In this paper, we present a general method to solve non-hermetic potentials with PT symmetry using the definition of two $η$-pseudo-hermetic and first-order operators. This generator applies to the Dirac equation which consists of two spinor wave functions and non-hermetic potentials, with position that mass is considered a constant value and also Hamiltonian hierarchy method and the shape invariance property are used to perform calculations. Furthermore, we show the correlation between the potential parameters with transmission probabilities that $η$-pseudo-hermetic using the change of focal points on Hamiltonian can be formalized based on Schrödinger-like equation. By using this method for some solvable potentials such as deformed Woods Saxon's potential, it can be shown that these real potentials can be decomposed into complex potentials consisting of eigenvalues of a class of $η$-pseudo-hermitic.
The scattering solutions of the one-dimensional Schrodinger equation for the Woods-Saxon potential are obtained within the position-dependent mass formalism. The wave functions, transmission and reflection coefficients are calculated in terms of Heun's function. These results are also studied for the constant mass case in detail.
By using the Pekeris approximation, the Schrödinger equation is solved for the nuclear deformed Woods-Saxon potential within the framework of the asymptotic iteration method (AIM). The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of hypergeometric function.
We obtain the energy eigenvalues and radial wave functions of the D-Dimensional Dirac equation in the case of spin symmetry for Woods-Saxon potential in minimal length formalism. The radial part of the D-Dimensional Dirac equation is solved by applied the supersymmetric quantum mechanics method using the Pekeris approximation to deal with the centrifugal term. The behavior of bound-state energy eigenvalues versus dimension and also quantum number is discussed for various minimal length parameters.
The analytical expressions for the eigenvalues and eigenvectors of the Klein-Gordon equation for q-deformed Woods-Saxon plus new generalized ring shape potential are derived within the asymptotic iteration method. The obtained eigenvalues are given in a closed form and the corresponding normalized eigenvectors, for any l, are formulated in terms of the generalized Jacobi polynomials for the radial part of the Klein-Gordon equation and associated Legendre polynomials for its angular one. When the shape deformation is canceled, we recover the same solutions previously obtained by the Nikiforov-U…
By using the Pekeris approximation, the Schrodinger equation is approximately solved for the nuclear deformed Woods-Saxon potential within the framework of the asymptotic iteration method. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of hypergeometric function.
The exact analytical solutions of the Schrödinger equation for the generalized symmetrical Woods-Saxon potential are examined for the scattering, bound and quasi-bound states in one dimension. The reflection and transmission coefficients are analytically obtained. Then, the correlations between the potential parameters and the reflection-transmission coefficients are investigated, and a transmission resonance condition is derived. Occurrence of the transmission resonance has been shown when incident energy of the particle is equal to one of the resonance energies of the quasi-bound states.
We solve the Duffin-Kemmer-Pétiau equation in the presence of a spatially one-dimensional symmetric potential well. We compute the scattering state solutions and we derive conditions for transmission resonances. The bound solutions are derived by a graphic study and the appearance of the antiparticle bound state is discussed.
The energy spectra and the wave function depending on the c-factor are investigated for a more general Woods-Saxon potential (MGWSP) with an arbitrary l - state. The wave functions are expressed in terms of the Jacobi polynomials. Two potentials are obtained from this MGWSP as special cases. These special potentials are Hulthen and the standard Woods-Saxon potentials. We also discuss the energy spectrum and wave function for the special cases.
We study the approximate analytical solutions of the Dirac equation for the generalized Woods-Saxon potential with the pseudo-centrifugal term. In the framework of the spin and pseudospin symmetry concept, the approximately analytical bound state energy eigenvalues and the corresponding upper- and lower-spinor components of the two Dirac particles are obtained, in closed form, by means of the Nikiforov-Uvarov method which is based on solving the second-order linear differential equation by reducing it to a generalized equation of hypergeometric type. The special cases $κ=\pm 1$ ($l=% \widetild…
The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulthén potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.
Recently, it has been shown that the generalized symmetric Woods-Saxon potential energy, in which surface interaction terms are taken into account, describes the physical processes better than the standard form. Therefore in this study, we investigate the scattering of Klein-Gordon particles in the presence of both generalized symmetric Woods-Saxon vector and scalar potential. In one spatial dimension we obtain the solutions in terms of hypergeometric functions for spin symmetric or pseudo-spin symmetric cases. Finally, we plot transmission and reflection probabilities for incident particles w…
The bound state solution of the radial Schrödinger equation with the generalized Woods-Saxon potential is carefully examined by using the Pekeris approximation for arbitrary $\ell$ states. The energy eigenvalues and the corresponding eigenfunctions are analytically obtained for different $n$ and $\ell$ quantum numbers. The obtained closed forms are applied to calculate the single particle energy levels of neutron orbiting around $^{56}$Fe nucleus in order to check consistency between the analytical and Gamow code results. The analytical results are in good agreement with the results obtained b…
In this manuscript, we present analytical solution of the Klein-Gordon equation with the multi-parameter q-deformed Woods-Saxon type potential energy under the spin symmetric limit in $(1+1)$ dimension. In the scattering case, we obtain the reflection and transmission probabilities and prove the conservation of the total probability. Moreover, we analyze the correlation between the potential parameters with the reflection and transmission probabilities. In the bound state case, we use the continuity conditions and derive a quantization scheme. To confirm our results numerically, in both cases …
Motivated by the observation of a recent renewal of rather strong interest in the description of bound states by (semi-) relativistic equations of motion, we revisit, for the example of the Woods-Saxon interactions, the eigenvalue problem posed by the spinless Salpeter equation and recall various elementary knowledge, considerations, and techniques that practitioners seeking solutions to this specific reduction of the Bethe-Salpeter equation may find helpful.
The bound state energy eigenvalues and the corresponding eigenfunctions of the generalized Woods-Saxon potential reported in [Phys. Rev. C 72, 027001 (2005)] is extended to the fractional forms using the generalized fractional derivative and the fractional Nikiforov-Uvarov (NU) technique. Analytical solutions of bound states of the Schrodinger equation for the present potential are obtained in the terms of fractional Jacobi polynomials. It is demonstrated that the classical results are a special case of the present results at Elfa=Beta=1 Therefore, the present results play important role in mo…
By employing the Pekeris approximation, the D-dimensional Schrödinger equation is solved for the nuclear deformed Woods-Saxon potential plus double ring-shaped potential within the framework of the Asymptotic Iteration Method (AIM). The energy eingenvalues are given in a closed form and the corresponding normalized eigenfunctions are obtained in terms of hypergeometric functions. Our general results reproduce many predictions obtained in the literature, using the Nikiforov-Uvarov method (NU) and the Improved Quantization Rule approach, particulary those derived by considering Woods-Saxon poten…
More recently, comprehensive application results of approximate analytical solutions of the Woods-Saxon potential in closed form for the 5-dimensional Bohr Hamiltonian have been appeared [14] and its comparison to the data for many different nuclei has clearly revealed the domains for the sucsess and failure in case of using such potential forms to analyse the data concerning with the nuclear structure of deformed nuclei within the frame of the collective model. Gaining confidence from this work, exact solvability of the Woods-Saxon type potentials in lower dimensions for the bound states havi…
We study the one-dimensional Dirac equation in the framework of a position dependent mass under the action of a Woods-Saxon external potential. We find that constraining appropriately the mass function it is possible to obtain a solution of the problem in terms of the hypergeometric function. The mass function for which this turns out to be possible is continuous. In particular we study the scattering problem and derive exact expressions for the reflection and transmission coefficients which are compared to those of the constant mass case. For the very same mass function the bound state proble…
Approximate scattering and bound state solutions of the one-dimensional effective-mass Dirac equation with the Woods-Saxon potential are obtained in terms of the hypergeometric-type functions. Transmission and reflection coefficients are calculated by using behavior of the wave functions at infinity. The same analysis is done for the constant mass case. It is also pointed out that our results are in agreement with those obtained in literature. Meanwhile, an analytic expression is obtained for the transmission resonance and observed that the expressions for bound states and resonances are equal…
In this work, the analytical solutions of the $D$-dimensional Schrödinger equation are studied in great detail for the Wood-Saxon potential by taking advantage of the Pekeris approximation. Within a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any angular momentum case within the context of the Nikiforov-Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions …
The quantum engine cycle serves as an analogous representation of classical heat engines for microscopic systems and the quantum regime of thermal devices is composed of a single element. In this work, the Quantum-Mechanical properties of a non-linear quantum oscillator described by the Woods-Saxon [WS] model are examined. The Quantum-Mechanical analogue of the Carnot cycle was constructed using changes in both the width L of the well and the quantum state of the potential well. The efficiency of the quantum engine, consisting of adiabatic and isothermal processes based on the Woods-Saxon [WS]…
We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial $δ$-$δ'$ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined …
In this study, we obtain an approximate solution of the Schrodinger equation in arbitrary dimensions for the generalized shifted Hulthen potential model within the framework of the Nikiforov-Uvarov method. The bound state energy eigenvalues were computed and the corresponding eigenfunction was also obtained. It is found that the numerical eigenvalues were in good agreement for all three approximations scheme used. Special cases were considered when the potential parameters were altered, resulting into Hulthen Potential and Woods-Saxon Potential respectively. Their energy eigenvalues expression…
Unified description of magic numbers of metal clusters in terms of the 3-dimensional q-deformed harmonic oscillator
用具有Uq（3）>SOq（3）对称性的三维q形变谐振子预言的幻数与碱金属原子团簇的实验数据进行了比较（Li，Na，K，Rb，Cs），贵金属（Cu，Ag，Au），二价金属（Zn，Cd）和三价金属（Al，In），以及凝胶模型，Woods-Saxon和酒瓶电势的理论预测，并且涉及使用3 n +1伪量子数的分类方案。在碱金属团簇和贵金属团簇中，三维q变形谐振子正确地预言了实验观测到的所有幻数，直到1500（这是基于电子壳层填充的理论的有效性的预期极限），而另外，它给出了二价金属和三价金属簇的幻数的令人满意的结果，从而表明Uq（3），它是球面（3维各向同性）谐振子的U（3）对称性的非线性扩展，是几个金属团簇系统对称性的良好候选者。与三维q变形谐振子近似产生相同频谱的角动量相关势的泰勒展开式被发现与对称化Woods-Saxon势和酒瓶对称化Woods-Saxon势的泰勒展开式相似，已知它们提供了Ekardt势的成功拟合。
In this paper, we present the exact solution of one dimensional Schrödinger equation for Wood-Saxon plus Rosen-Morse plus symmetrical double well potential via Nikiforov-Uvarov mathematical method. The eigenvalues and eigenfunctions of this potential are obtained. The energy equations and the corresponding wave function for special cases of this potential are briefly discussed. The PT-symmetry and Hermiticity for this potential are also considered.
Analytical Solutions of One-Dimensional ($1\mathcal{D}$) Potentials for Spin-0 Particles via the Feshbach-Villars Formalism
Echahid Cheikh Larbi Tebessi University（埃查希德·谢赫·拉比·特贝西大学）；Research Center of Astrophysics and Cosmology, Khazar University（哈萨克斯坦大学天体物理学与宇宙学研究中心）；Departamento de Física, Universidade Federal do Maranhão（马拉尼昂联邦大学物理系）
Observing renewed interest in long-standing (semi-) relativistic descriptions of bound states, we would like to make a few comments on the eigenvalue problem posed by the spinless Salpeter equation and, illustrated by the examples of the nonsingular Woods-Saxon potential and the singular Hulthén potential, recall elementary tools that practitioners looking for analytic albeit approximate solutions might find useful in their quest.
