Cevdet Tezcan, Ramazan Sever
The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.
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We systematically describe and classify 1-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.
By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schrödinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.
We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrödinger equations. The connection between them is stablished through the biconfluent Heun equation. We found that each invariant $n$-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to $n$ potentials, each with one known solution, or to one potential with $n$-known solutions. Among these Schrödinger potentials appear the quarkonium and the sextic oscillator.
In this paper we present exact solutions of Schrodinger equation (SE) for a class of non central physical potentials within the formalism of position-dependent effective mass. The energy eigenvalues and eigenfunctions of the bound-states for the Schrodinger equation are obtained analytically by means of asymptotic iteration method (AIM) and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates. Our results are in excellent agreement with other works in the literature.
The Fokker-Planck equation associated with the two - dimensional stationary Schrödinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schrödinger equations. The examples of exactly solvable two - dimensional…
We review an exact analytical resolution method for general one-dimensional (1D) quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving spectral (usual) vs "classical" (newer) zeta-regularisations in parallel. The central results are a set of Bohr--Sommerfeld-like but exact quantisation conditions, directly drawn from Wronskian identities, and appearing to extend Bethe-Ansatz formulae of integrable systems. Such exact quantisation conditions do not just select the eigenvalues; some evaluate the spectral determ…
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.
We address the problem of obtaining the exact reduced dynamics of the spin-half (qubit) immersed within the bosonic bath (enviroment). An exact solution of the Schrodinger equation with the paradigmatic spin-boson Hamiltonian is obtained. We believe that this result is a major step ahead and may ultimately contribute to the complete resolution of the problem in question. We also construct the constant of motion for the spin-boson system. In contrast to the standard techniques available within the framework of the open quantum systems theory, our analysis is based on the theory of block operato…
We write Schrödinger equation for the Coulomb potential in both de Sitter and Anti-de Sitter spaces using the Extended Uncertainty Principle formulation. We use the Nikiforov-Uvarov method to solve the equations. The energy eigenvalues for both systems are given in their exact forms and the corresponding radial wave functions are expressed in associated Jacobi polynomials for de Sitter space, while those of Anti-de Sitter space are given in terms of Romanovski polynomials. We have also studied the effect of the spatial deformation parameter on the bound states in the two cases.
In this paper an exact transparent boundary condition (TBC) for the multidimensional Schrödinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for 2D and 3D equations reported before. A new exact fully discrete (i.e. derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and p…
We construct a Madelung fluid model with specific time variable parameters as dissipative quantum fluid and linearize it in terms of Schrodinger equation with time dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrodinger equation and the corresponding classical linear ODE with variable frequency and damping. For the complex velocity field the Madelung system takes the form of a nonlinear complex Schrodinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we d…
We present a class of confining potentials which allow one to reduce the one-dimensional Schroodinger equation to a named equation of mathematical physics, namely either Bessel's or Whittaker's differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.
Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential
该研究利用 Dunkl 导数与球坐标变量分离，解析求解三维自由粒子、赝谐振子及 Mie 型势的 Schrödinger 方程，给出波函数与能谱，并验证参数消失时结果回归已有结论。
Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well
本文研究自由粒子在圆柱形势阱中Schrödinger-Dunkl方程的解，考虑有限和无限高情况，利用Dunkl形式主义得到径向和轴向波函数的解析表达式，基于反射算子的本征值分类能量谱和解，并讨论波函数获得确定奇偶性的条件及对Dunkl参数的约束。
Exactly solved Schrödinger equations with time-dependent Hamiltonians
该研究给出与量子自旋电池相关的含时哈密顿量演化算符的精确公式，通过新颖数学技术组合获精确解，可恢复并超越现有展开近似，此方法广泛适用于非自治微分系统。
In this paper, three plausible axioms together with two definitions are employed to build an axiomatic framework, and then with the help of the Dirac formalism, it is demonstrated that the time-dependent Schrödinger wave equation is no longer a postulate for the whole theory, but a theorem. Subsequently, a proof for the theorem is presented. The result implies that the other remaining axioms of the picture essentially involve the Schrödinger equation, and this consequence lets the whole theory become stronger.
On the exact solutions of a one-dimensional Schrödinger equation with a rational potential
本文研究具有有理势的一维条件可解薛定谔方程，通过精确本征函数的节点建立了孤立精确本征值与哈密顿算符实际本征值之间的联系。
A novel exactly solvable Schrödinger equation with a position-dependent mass (PDM) describing a Coulomb problem in $D$ dimensions is obtained by extending the known duality relating the quantum $d$-dimensional oscillator and $D$-dimensional Coulomb problems in Euclidean spaces for $D = (d+2)/2$. As an intermediate step, a mapping between a quantum $d$-dimensional nonlinear oscillator of Mathews-Lakshmanan type (or oscillator in a space of constant curvature) and a quantum $D$-dimensional Coulomb-like problem in a space of nonconstant curvature is derived. It is finally reinterpreted in a PDM b…
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical w…
The exact solutions of Schrodinger equation are obtained for a noncentral potential which is a ring-shaped potential. The energy eigenvalues and corresponding eigenfunctions are obtained for any angular momentum l. Nikiforov-Uvarov method is used in the computations.
Exact bound state solutions and corresponding normalized eigenfunctions of the radial Schrödinger equation are studied for the pseudoharmonic and Mie-type potentials by using the Laplace transform approach. The analytical results are obtained and seen that they are the same with the ones obtained before. The energy eigenvalues of the inverse square plus square potential and three-dimensional harmonic oscillator are given as special cases. It is shown the variation of the first six normalized wavefunctions of the above potentials. It is also given numerical results for the bound states of two d…
Effective mass Schrodinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrodinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrodinger equation starting from the constant mass Schrodinger equation.
Exact boundary conditions at finite distance for the solutions of the time-dependent Schrodinger equation are derived. A numerical scheme based on Crank-Nicholson method is proposed to illustrate its applicability in several examples.
We deal with the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, Poschl-Teller and Hulthen type potentials are considered respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.
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.
The polynomial solution of the N-dimensional space Schrodinger equation for a special case of Mie potential is obtained for any arbitrary $% l-state. The exact bound-state energy eigenvalues and the corresponding eigenfunctions are calculated for diatomic molecular systems in the Mie-type potential. Keywords: Mie potential, Schrodinger equation, Eigenvalue, Eigenfunction, Diatomic molecules
A general form of the effective mass Schrodinger equation is solved exactly for Hulthen potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function.
The Schrödinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schrödinger equation for the radial wave function through a simple transformation. As an example, applying an ${\it ansatz}$ to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schrödinger equation with the octic potential, $V(r)=ar^2-br^4+cr^6-dr^4+er^{10}$.
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.
Exact solutions of Schrodinger equation for PT-/non-PT-symmetric and non-Hermitian Morse and Poschl-Teller potentials are obtained with the position-dependent effective mass by applying a point canonical transformation method. Three kinds of mass distributions are used in order to construct exactly solvable target potentials and obtain energy spectrum and corresponding wave functions.
We employ the point canonical transformation (PCT) to solve the D-dimensional Schrödinger equation with position-dependent effective mass (PDEM) function for two molecular pseudoharmonic and modified Kratzer (Mie-type) potentials. In mapping the transformed exactly solvable D-dimensional ($D\geq 2$) Schrödinger equation with constant mass into the effective mass equation by employing a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of vario…
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.
The aim of this paper is to find the exact solutions of the Schrodinger equation. As is known, the Schrodinger equation can be reduced to the continuum equation. In this paper, using the non-linear Legendre transform the equation of continuity is linearized. Particular solutions of such a linear equation are found in the paper and an inverse Legendre transform is considered for them with subsequent construction of solutions of the Schrodinger equation. Examples of the classical and quantum systems are considered.
Exact solutions of the inhomogeneous nonlinear Schrödinger equation through supersymmetric potentials
本文通过超对称方法构造非均匀非线性薛定谔方程的精确解，利用波斯-泰勒势作为示例展示其应用。
We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.
Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n<5 for some diatomic molecules.
Exactly solvable models of nonlinear extensions of the Schrödinger equation
本文提出一种构造Schrödinger方程精确可解非线性扩展的方法，通过建立精确可解的常微分Schrödinger方程与非线性理论间的对应关系，重新推导了孤子解并发现了新的精确可解非线性理论，可推广至相对论性孤子理论。
Utilizing an ${\it ansatz}$ for the eigenfunctions, we arrive at an exact closed form solution to the Schrödinger equation with the inverse-power potential, $V(r)=ar^{-4}+br^{-3}+cr^{-2}+dr^{-1}$ in two dimensions, where the parameters of the potential $a, b, c, d$ satisfy a constraint.
In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 {\sl J.\ Phys.\ A} {\bf 53} 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen-Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension t…
Stationary 1D Schrödinger equations with polynomial potentials are reduced to explicit countable closed systems of exact quantization conditions, which are selfconsistent constraints upon the zeros of zeta-regularized spectral determinants, complementing the usual asymptotic (Bohr--Sommerfeld) constraints. (This reduction is currently completed under a certain vanishing condition.) In particular, the symmetric quartic oscillators are admissible systems, and the formalism is tested upon them. Enforcing the exact and asymptotic constraints by suitable iterative schemes, we numerically observe ge…
Exact solutions of Schrodinger equation are obtained for the modified Kratzer and the corrected Morse potentials with the position-dependent effective mass. The bound state energy eigenvalues and the corresponding eigenfunctions are calculated for any angular momentum for target potentials. Various forms of point canonical transformations are applied. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd Keywords: Morse potential, Kratzer potential, Position-dependent mass, Point canonical transformation, Effective mass Schrödinger equation.
An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrodinger equation. Thus there is an exact formulation of the uncertainty principle which precisely captures the essence of what is "quantum" about quantum mechanics.
New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.
We introduce a new exactly integrable potential for the Schrödinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients.
By applying an ansatz to the eigenfunction, an exact closed form solution of the Schrödinger equation in 2D is obtained with the potentials, $V(r)=ar^2+br^4+cr^6$, $V(r)=ar+br^2+cr^{-1}$ and $V(r)=ar^2+br^{-2}+cr^{-4}+dr^{-6}$, respectively. The restrictions on the parameters of the given potential and the angular momentum $m$ are obtained.
Making use of an ${\it ansatz}$ for the eigenfunctions, we obtain an exact closed form solution to the non-relativistic Schrödinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$ in two dimensions, where the parameters of the potential $a, b, c$ satisfy some constraints.
We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schrödinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential, which belongs to a general Heun family, is a short-range one that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. We derive the exact spectrum equation for the energy and discuss the bound states supported by the potential.
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.
We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schrödinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. S…
It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrodinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3)-space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentra…
Exact solutions of the Schrodinger equation are obtained for the Rosen-Morse and Scarf potentials with the position-dependent effective mass by appliying a general point canonical transformation. The general form of the point canonical transformation is introduced by using a free parameter. Two different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
Quasi-Exactly Solvable Schrödinger Equations occupy an intermediate place between exactly-solvable (e.g. the harmonic oscillator and Coulomb problems etc) and non-solvable ones. Their major property is an explicit knowledge of several eigenstates while the remaining ones are unknown. Many of these problems are of the anharmonic oscillator type with a special type of anharmonicity. The Hamiltonians of quasi-exactly-solvable problems are characterized by the existence of a hidden algebraic structure but do not have any hidden symmetry properties. In particular, all known one-dimensional (quasi)-…
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.
Exact solvability (ES) of one-dimensional quantum potentials $V(x)$ is a vague concept. We propose that beyond its most conventional range the ES status should be attributed also to many less common interaction models for which the wave functions remain piecewise proportional to special functions. The claim is supported by constructive analysis of a toy model $V(x)= -g^2\exp (-|x|)$. The detailed description of the related bound-state and scattering solutions of Schrödinger equation is provided in terms of Bessel functions which are properly matched in the origin.
We use the optimized trigonometric finite basis method to find energy eigenvalues and eigenfunctions of the time-independent Schrodinger equation with high accuracy. We apply this method to the quartic anharmonic oscillator and the harmonic oscillator perturbed by a trigonometric anharmonic term as not exactly solvable cases and obtain the nearly exact solutions.
We outline a general method for obtaining exact solutions of Schrödinger equations with a position dependent effective mass and compare the results with those obtained within the frame of supersymmetric quantum theory. We observe that the distinct effective mass Hamiltonians proposed in the literature in fact describe exactly equivalent systems having identical spectra and wave functions as far as exact solvability is concerned. This observation clarifies the Hamiltonian dependence of the band-offset ratio for quantum wells.
A prescription is given to obtain some exact results for certain external potentials $V\left({\vec r}\right)$ of the time-independent Gross-Pitaevskii and Schrödinger equations. The study motivation is the ability to program $V\left({\vec r}\right)$ experimentally in Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given $V\left({\vec r}\right)$, we ask which $V\left({\vec r}\right)$ will have a given pdf (probability density function) $P\left({\vec r}\right)$. Several examples in 1D and 2D are presented for well-known pdfs and for the hydrogen atom in momen…
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.
We consider the radial Schr\" odinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.
The Darboux transformation operator technique in differential and integral forms is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations of an arbitrary order. A relation between supersymmetry and the generalized Darboux transformation is considered. The method is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials with…
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