Exact solutions of dirac and schrodinger equations for a. Dec 27, 2016 a need for an exactly solvable model a. As such, this equation corresponds to the mindependent schrodinger equation for the square well that. The third exactly solvable hypergeometric potential. Beside the wellknown families, we explicitly describe two new classes of exactly solvable schrodinger equations that can be reduced to the hermite equation. Effective mass schrodinger equation for exactly solvable. The lambert w steppotential an exactly solvable confluent hypergeometric potential the lambert w steppotential an exactly solvable confluent hypergeometric potential ishkhanyan, a. By exploiting the hidden algebraic structure of the schrodinger hamiltonian, namely the sl2, we propose a unified approach of generating both exactly solvable and quasi exactly solvable quantum potentials. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. That is, the proposed exactly solvable multiparameter exponential. The latter is solved by point canonical transformation of the exactly.
The present paper discusses the connectionbetween exactly solvable schrodinger equations and theliouville transformation. There are two methods of obtaining exactly solvable potentials in quantum mechanics. Mathematically, the time independent schrodinger equation together. However, there exists a subset of physical systems for which the form of the eigenfunctions and their. We are looking for solutions to the corresponding equation for the intertwined hamiltonian hq, hq qdeq q. We study the schrodinger equation with spatial mass distributions for a class of exactly solvable potentials. In addition, this class is shown to containtwo new families of exactly solvablepotentials. The solvable potentials considered to date are given in terms of polynomials of the spatial coordinate, and in terms of exponential, trigonometric, hyperbolic, and, recently, even elliptic functions of. The boundstate solutions are easily calculated from this eigenfunction ansatz.
Hermitian potential functions by using the transforma tion. Exactly solvable potentials in quantum mechanics university of. However, it had to come from somewhere, and it is indeed possible to derive the schr odinger equation using. Point canonical transformation versus deformed shape. Exactly solvable schrodinger equation for a class of multiparameter exponentialtype potentials article in international journal of quantum chemistry 1121 january 2012 with 91 reads. The hydrogen atom, consisting of an electron and a proton, is a twoparticle system, and the internal motion of two particles around their center of mass is equivalent to the motion of a single particle with a reduced mass.
We systematically describe and classify onedimensional schrodinger equations that can be solved in terms of hypergeometric type functions. We deal with the exact solutions of schrodinger equation characterized by positiondependent effective mass via point canonical transformations. We obtain, in this way, two new classes of quasi exactly solvable systems one of which is of periodic type while the other hyperbolic. Dirac equation with positiondependent effective mass and. Supersymmetry, operator transformations and exactly solvable. The 1d schrodinger equation with elliptic potentials. These quantummechanical systems with analytical solutions are listed below. Schrodinger equation with the hyperbolic scarf potential defines the polar angle part of the re spective. We have derived a general expression for the potentials also including natanzon confluent potentials. To show the usefulness of the proposed approach, we consider explicitly. Before we take the giant leap into wonders of quantum mechanics, we shall start with a brief. Exactly solvable potentials of calogero type for qdeformed.
Exactly solvable potentials and romanovski polynomials in. Introduction though much insight in quantum mechanics is gained from understanding the solutions of the schrodinger equation in terms of special functions, such solutions are rare. The supersymmetric quantum mechanics manages the family of exactly solvable potentials presented in table 1. International journal of recent advances in physics ijrap. Complex square wella new exactly solvable model 6775 and we have used the identity lim. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found. The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. Effective mass schrodinger equation for exactly solvable class of onedimensional potentials article pdf available in journal of mathematical chemistry 431. 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. Solution of the radial schrodinger equation for the. Complex square wella new exactly solvable quantum mechanical.
This solution yields a class of exactly solvable potentials and can be used to calculate the ground state for the class of these potentials. The threedimensional schrodinger equation with an effective mass is solved for a new class of angular momentum dependent potentials with varying depths and shapes. With the choice of positiondependent mass forms, exactly solvable target potentials are. Sep 28, 2010 the classes of exactly solvable multiline soliton potentials and corresponding wave functions of twodimensional stationary schrodinger equation via. Introduction the study of exactly solvable schr6dinger equations has a long history 11.
Exactly solvable schrodinger equation for a class of. Each quantum mechanical system is characterised by the potential energy function v r. The hamiltonjacobische differential equation is a partial differential equation. Solution of a novel quasiexactly solvable potential via. Supersymmetry, operator transformations and exactly. Exceptional orthogonal polynomials, exactly solvable. A large class of potentials can be solved algebraically by using supersymmetry. Depending on an involved parameter, the potential presents either a shortrange singular well which behaves as inverse. The solvable potentials considered to date are given in terms of polynomials of the spatial coordinate, and in terms of exponential, trigonometric, hyperbolic, and, recently, even elliptic functions of the spatial coordinate. As an illustrative example, we discuss the interrelation between the generalized coulomb and oscillator systems. Frobenius series solutions of the schrodinger equation with. Note that all solvable potentials that we consider are holomorphic on the complex plane, apart from some isolated singularities. We consider again the time dependent schrodinger equation prop.
Their energy of the bound states and corresponding. Exact solutions of the radial schrodinger equation for some. Lie symmetry and the bethe ansatz solution of a new quasi exactly solvable. Twodimensional stationary schrodinger equation via the. Their energy of the bound states and corresponding wavefunctions.
With the choice of positiondependent mass forms, exactly solvable target potentials are constructed. Exact solutions of a class of doublewell potentials. The corresponding normalized wavefunctions are also obtained. Dirac equation with position dependent effective mass and. Pdf effective mass schrodinger equation for exactly solvable.
Annals of physics 152, 203219 1984 a class of exactly solvable potentials. Solutions of onedimensional effective mass schrodinger equation. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with nonconstant coefficients of two confluent. The solution to a spectral problem involving the schrodinger equation for a particular class of multiparameter exponential. Though the original derivation of this result is a bit. A unified treatment of exactly solvable and quasiexactly. Stationary schrodinger equation, exactly solvable potential, steppotential, general heun potentials 1. In the context of the gf technique, steiner 10, 11, in the course of his work on radial path. The method provides us with direct way of calculating the wave func tions, energy eigenvalues and mass function for the gen erated system from the already solved potential. Though the paper by lamieux and bose was not noticed much, however, it contains a. Exactly solvable schrodinger equation for a class of multiparameter. A new class of quasiexactly solvable potentials with. The darboux transformation operator method is extended to the onedimensional stationary dirac equation. Pdf effective mass schrodinger equation for exactly.
An exactly solvable doublewell potential model for the onedimensional schrodinger equation is. The proposal is based on the exactly solvable constant mass schrodinger equation cmse for a class of multiparameter exponentialtype potentials, adapted to the positiondependentmass pdm. Exactly solvable schrodinger operators pdf paperity. Conservation laws and associated lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins.
Finally, in section 6 we will consider the extension of the theory to the 3d case. Mass function of parent system gets transformed to that of generated system. Solution of a novel quasi exactly solvable potential via asymptotic iteration method. We will describe both the complex and the real classi cation of such potentials. Rosenmorse ii, scarf ii and coulomb potentials using nu method 3. We obtain the general solution of a system of differential equations introduced by ge et al. One of the main problems in condensed matter physics is the con. It is natural to consider rst a classi cation of exactly solvable schr odinger equations of the complex variable. Exactly solvable noncentral potentials using nikiforovuvarov method. Liouville transformation and exactly solvable schrodinger.
Exactly solvable potentials for the schrdinger equation. Besides the relativistic coulomb problem, the dirac equation seems. Recently, the study of higher order anharmonic potentials has been much more desirable in di. This transformation yields alarge class of exactly solvable potentials, including the exactly solvable potentials introduced bynatanzon. Schrodinger potentials solvable in terms of the general. The eigenfunctions and scattering function are also given in closed form.
The schrodinger equation for the hydrogen atom can be solved exactly last updated. List of quantummechanical systems with analytical solutions. Exactly solvable schrodinger operators faculty of physics. A new exactly integrable hypergeometric potential for the. Exactly solvable potentials of calogero type for qdeformed coxeter groups andreas fring1 and christian korff2 1 centre for mathematical science, city university, northampton square, london ec1v 0hb, uk 2 school of mathematics, university of edinburgh, may. New potentials can be isospectral with the initial one or their spectra differ by one or two levels. A need for an exactly solvable model physics forums.
Mapping of noncentral potentials under point canonical. The schrodinger equation with general noncentral potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions for these. Related content exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics c quesne. For nontrivial potentials that have no exact solutions, kis always greater than nin these numerical solutions and the approximate.
The schrodinger equation is a linear partial differential equation that describes the wave function or state function of a quantummechanical system 12 it is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. Transformation to shape invariant potentials algebraic approach, mapping by canonical transformation, lie. The two potentials can be considered to be superpartners of each other. Algebraic bethe ansatz exact solutions of a class of doublewell potentials. The bound state energies are given as zeros of a calculable function. Exact solutions of schrodinger equation with solvable potentials in this section we will obtain the solutions of three exactly solvable potentials. Exactly solvable hydrogenlike potentials 10165 let h be a hamiltonian with known solutions to the timeindependent schrodinger. Exactly solvable potentials for the onedimensional.
One of the recently introduced methods for solving second order differential equations such as schrodinger, kleingordon, dirac equations, etc. The energy eigen, values and resonances are given in algebraic form as a function of the effective mass and depth and shape of the potential. Generation of exactly solvable potentials of position. The morse, poschlteller and hulthen type potentials are considered respectively. By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial schrodinger equation for the pseudoharmonic and the kratzer potentials in two dimensions. Therefore, the solution of the dirac equation in terms of orthogonal polynomials, with solvable non hermitian potentials are investigated within a novel formalism by transforming the relativistic equation into a schrodinger like one. The onedimensional schrodinger equation is solved for a new class of potentials with varying depths and shapes. Exactly solvable potentials from decemvirate power potential. We obtain the analytical solutions by transforming the mass distribution function into the effective potential, where both the original and the modified potential are physically relevant and exactly solvable. The spirit of et method is similar to that of susy quantum. Conditions governing the generalisation of threshold bound states by n attractive delta potentials in one and three dimensions.
This permits one, starting from an exactly solvable dirac equation, to obtain a family of matrixvalued exactly solvable dirac potentials. Exactly solvable potentials esps of positiondependent mass pdm schrodinger equation are generated from hulthen potential parent system by using extended transformation et method. The method includes a coordinate transformation ct followed by functional transformation ft of wave function. Schrodinger equation with a new potential, is discussed. These eigenvalues are the roots of a jthdegree polynomial. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Quantummechanical potentials are said to bequasiexactly solvableqes if a. Our proposal is based on an auxiliary function gx which determines the transformation needed to find exactlysolvable potentials associated.
Motivated by the observation that all known exactly solvable shape invariant central potentials are interrelated via point canonical transformations, we develop an algebraic framework to show that a similar mapping procedure is also exist between a class of noncentral potentials. The schrodinger equation for the hydrogen atom can be. Pdf exactly solvable schrodinger equation with hypergeometric. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry to cite this article.
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