Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

The study involves finding exact eigenvalues of the radial Schrödinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.     

A new method of solving the many-body Schrödinger equation

A new method of solving the many-body Schrödinger equation is proposed. It is based on the use of constant particle-particle interaction potential surfaces (IPSs) and the representation of the many-body wave function in a configuration interaction form with coefficients depending on the total interaction potential. For these coefficients the corresponding set of linear ordinary differential equations is obtained. A hierarchy of approximations is developed for IPSs. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with increasing basis set.     

Comment on “a new approach to solve the Schrödinger equation with an anharmonic sextic potential”

We analyze a recent application of the Nikiforov-Uvarov (NU) method to an N -dimensional anharmonic oscillator with a central-field sextic potential-energy function. We show that most of the equations derived by the author exhibit errors (or typos) and that his interpretation of the results may not be correct. By means of the Frobenius (power-series) method we derive exact particular solutions to the Schrödinger equation and compare them with those coming from the NU method.

     

Pseudospectral methods of solution of the Schrödinger equation

Several different pseudospectral methods of solution of the Schrödinger equation are applied to the calculation of the eigenvalues of the Morse potential for I₂ and the Cahill–Parsegian potential for Ar₂ [Cahill, Parsegian, J. Chem. Phys. 121, 10839 (2004)]. The calculation of the eigenvalues for the Woods–Saxon potential are also considered. The convergence of the eigenvalues with a quadrature discretization method is found to be very fast owing to the judicious choice for the weight function, basis set and quadrature points. The weight function used is either related to the exact ground state wavefunction, if known, or an approximation to it from some reference potential. We compare several different pseudospectral methods.     

A new two-step hybrid method for the numerical solution of the Schrödinger equation

With this paper, a new algorithm is developed for the numerical solution of the one-dimensional Schrödinger equation. The new method uses the minimum order of the phase-lag and its derivatives. Error analysis and the numerical results illustrate the efficiency of the new algorithm.     

Novel bound states treatment of the two dimensional Schrödinger equation with pseudocentral potentials plus multiparameter noncentral potential

By converting the rectangular basis potential V(x, y) into the form as \({V({r}) + V({r},\varphi)}\) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrödinger equation in plane-polar coordinates have been carried out through the analytic approaching technique of the Nikiforov and Uvarov. Both the exact bound state energy spectra and the corresponding bound state wavefunctions of the complete system are determined explicitly and in closed forms. Our presented results are identical to those of the previous works and they may also be useful for investigation and analysis of structural characteristics in a variety of quantum systems.     

New optimized two-derivative Runge-Kutta type methods for solving the radial Schrödinger equation

This paper focuses on adapted two-derivative Runge-Kutta (TDRK) type methods for solving the Schrödinger equation. Two new TDRK methods are derived by nullifying their phase-lags and the first derivatives of the phase-lags. Error analysis is carried out by means of asymptotic expressions of the local errors. Numerical results are reported to show the efficiency and robustness of the new methods in comparison with some RK type methods specially tuned to the integration of the radial time-independent Schrödinger equation with the Woods–Saxon potential.     

Extended Sobolev and Hilbert spaces and approximate stationary solutions for electronic systems within the non-linear Schrödinger equation

The definition of Sobolev spaces, which has already been shown to be a convenient way to set up the Schrödinger equation for approximate stationary solutions within extended Hilbert spaces, is readily generalized in order to express, in a similar way, the so-called non-linear Schrödinger equation (NLSE). The unavoidable theory, related to extended Hilbert and Sobolev spaces, is previously described in order to design the formalism inherent to the approximate NLSE. Afterwards the nature of the NLSE stationary solutions is discussed. The procedure uses as a basic tool an implied N-electron quantum self-similarity measure, provided with the structure of an overlap-like measure form, involving the integral of the fourth power of the N-electron wavefunction. Computation of this theoretical element is sketched and a two-electron case is developed as an illustrative example within the LCAO MO framework. The N-electron Slater determinant situation is also presented under the additional help of the nested sums formalism. It is shown afterwards that addition of second order gradient terms on the extended wavefunction provides variation of mass with velocity corrections into the energy expression. Finally, use into the Hamilton operator of exponential terms depending on the density functions in the extended Hilbert spaces formalism provides the theory with a general structure.     

Exact solution of Schrödinger equation with deformed ring-shaped potential

Exact solution of the Schrödinger 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.AMS Subject Classification: 03.65.–w, 12.39.Jh, 21.10.–k     

Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods

The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and Numerov type methods are used to solve it. Specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe et al. [Int. J. Comp. Math 32 (1990) 233–242], and the minimum phase-lag method of Rao et al. [Int. J. Comp. Math 37 (1990) 63–77] are applied to this problem. All methods are applied for the computation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon–Heils potential. The results are compared with the results produced by full discterization.     

Exact solution of the Schrödinger equation with a Lennard–Jones potential

The Schrödinger equation with a Lennard–Jones potential is solved by using a procedure that treats in a rigorous way the irregular singularities at the origin and at infinity. Global solutions are obtained thanks to the computation of the connection factors between Floquet and Thomé solutions. The energies of the bound states result as zeros of a function defined by a convergent series whose successive terms are calculated by means of recurrence relations. The procedure gives also the wave functions expressed either as a linear combination of two Laurent expansions, at moderate distances, or as an asymptotic expansion, near the singular points. A table of the critical intensities of the potential, for which a new bound state (of zero energy) appears, is also given.     

Trigonometrically fitted high-order predictor–corrector method with phase-lag of order infinity for the numerical solution of radial Schrödinger equation

In this paper, we present a new optimized symmetric ten-step predictor–corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric eight-step predictor–corrector method of Simos and et al, that is constructed to solve numerically the radial Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.     

Unique continuation for the magnetic Schrödinger equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.     

A Laplace transform approach to find the exact solution of the $$$$-dimensional Schrödinger equation with Mie-type potentials and construction of Ladder operators

The second order \(N\)-dimensional Schrödinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution theorem. The Ladder operators are also constructed for the Mie-type potentials in \(N\)-dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.     

An efficient six-step method for the solution of the Schrödinger equation

In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are:

• It is of twelfth algebraic order.
• It has three stages.
• It has vanished phase-lag.
• It has vanished its derivatives up to order two.
• All the stages of the scheme are approximations on the point \(x_{n+3}\).

This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature.

     

Sixth algebraic order trigonometrically fitted predictor–corrector methods for the numerical solution of the radial Schrödinger equation

Our new trigonometrically fitted predictor–corrector (P–C) schemes presented here are based on the well known Adams–Bashforth–Moulton methods: the predictor is based on the fifth order Adams–Bashforth scheme and the corrector on the sixth order Adams–Moulton scheme. We tested the efficiency of our newly developed schemes against well known methods, with excellent results. The numerical experiments showed that at least one of our schemes is noticeably more efficient compared to other methods, some of which are specially designed for this type of problem. It is also worth mentioning that this is the first time that sixth algebraic order trigonometrically fitted Adams–Bashforth–Moulton P–C schemes are used to efficiently solve the radial Schrödinger equation.Active Member of the European Academy of Sciences and Arts     

Bound state solution of the Schrödinger equation for Mie potential

Exact solution of Schr?dinger equation for the Mie potential is obtained for an arbitrary angular momentum. The energy eigenvalues and the corresponding wavefunctions are calculated by the use of the Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The bound states are calculated numerically for some values of ℓ and n with n ≤ 5. They are applied to several diatomic molecules.      

A new fourteenth algebraic order finite difference method for the approximate solution of the Schrödinger equation

In the present paper we will develop and analyse a new five-stages symmetric two-step method of high algebraic order with vanished phase-lag and its first, second, third, fourth and fifth derivatives. We will construct the new method. We will compute its local local truncation error (LTE). We will produce the asymptotic form of the LTE applying the new method to the radial time independent Schrödinger equation and we will compare it with other asymptotic forms of LTE of similar methods. Applying the new method to a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, we will investigate the stability and the interval of periodicity of the new method based and we will compare the produced interval of periodicity with other intervals of similar methods. Finally, we will examine the effectiveness of the new method applying it to the coupled Schrödinger equations.