In this paper, we solve analytically the Schrodinger equation for s-wave and arbitrary angular momenta with the Hua potential is investigated respectively. The wave function as well as energy equation are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.
We present arguments proving that the results obtained by Hassanabadi and coworkers in the study of the D-dimensional Schrödinger equation with molecular Hua potential through the supersymmetry method in quantum mechanics are incorrect. We identified the inconsistencies in their reasoning on the allowed values of the parameter q and we constructed the correct energy spectrum.
The Tietz-Hua (TH) potential is one of the very best analytical model potentials for the vibrational energy of diatomic molecules. By using the Nikiforov-Uvarov (NU) method, we have obtained the exact analytical s-wave solutions of the radial Schrödinger equation (SE) for the TH potential. The energy eigenvalues and the corresponding eigenfunctions are calculated in closed forms. Some numerical results for diatomic molecules are also presented.
We present arguments demonstrating that the application of the Nikiforov-Uvarov polynomial method to solve the Schrödinger equation with the Tietz-Hua potential is valid only when $e^{-b_{h}r_{e}}\leq c_{h}<1$ and $r_{0}<r<+\infty $. In particular, it is pointed out that the numerical results with $c_{h}\neq 0$ for the diatomic molecules $\mathrm{HF}$, $\mathrm{N}_{2}$, $\mathrm{I}_{2}$, $\mathrm{H}_{2}$, $\mathrm{O}_{2}$ and $\mathrm{O}_{2}^{+}$ given in Tables 3-5 by Hamzavi and co-workers are wrong. When $-1\leq c_{h}<0$ or $0<c_{h}<e^{-b_{h}r_{e}}$, this approach is not suitable. In both c…
The time-dependent quantum system of two laser-driven electrons in a harmonic oscillator potential is analysed, taking into account the repulsive Coulomb interaction between both particles. The Schrodinger equation of the two-particle system is shown to be analytically soluble in case of arbitrary laser frequencies and individual oscillator frequencies, defining the system. Quantum information processing could be a possible field of application
The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schrödinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schrödinger equation with exponential potentials of the form $-αr^λ\exp(-βr)$ can also be analytically solved by using the auxiliary field method. Formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn on the Yukawa potential an…
In this work we discuss in detail the known solutions of the stationary Schrödinger equation subject to a deformable hyperbolic tangent potential exactly soluble $ V(x) = \frac {V_0} {2} (1+ \tanh (δx)) $. We find the analytical solutions in terms of Gauss hypergeometric functions for the scattering states with energy greater than the maximum value of the potential. We also discussed the case for the energy lower than the maximum and the similarities and differences with the abrupt step potential in both cases. We graphically illustrate the relevant physical situations to the problem.
It is now common practice to solve the Schrödinger equation to estimate the tunneling current between two metal electrodes at specified potentials, or the transmission through a potential barrier by assuming an incident, reflected, and transmitted wave. However, we suggest that these methods may not be appropriate for nanoscale circuits. The electron man-free path may be as long as 68.2 nm in metallic elements so we consider the possibility that quantum effects may occur throughout a nanoscale circuit, including the connections. Analytical methods are presented for modeling the coherent transf…
Approximate analytical solutions of the Dirac equation are obtained for the Hellmann potential, Wei Hua potential and Varshni potential with any $κ$-value for the cases having the Dirac equation pseudospin and spin symmetries. Closed forms of the energy eigenvalue equations and the spinor wave functions are obtained by using the Nikiforov-Uvarov method and some tables are given to see the dependence of the energy eigenvalues on different quantum number pairs ($n, κ$).
Analytical solutions are presented for eigenvalues, eigenfunctions of {\color{red} D-dimensional Schrodinger equation having Eckart potential} within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term, from a combination of Greene-Aldrich and Pekeris approximations. Solutions are obtained in terms of hypergeometric functions. It facilitates an accurate representation in entire domain. Its validity is illustrated for energies in an arbitrary $\ell \neq 0$ quantum state. Results are compared for a chosen set of potential parameters in different dimensions. In s…
A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ bound-state solutions of the Schrödinger equation for an exponential-type potential in the framework of the hypergeometric method. The corresponding normalized wave functions are also found in terms of the Jacobi polynomials. To show the accuracy of the new proposed approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $σ_{\text{0}}.$ Our numerical results are of high accuracy like the other numeric…
The bound state solutions of the $D$-dimensional Schrödinger equation for new mixed class of potential, $V(r)=\frac{V_1}{r^2}+\frac{V_2e^{-αr}}{r}+V_3cothαr+V_4\,,$ are studied within the framework of the Pekeris approximation for any arbitrary $\ell$-state. Asymptotic iteration method (AIM) is used for the work. The energy spectrum are obtained as well as their corresponding normalized eigenfunctions are derived in terms of generalized hypergeometric functions $\,_{2}F_{1}(a,b,c;z)$. It is shown that using the Pekeris approximation, present potential model is very much capable of deriving oth…
The analytical solutions of the N-dimensional Schrodinger equation with position-dependent mass for a general class of central potentials is obtained via the series expansion method. The position-dependent mass is expanded in series about origin. As a special case, the analytical bound-state series solutions and the recursion relation of the linear-plus-Coulomb (Cornell) potential with the decaying position-dependent mass m=m_{0}e^{-λr} are also found.
An analysis of the analytical solution of the Schrödinger equation (which is a second order differential equation) for $H_2^+$ shows that the second linear independent solution of this equation is a square integrable function and therefore the ground state total wave function is a linear combination of two linear independent wave functions of different space symmetry: cylindrical and spherical. The wave function of cylindrical symmetry is well known. It has maxima at the positions of nuclei. The wave function of spherical symmetry and the corresponding spherical electron distribution, which ex…
We obtain the quantized momentum eigenvalues, $P_n$ , and the momentum eigenstates for the space-like Schrödinger equation, the Feinberg-Horodecki equation, with the general potential which is constructed by the temporal counterpart of the spatial form of these potentials. The present work is illustrated with two special cases of the general form: time-dependent Wei-Hua Oscillator and time-dependent Manning-Rosen potential. We also plot the variations of the general molecular potential with its two special cases and their momentum states for few quantized states against the screening parameter…
In this study, we develop a semi-analytical framework to solve generalized Jaynes-Tavis-Cummings Hamiltonians describing multi-qudit systems coupled via EM resonators. Besides the multi-level generalization we allow for an arbitrary number of resonators and/or modes, with nonidentical couplings to the qudits. Our method is based on generic excitation-number operators which commute with the respective Hamiltonians in the rotating wave approximation (RWA). The validity of the RWA is assessed explicitly. The formalism enables the study of eigenstates, eigenenergies and corresponding time evolutio…
Analytical solution of the Schrodinger equation for the neutral helium atom in the ground state considering the uncertainty principle, vibrational modes and quantum-electrodynamical effects
Bound state solutions of the Schrödinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods
University of Batna I（巴特纳第一大学）；University of Biskra（比斯克拉大学）；University of Khenchela
A method is proposed to transform any analytic solution of the Bloch equation into an analytic solution of the Landau-Lifshitz-Gilbert equation. This allows for the analytical description of the dynamics of a two level system with damping. This method shows that damping turns the linear Schrödinger equation of a two-level system into a nonlinear Schrödinger equation. As applications, it is shown that damping has a relatively mild influence on self-induced transparency but destroys dynamical localization.
Schrödinger equation for two center Coulomb plus harmonic oscillator potential is solved by the method of ethalon equation at large intercenter separations. Asymptotical expansions for energy term and wave function are obtained in the analytical form.
Exact solution of the Schrodinger equation with deformed ring shaped potential is obtained in the parabolic and spherical coordinates. The Nikiforov-Uvarov method is used in the solution. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. The agreement of our results is good.
By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrodinger equation for the pseudoharmonic and Kratzer potentials in two dimensions. The energy levels of all the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained.
We derive approximate analytical solutions for a particle in a homogenous gravitational field and confined between two independently vibrating mirrors. This constitutes an extension of the qBOUNCE experiment in which ultra-cold neutrons are employed for Gravity Resonance Spectroscopy.
Exact bound state solutions of the Schrodinger equation for generalized noncentral potential are examined by means of the Nikiforov-Uvarov method. The wavefunctions and the corresponding energy eigenvalues of the system are obtained analytically. The results examined for the potential are compatible with those obtained by the other methods.
We present analytical solutions of the Schrodinger equation for the one-dimensional hydrogen molecular ion. In particular, we present closed form expressions for the electronic energy curves of this system that correspond to the ground state and the first excited state. Our results agree with numerical solutions obtained before.
Analytical solutions of the Schrodinger equation are obtained for some diatomic molecular potentials with any angular momentum. The energy eigenvalues and wave functions are calculated exactly. The asymptotic form of the equation is also considered. Algebraic method is used in the calculations.
We propose a new method to obtain approximate solutions for the Schrödinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows in many cases to find analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which improve a lot th…
We investigate the presence of localized analytical solutions of the Schrödinger equation with logarithm nonlinearity. After including inhomogeneities in the linear and nonlinear coefficients, we use similarity transformation to convert the nonautonomous nonlinear equation into an autonomous one, which we solve analytically. In particular, we study stability of the analytical solutions numerically.
Using a recently developed technique to solve Schrödinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrödinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass $m(x)$ particle scattered by a potential $\mathcal{V}(x)$. We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
In this paper, the Schrodinger equation for s-wave and arbitrary angular momenta with the Modified Mobuis Square potential is investigated respectively. The eigenfunctions as well as energy eigenvalues are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.
The general solutions of Schrodinger equation for non central potential are obtained by using Nikiforov Uvarov method. The Schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions for these potentials are derived analytically. Non central potential is reduced to Coulomb and Hartmann potential by making special selections, and the obtained solutions are compared with the solutions of Coulomb and Hartmann ring shaped potentials given in literature.
The polynomial solution of the Schrodinger equation for the Pseudoharmonic potential is found for any arbitrary angular momentum $l$. The exact bound-state energy eigenvalues and the corresponding eigen functions are analytically calculated. The energy states for several diatomic molecular systems are calculated numerically for various principal and angular quantum numbers. By a proper transformation, this problem is also solved very simply by using the known eigensolutions of anharmonic oscillator potential.
The study of the Schrödinger equation with the position-dependent effective mass has attracted a lot of attention, due to its applications in many fields of physics, including the properties of the semiconductors, semiconductor heterostructures, graded alloys, quantum liquids, Helium-3 clusters, quantum wells, wires and dots etc. In the present work we obtain several classes of solutions of the one-dimensional Schrödinger equation with position-dependent particle mass. As a first step the single particle Schrödinger equation with position-dependent mass is transformed into an equivalent Riccat…
We propose an exact method for solving a one-dimensional Schrödinger equation. An arbitrary potential is represented by the collection of short-width potentials. For building the collection scheme, a new solvable potential is introduced. It is based on the simple expansion of the wavefunction of the introduced potential. The illustration of the scheme is done by reproducing the results of the rectangular potential. The scheme has computational advantages and the transmission properties, eigenenergies can be calculated efficiently. The presented scheme is compared with the other similar schemes…
In this work, we obtain the Schrödinger equation solutions for the Kratzer potential plus screened Coulomb potential model using the series expansion method. The energy eigenvalues is obtained in non-relativistic regime and the corresponding unnormalized eigenfunction. Three special cases were obtained. We applied the present results to calculate heavy-meson masses of charmonium and bottomonium , and we got the numerical values for 1S, 2S, 1P,2P, 3S, 4S ,1D,2D and 1F states. The results are in good agreement with experimental data and the work of other researchers.
The full analytical solution of the Schrödinger equation for the hydrogen molecular ion $H_2^+$ (special case of the quantum tree-body problem with the Coulomb interaction) is obtained first. The solution shows that the total wave function is a two-term function in the sense that it is a linear combination of the two linear independent wave functions. The two-term character of the total wave function was visualized in calculations of the total electron density of $H_2^+$ at different internuclear separations.
In this paper, we study the nonrelativistic quantum mechanics of an electron in a spacetime containing a topological defect. We also consider that the electron is influenced by the Hulthén potential. In particular, we deal with the Schrödinger equation in the presence of a global monopole. We obtain approximate solutions for the problem, determine the scattering phase shift and the $S$-matrix, and analyze bound states.
In this work, aiming to solve numerically the Schrödinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schrödinger equation with potentials of the form V(x) + deltap(x), where deltap(x) is a pseudo-delta function, a very high and thin barrier. The numerical results show good agreement with analytical results found in the literature. Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta b…
Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples - the damped harmonic oscillator and a two-level atom with homodyne photodetection. Then, we consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markov…
Making an ansatz to the wave function, the exact solutions of the $D$% -dimensional radial Schrodinger equation with some molecular potentials like pseudoharmonic and modified Kratzer potentials are obtained. The restriction on the parameters of the given potential, $δ$ and $η$ are also given, where $η$ depends on a linear combination of the angular momentum quantum number $\ell $ and the spatial dimensions $D$ and $δ$ is a parameter in the ansatz to the wave function. On inserting D=3, we find that the bound state eigensolutions recover their standard analytical forms in literature.
We present the non-Markovian generalization of the widely used stochastic Schrodinger equation. Our result allows to describe open quantum systems in terms of stochastic state vectors rather than density operators, without approximation. Moreover, it unifies two recent independent attempts towards a stochastic description of non-Markovian open systems, based on path integrals on the one hand and coherent states on the other. The latter approach utilizes the analytical properties of coherent states and enables a microscopic interpretation of the stochastic states. The alternative first approach…
We tried to determine the range of validity of a recently proposed modification of the Hellmann potential that leads to analytical eigenvalues and eigenfunctions. We discuss the difficulties that we found in the analysis of the main equations and results. We conclude that the eigenvalues reported by the authors do not exhibit the same order as those of the Hellmann potential thus leading to a different underlying physics. What is more: the spectrum of the modified model is qualitatively different from the one supported by the Hellmann potential.
We present analytically the exact energy bound-states solutions of the Schrodinger equation in D-dimensions for a recently proposed modified Kratzer potential plus ring-shaped potential by means of the conventional Nikiforov-Uvarov method. We give a clear recipe of how to obtain an explicit solution to the wave functions in terms of orthogonal polynomials. The results obtained in this work are more general and true for any dimension which can be reduced to the standard forms in three-dimensions given by other works.
In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schrödinger equation, which is given by the Airy function. The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation. Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential. The method is shown to work in both one and two dimensions, and can be easily extended if required. The applicability of the method is discussed in relation t…
Using a variational approach based on a Lagrangian formulation and Gaussian trial functions, we derive a simple dynamical system that captures the main features of the time-dependent Schrödinger-Newton equations. With little analytical or numerical effort, the model furnishes information on the ground state density and energy eigenvalue, the linear frequencies, as well as the nonlinear long-time behaviour. Our results are in good agreement with those obtained through analytical estimates or numerical simulations of the full Schrödinger-Newton equations.
A novel method is proposed to determine an analytical expression for eigenfunctions and numerical result for eigenvalues of the Schrödinger type equations, within the context of Taylor expansion of a function. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical result for eigenvalues.
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schrödinger equation. This technique can generate the spectrum associated with an arbitrary potential $V(r)$ starting from the analytically known spectrum of a particular potential $P(r)$. In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of $P(r)$. The method is extended in order to find accurate analytical energy formulae for radial potentials of the for…
Analytical solution of the Schrödinger equation with $1/r^3$ and attractive $1/r^2$ potentials: Universal three-body parameter of mixed-dimensional Efimov states
University of Electro-Communications（电气通信大学）；Tohoku University（东北大学）
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…
We study the interpolation from occupation number Fock states to Schrödinger cat states on systems modeled by two-mode Bose-Hubbard Hamiltonian, like, for instance, bosons in a double well or superconducting Cooper pair boxes. In the repulsive interaction regime, by a simplified single particle description, we calculate, analytically, energy, number fluctuations, stability under coupling to a heat bath, entanglement entropy and Fisher information, all in terms of hypergeometric polynomials of the single particle overlap parameter. Our approach allows us to find how those quantities scale with …
The accuracy of different transfer matrix approaches, widely used to solve the stationary effective mass Schrödinger equation for arbitrary one-dimensional potentials, is investigated analytically and numerically. Both the case of a constant and a position dependent effective mass are considered. Comparisons with a finite difference method are also performed. Based on analytical model potentials as well as self-consistent Schrödinger-Poisson simulations of a heterostructure device, it is shown that a symmetrized transfer matrix approach yields a similar accuracy as the Airy function method at …
A system of two bilinearly coupled harmonic oscillators has been solved analytically by using the Born-Oppenheimer (BO) product wavefunction ansatz and the phase-space bound trajectory approach [J. S. Molano et al., Chem. Phys. Lett. \textbf{76}(12), 138171 (2021)]. The bilinearly coupled oscillator system allows to obtain the analytical expression of the quantum system, facilitating comparison with the results of using the BO ansatz product. The analytical and BO wavefunctions are obtained as a product of parabolic cylinder functions. The arguments of the parabolic cylinder functions of the e…
In this paper it is shown how to solve numerically eigenvalue problems associated to second order linear ordinary differential equations, containing also terms which depend on the variable. A didactic presentation of the Numerov Method is given and, in the sequel, it is applied to two quantum non-relativistic problems with well known analytical solutions: the simple harmonic oscillator and the hydrogen atom. The numerical results are compared to those obtained analytically.
Analytical solutions to the time-dependent Schrodinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presentin…
