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Abstract The focus of this paper is to investigate the behavior of a codimension-one thick brane in the modified symmetric teleparallel gravity structure $$f(Q, {\mathcal {T}})$$ f ( Q , T ) , where Q is the nonmetricity scalar and $${\mathcal {T}}$$ T is the trace of the energy–momentum tensor. Additionally, we examine whether braneworld scenarios are stable under small tensor perturbations and identify the associated massive and resonant modes. Our results indicate that the parameters controlling the modifications to the gravitational model-specifically those affecting nonmetricity and the energy–momentum tensor-significantly influence both the location and existence of these modes.

Abstract This study aims into brane-world scenarios within the framework of f(R, A) gravity called Ricci-inverse modified theory, exploring the dynamics of spin-1/2 fermions coupled non-minimally to background scalar field functions. Through theoretical analysis and numerical techniques, we investigate the localization of both massless and massive fermionic modes on the brane. Our findings shows intricate phenomena, including unexpected configurations of the effective potentials and the identification of resonance modes in massive fermionic spectra. These insights shed light on the interplay between modified gravity and matter in extra-dimensional models, offering promising avenues for experimental validation and furthering our understanding of fundamental physics.

Abstract In our research, we explore the impact of a geometric non-minimal coupling between fermions and curvature scalars on the positioning of fermions within a brane structure. Our investigation encompasses massless fermionic modes and massive modes, revealing that specifically fermionic modes with left chirality are situated within the brane. This insight allows us to gain a more direct understanding of how the system’s geometry influences the placement of fermion fields on the brane. To delve deeper into the precise localization of massless fermionic modes, we employ entropic information measurements. By varying the parameters that govern the curvature scalars, we observe the ability to adjust the localization of these fermionic modes on the brane. Importantly, the BBM uncertainty relation remains satisfied in all scenarios, particularly for $$f_{1,2,3}$$ f 1 , 2 , 3 . We leverage the Shannon entropy as a highly effective metric to yield accurate results regarding the positioning of massless fermionic modes within our model. In the case of massive fermionic modes, we utilize relative probability analysis to pinpoint the resonant modes, and we identify the existence of these resonant modes in both $$f_{1,2,3}$$ f 1 , 2 , 3 cases. Ultimately, we ascertain that these parameters influence the placement of resonant modes by increasing the amplitude of the massive modes in the core of the brane.

We delve into the intricacies of the hyperbolic Pöschl–Teller potential, often referred to as the 2nd Pöschl–Teller potential (PTP), and explore its implications on the Schrödinger equation for arbitrary l state case. This exploration encompasses a comprehensive review of various methodologies employed in studying the exact solutions of the S wave scenario, along with approximations concerning the centrifugal term within the radial equation for the arbitrary l setup. Our investigation yields exact solutions for the quasi-exact arbitrary l state model. We achieve this by employing a novel approach rooted in Fuchsian differential equations, thereby presenting a potential solution for the S wave case as well. This innovative method holds promise, particularly in seeking solutions for the Schrödinger equation involving non-trivial potentials where exact solutions remain elusive.

In this work, we use a deformed version of the Heisenberg–Weyl algebra in the Dicke Model to incorporate a Kerr medium and nonlinear interaction between matter and radiation fields. The deformed algebra allows us to define two distinct nonlinear coherent states for the radiation field. These coherent states, along with the conventional Heisenberg–Weyl coherent states and their symmetry-adapted versions, are employed as trial variational states in the energy-surface minimisation method. The obtained results reveal that the Kerr medium introduces a delay in reaching the super-radiant region, with the delay becoming more pronounced as the nonlinear parameter increases. To evaluate the different approaches, we analyse the behaviour of the critical coupling, ground-state energy, entropy of entanglement between matter and radiation, and fidelity between the numerical solution and the symmetry-adapted solution. We find that one particular deformed coherent state outperforms the other trial states in approximating the ground state of the system, providing an accurate description of the critical behaviour in the proposed nonlinear model.

The He–McKellar–Wilkens (HMW) effect in noncommutative space has been explored through two distinct methodologies. One approach treats the neutral particle, which harbors a permanent electric dipole moment, as an unstructured entity, while the other approach considers the neutral particle as a composite system consisting of a pair of oppositely charged particles. To preserve gauge symmetry, we apply the Seiberg–Witten map. Surprisingly, both of these approaches converge on the same result. They independently confirm that, up to the first order of the noncommutative parameter (NCP), no corrections are observed in the phase of the HMW effect. Remarkably, these two approaches, although founded on fundamentally different mechanisms, yield identical conclusions.

In this paper, we employ PCA and t-SNE analyses to gain deeper insights into the behavior of entangled and non-entangled mixing operators within the Quantum Approximate Optimization Algorithm (QAOA) at various depths. We utilize a dataset containing optimized parameters generated for max-cut problems with cyclic and complete configurations. This dataset encompasses the resulting RZ, RX, and RY parameters for QAOA models at different depths (1L, 2L, and 3L) with or without an entanglement stage within the mixing operator. Our findings reveal distinct behaviors when processing the different parameters with PCA and t-SNE. Specifically, most of the entangled QAOA models demonstrate an enhanced capacity to preserve information in the mapping, along with a greater level of correlated information detectable by PCA and t-SNE. Analyzing the overall mapping results, a clear differentiation emerges between entangled and non-entangled models. This distinction is quantified numerically through explained variance in PCA and Kullback–Leibler divergence (post-optimization) in t-SNE. These disparities are also visually evident in the mapping data produced by both methods, with certain entangled QAOA models displaying clustering effects in both visualization techniques.

Recent investigations have suggested that the use of non-classical states of light, such as entangled photon pairs, may open new and exciting avenues in experimental two-photon absorption spectroscopy. Despite several experimental studies of entangled two-photon absorption (eTPA), there is still a heated debate on whether eTPA has truly been observed. This interesting debate has arisen mainly because it has recently been argued that single-photon-loss mechanisms, such as scattering or hot-band absorption, may mimic the expected entangled-photon linear absorption behavior. In this work, we focus on transmission measurements of eTPA and explore three different two-photon quantum interferometers in the context of assessing eTPA. We demonstrate that the so-called N00N-state configuration is the only one among those considered insensitive to linear (single-photon) losses. Remarkably, our results show that N00N states may become a potentially powerful tool for quantum spectroscopy, placing them as a strong candidate for the certification of eTPA in an arbitrary sample.

The Gaussian solitary wave solutions are obtained for a class of logarithmic nonlinear Schrödinger equation (NLSE) with a standard harmonic oscillator potential. We find that the Gaussian solution does not exist at all in the straight line x=vt, but it does in the curve x=vt. We illustrate the behavior of the Gaussons and analyze the correlations with respect to the relevant parameters. The logarithmic nonlinearity as an imaginary term enables us to get the associated advection equation, thus showing in a novel way the existence of a Gaussian solutions in the curve x=vt. Moreover, we generalize the present study to the high-dimensional (1 + n) NLSE case. These results will enrich the current literature on this nonlinear equation.

In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials U0,3 exhibit greater localization compared to the double-well potentials U1,2. This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki–Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths u¯. An intriguing observation is that the sum of position and momentum entropies increases with the variable u¯ for potentials U1,2,3, while for U0, the sum decreases with u¯. Additionally, the sum of the cases U0 and U3 almost remains constant within the relative value 0.01 as u¯ increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy F¯x and find that it increases with the depth u¯ of the potential wells but F¯p decreases with the depth.

Wigner–Dunkl quantum mechanics can be viewed as deformed quantum mechanics, where the commutator between canonical momentum and position operators contains in addition a reflection operator. In the present work we carry over the general concepts of supersymmetric (SUSY) quantum mechanics to the Wigner–Dunkl quantum formulation. We investigate the SUSY ground states, the SUSY transformations and introduce a generalized shape invariance allowing for explicit solutions. In this way, exact solutions for a certain class of SUSY potentials can be provided. As example the harmonic oscillator is considered. We also take a look into a singular oscillator model, which does not belong to the shape-invariance class. Nevertheless, exact solutions can be provided, showing the peculiarities of that model in the SUSY Wigner–Dunkl quantum mechanics. A modification of that singular model is presented which restores shape invariance

In this study, we investigate the position and momentum Shannon entropy, denoted as Sx and Sp, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by k in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, ρs(x), and the momentum entropy density, ρs(p), for low-lying states. Specifically, as the fractional derivative k decreases, ρs(x) becomes more localized, whereas ρs(p) becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy Sx decreases, while the momentum entropy Sp increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative k. It is noteworthy that, despite the increase in position Shannon entropy Sx and the decrease in momentum Shannon entropy Sp with an increase in the depth u of the HDWP, the Beckner–Bialynicki-Birula–Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth u of the HDWP and the fractional derivative k. Our results indicate that the Fisher entropy increases as the depth u of the HDWP is increased and the fractional derivative k is decreased.

The Schrödinger equation under the application of a position-dependent mass (PDM) with an exponential form is presented. Several physical models are carried out by choosing different external potential fields including the free field or a confined hard-all potential, the linear potential plus an attractive centrifugal-like term and harmonic oscillator. The eigenfunction of the first case is given by a Bessel function. The calculations of the second case are found to have the Airy function and we are able to get a general form of energy levels based on the zeros of Airy function of the first kind. The last case is found that the eigenfunctions are given by the popular associated Laguerre function. To provide a better physical insight into the solutions, some figures are plotted graphically.

In this paper we propose a new method for accurately solving the energy levels and analytical wave functions of a rigid planar rotor in an electric field. First, we use different forms of variable substitution to transform the corresponding stationary Schrödinger equations into a confluent Heun differential equation, and then according to the characteristics of confluent Heun differential equations, confluent Heun functions as well as the studied quantum system, we obtain the analytical solutions of two even-parity functions and two odd-parity functions simultaneously at φ = 0 (2π) and π. Due to symmetry, two analytical solutions corresponding to the same parity should be linearly dependent, so that the Wronskian determinant can be constructed separately and the energy spectrum equations for calculating odd and even parity states can be found. Using the plotting method determines each respective energy and the corresponding quantum number. Then relying on the Maple software to calculate the energy spectra of different eigenstates, we finally obtain the analytical wave function of the bound states expressed by the confluent Heun function and also discuss their linear dependencies.

We present the Shannon entropy of the Schrödinger equation with an asymmetric rectangular multiple well, in which the potential well width is equal but the barrier width is in an arithmetic sequence. When the confined potential is small, we notice that the wave function of the particle is distributed inside the entire interval, but its distribution is mainly confined inside the wells as the confined potential increases. For the higher excited state, we find that the magnitude inside the barrier is always bigger than that inside the well. However, in the double-well case the magnitude is the biggest in the leftmost second barrier for Vconf=1000meV, but it is in the rightmost barrier in the triple well case. For large confined potential, the momentum entropy density (MED) will be concentrated to the center of the total rectangular multiple well with the increasing confined potential. Likewise, we see that the momentum entropy densities of the ground, first and second excited states are overlapped in the triple well case, but they are all overlapped in the four well case. Particularly, in the double well case, the position entropy densities of ground state and 2nd excited state as well as the 1st and 3rd excited states occupy each well. In the triple well case, however, the position entropy densities of ground state and 2nd excited state occupy each one of three wells, but the 1st and 3rd excited states occupy one of four wells. The position entropy density (PED) of each state in four well case is distributed in each well of them as the confined potential is very large. When the number of wells increases, there exist some additional oscillations in the PED. For the higher excited state, the oscillator frequency of the PED increases inside the well but decreases inside the barrier as the confined potential depth increases. For the lower states, as the potential well depth increases the MED extends outward from the original central distribution, but the main distribution is concentrated to the center of the total system for the higher excited state. We also note that both the position entropy Sxand the momentum entropy Spdo not decrease or increase as the parameter Vconfincreases, but their sum always satisfies the BBM inequality.

In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0

In this work we present the entanglement measures of a tripartite W-State entangled system in noninertial frame through the coordinate transformation between Minkowski and Rindler. Two cases are considered, i.e., when one qubit goes in a uniform acceleration a and the others remain stationary and when two qubits undergo in a uniform acceleration and while the other is stationary. The analytical negativities for one-tangle, two-tangle and π-tangle in total are not written out explicitly except for some special cases due to complicated expressions, but we illustrate them in graphics and study their dependencies on the acceleration parameters rb and rc. We find that the negativities of the one-tangle, two-tangle and π-tangle decrease with the acceleration parameters except for the constant NAB=NBA. The negativity NCI(AB) and the πCI decrease faster than NA(BCI) and πA(πB). The negativities NACI (NB(ACI))and NBICI decrease faster than those NAB and NABI, respectively. It is interesting to see there exist turning points for the negativity NA(BICI) at the coordinate position (rb,rc) with rb=rc, which implies that the NA(BICI) has minimum value when Bob and Charlie are on the same position and the degree of the entanglement of the subsystem ρA(BICI) becomes smallest. The von Neumann entropy of tripartite system densities ρABCI and ρABICI are obtained analytically. We notice that they all increase with the acceleration parameters rb and rc. Keywords: W-state, Entanglement, Dirac field, Noninertial frames

Abstract Inspired by the electromagnetic duality, we propose an approach to realize the fractional angular momentum by using a cold atom which possesses a permanent magnetic dipole momentum. This atom interacts with two electric fields and is trapped by a harmonic potential which enable the motion of the atom to be planar and rotationally symmetric. We show that eigenvalues of the canonical angular momentum of the cold atom can take fractional values when the atom is cooled down to its lowest kinetic energy level. The fractional part of canonical angular momentum is dual to that of the fractional angular momenta realized by using a charged particle. Another approach of getting the fractional angular momentum is also presented. The differences between these two approaches are investigated.

According to the single-mode approximation applied to two different mo des, each associated with different uniformly accelerating reference frames, we present analytical expression of the Minkowski states for both the ground and first excited states. Applying such an approximation, we study the entanglement property of Bell and Greenberger–Horne–Zeilinger (GHZ) states formed by such states. The corresponding entanglement properties are described by studying negativity and von Neumann entropy. The degree of entanglement will be degraded when the acceleration parameters increase. We find that the greater the number of particles in the entangled system, the more stable the system that is studied by the von Neumann entropy. The present results will be reduced to those in the case of the uniformly accelerating reference frame.

In this work we solve the angular Teukolsky equation in alternative way for a more general case τ≠0 but m=0,s=0. We first transform this equation to a confluent Heun differential equation and then construct the Wronskian determinant to calculate the eigenvalues and normalized eigenfunctions. We find that the eigenvalues for larger l are approximately given by 0Al0≈[l(l+1)-τR2/2]-iτI2/2 with an arbitrary τ2=τR2+iτI2. The angular probability distribution (APD) for the ground state moves towards the north and south poles for τR2>0, but aggregates to the equator for τR2≤0. However, we also notice that the APD for large angular momentum l always moves towards the north and south poles, regardless of the choice of τ2.

In this paper, a trial function method is employed to find exact solutions to the nonlinear Schrödinger equations with high-order time-dependent coefficients. This system might be used to describe the propagation of ultrashort optical pulses in nonlinear optical fibers, with self-steepening and self-frequency shift effects. The new general solutions are found for the general case a0≠0 including the Jacobi elliptic function solutions, solitary wave solutions, and rational function solutions which are presented in comparison with the previous ones obtained by Triki and Wazwaz, who only studied the special case a0=0.

We study both pentapartite GHZ and W-class states in the noninertial frame and explore their entanglement properties by carrying out the negativities including 1-4, 2-3, and 1-1 tangles, the whole entanglement measures such as algebraic and geometric averages π5 and Π5, and von Neumann entropy. We illustrate graphically the difference between the pentapartite GHZ and W-class states. We find that all 1-4, 2-3 tangles and the whole entanglements, which are observer dependent, degrade more quickly as the number of accelerated qubits increases. The entanglements of these quantities still exist even at the infinite acceleration limit. We also notice that all 1-1 tangles of pentapartite GHZ state Nαβ=NαIβ=NαIβI=0 where α,β∈(A,B,C,D,E), whereas all 1-1 tangles of the W-class state Nαβ,NαIβ and NαIβI are unequal to zero, e.g., Nαβ=0.12111 but NαIβ and NαIβI disappear at r>0.61548 and r>0.38671, respectively. We notice that the entanglement of the pentapartite GHZ and W-class quantum systems decays faster as the number of accelerated particles increases. Moreover, we also illustrate the difference of von Neumann entropy between them and find that the entropy in the pentapartite W-class state is greater than that of GHZ state. The von Neumann entropy in the pentapartite case is more unstable than those of tripartite and tetrapartite subsystems in the noninertial frame.

In this work, we study the quantum information entropies for two different types of hyperbolic single potential wells. We first study the behaviors of the moving particle subject to two different hyperbolic potential wells through focusing on their wave functions. The shapes of these hyperbolic potentials are similar, but we notice that their momentum entropy densities change along with the width of each potential and the magnitude of position entropy density decreases when the momentum entropy magnitude increases. On the other hand, we illustrate the behaviors of their position and momentum entropy densities. Finally, we show the variation of position and momentum entropies Sx and Sp with the change of the potential well depth u and verify that their sum still satisfies the BBM inequality relation.

We revisit time-independent and time-dependent Aharonov-Bohm effects in the noncommutative space-time by investigating a relativistic spin-half particle in the background of a long-thin magnetic-flux-carried solenoid. Our results show that, up to the first order of the noncommutative parameter, there are no noncommutative corrections on the AB effects for both cases.

The Aharonov–Bohm effect in the background of a time-dependent vector potential is re-examined for both non-relativistic and relativistic cases. Based on the solutions to the Schrodinger and Dirac equations which contain the time-dependent magnetic vector potential, we find that contrary to the conclusions in a recent paper (Singleton and Vagenas 2013 [4]), the interference pattern will be altered with respect to time because of the time-dependent vector potential.

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented. The visualizations of the space probability distribution (SPD) are illustrated for the two- (contour) and three-dimensional (isosurface) cases. The quantum numbers (n,l,m) are mainly relevant for those quasi-quantum numbers (n′,l′,m′) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n,l,m)=(6,5,0), the SPD moves towards two poles of z-axis when P increases. Finally, we discuss the different cases for the potential parameter b, which is taken as negative and positive values for c>0. Compared with the particular case b=0, the SPDs are shrunk for b=-0.5, while they are spread out for b=0.5.

We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a.

In this work, we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on m are illustrated graphically for a given potential parameter ξ. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We also notice that the energy levels ϵi decrease with the increasing potential parameter m.

The Schrödinger equation ψ′′(x)+κ2ψ(x)=0 where κ2=k2-V(x) is rewritten as a more popular form of a second order differential equation by taking a similarity transformation ψ(z)=ϕ(z)u(z) with z=z(x). The Schrödinger invariant IS(x) can be calculated directly by the Schwarzian derivative z,x and the invariant I(z) of the differential equation uzz+f(z)uz+g(z)u=0. We find an important relation for a moving particle as ∇2=-IS(x) and thus explain the reason why the Schrödinger invariant IS(x) keeps constant. As an illustration, we take the typical Heun’s differential equation as an object to construct a class of soluble potentials and generalize the previous results by taking different transformation ρ=z′(x) as before. We get a more general solution z(x) through integrating (z′)2=α1z2+β1z+γ1 directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail. The results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers. It should be recognized that a subtle and different choice of the transformation z(x) also related to ρ will lead to difficult connections to the results obtained from other different approaches.

We first present the exact solutions of the single ring-shaped Coulomb potential and then realize the visualizations of the space probability distribution for a moving particle within the framework of this potential. We illustrate the two-dimensional (contour) and three-dimensional (isosurface) visualizations for those specifically given quantum numbers (n, l, m) essentially related to those so-called quasi-quantum numbers (n′, l′, m′) through changing the single ring-shaped Coulomb potential parameter b. We find that the space probability distributions (isosurface) of a moving particle for the special case l=m and the usual case l≠m are spherical and circularly ring-shaped, respectively, by considering all variables r→=(r,θ,φ) in spherical coordinates. We also study the features of the relative probability values P of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (n, l, m) = (6, 5, 1), we notice that the space probability distribution for a moving particle will move towards the two poles of the z-axis as the relative probability value P increases. Moreover, we discuss the series expansion of the deformed spherical harmonics through the orthogonal and complete spherical harmonics and find that the principal component decreases gradually and other components will increase as the potential parameter b increases.

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, where t∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument ∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radii R sharing energy levels with a common eigenvalue. In circular well, the momentum operators P±=Px±iPy play the role of shift operators. The Px and Py operators, the third projection of the orbital angular momentum operator Lz, and the Hamiltonian H form a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation between ψlm(r) and ψ(l ± 1)(m±1)(r).