper, we show how one may remove the assumption of large $N_{\rm c}$ and so begin investigation of $N_{\rm c}{=}3$ without soft-emission approximations. Treating finite $N_{\rm c}$ requires (i) classifying different ways that four gluons can form a color singlet and (ii) calculating medium-induced transitions between those singlets, for which we find application of results for the generalization of Wigner 6-$j$ symbols from angular momentum to SU($N_{\rm c}$). Throughout, we make use of the multiple scattering ($\hat q$) approximation for high-energy partons crossing quark-gluon plasmas, and we find that this approximation is self-consistent only if the transverse-momentum diffusion parameter $\hat q$ for different color representations satisfies Casimir scaling (even for strongly-coupled, and not just weakly-coupled, quark-gluon plasmas). We also find that results for $N_{\rm c}{=}3$ depend, mathematically, on being able to calculate the propagator for a coupled non-relativistic quantum harmonic oscillator problem in which the spring constants are operators acting on a 5-dimensional Hilbert space of internal color states. Those spring constants are represented by constant $5{\times}5$ matrices, which we explicitly construct. We are unaware of any closed form solution for this type of harmonic oscillator problem, and we discuss prospects for using numerical evaluation.