Supersymmetric Solutions of PT-/non-PT-symmetric and Non-Hermitian Central Potentials via Hamiltonian Hierarchy Method

The supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian deformed Morse and Pöschl-Teller potentials are obtained by solving the Schrödinger equation. The Hamiltonian hierarchy method is used to get the real energy eigenvalues and corresponding eigenfunctions.     

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.     

Perturbational and variational treatments of the Morse oscillator

The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The Schrödinger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.     

Two-dimensional Darboux transformations for non-separable angular equations and solvable non-central potentials

We construct a two-dimensional Darboux transformation for the angular equation that arises from variable separation of a Schrödinger equation in spherical coordinates. Since we do not assume the angular equation to be further separable, our Darboux transformation allows for the systematic generation of solvable non-central potentials that are not accessible through the usual separation of variables.     

Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential

We solve the D‐dimensional Schrödinger equation under the Hua potential by using a Pekeris‐type approximation and the supersymmetry quantum mechanics. The reliability of the spectrum is checked via a comparison with the finite difference method. This interaction resembles Eckart, Morse, and Manning–Rosen potentials. Some useful quantities are reported via the Hellmann–Feynman Theorem. © 2012 Wiley Periodicals, Inc.     

An improvement to the comparison equation method for solving the Schrödinger equation

A systematic improved comparison equation method to solve the Schrödinger equation is described. The method is useful in quantum mechanical calculations involving two or more transition or turning points and is applicable to real potentials with continuous derivatives. As a computational example of the method, a study of the bound-state problem using the Morse potential is given.     

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.     

Solutions of N-dimensional Schrödinger equation with Morse potential via Laplace transforms

A study is undertaken to investigate an analytical solution for the N-dimensional Schrödinger equation with the Morse potential based on the Laplace transformation method. The results show that in the Pekeris approximation, the radial part of the Schrödinger equation reduces to the corresponding equation in one dimension. In addition, a comparison is made between the energy spectrum resulted from this method and the spectra that are obtained from the two-point quasi-rational approximation method and the Nikiforov–Uvarov approach.     

A new approach to the determination of good action-angle variables for coupled oscillator systems

A theory of action-angle variables for coupled oscillator systems is developed which involves solving the Schrödinger equation using a basis of WKB eigenfunctions, then using the logarithm of the resulting wavefunction to define the generator for the canonical transformation which determines the action-angle variables. This theory is based on the marriage between Miller's method for solving the Hamilton-Jacobi equation using the logarithm of the generating function, and the Ratner-Buch-Gerber method for solving the Schrödinger equation using WKB basis functions. A perturbation-theory analysis of this theory indicates that the semiclassical eigenvalues and canonical transformations obtained from it should become identical to their exact classical counterparts in the limit of large actions for each vibrational mode. Two methods for systematically improving the theory for the lower eigenstates are also proposed. Numerical applications of the theory are presented for two systems, the Morse oscillator and the Henon-Heiles two-mode hamiltonian. The resulting semiclassical eigenvalues are in excellent agreement with their exact quantum counterparts, with the magnitude of the error roughly independent of the energy of the eigenstate. Analogous good agreement is found in comparing the approximate and exact classical canonical transformations. In particular, for the Morse oscillator, good results are obtained for certain higher energy states where second-order classical perturbation theory makes serious errors. Other information examined includes surfaces of section for the Henon-Heiles system (comparing the analytical functions obtained from the present theory with results based on exact trajectory calculations) and vibrational distributions chosen to simulate trajectory calculations (using the present theory to determine bin boundaries for a histogram calculation). Again, the comparison in each case with accurate results is excellent, with maximum errors in action calculations of 0.02 h, and in angle calculations of 0.01 rad.     

Sequential versus direct multiphoton absorption

We present results of exact calculations on a three-level Morse oscillator modeling HF which suggest that multiphoton absorption proceeds by sequential single-level transitions, transitions arising from coupling between non-adjacent states being dynamically negligible. The time-dependent Schrödinger equation is integrated in the Floquet formalism.     

Modified <Emphasis Type="Italic">ℓ</Emphasis>-states of diatomic molecules subject to central potentials plus an angle-dependent potential

We present modified ?-states of diatomic molecules by solving the radial and angle-dependent parts of the Schrödinger equation for central potentials, such as Morse and Kratzer, plus an exactly solvable angle-dependent potential V _θ (θ)/r ² within the framework of the Nikiforov–Uvarov (NU) method. We emphasize that the contribution which comes from the solution of the Schrödinger equation for the angle-dependent potential modifies the usual angular momentum quantum number ?. We calculate explicitly bound state energies of a number of neutral diatomic molecules composed of a first-row transition metal and main-group elements for both Morse and Kratzer potentials plus an angle-dependent potential. We also compare the bound state energies for both potentials, taking into account spectroscopic parameters of diatomic molecules and arbitrary values of potential constants.     

The algebraic approach for the derivation of ladder operators and coherent states for the Goldman and Krivchenkov oscillator by the use of supersymmetric quantum mechanics

The ladder operators for the Goldman and Krivchenkov anharmonic potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The coherent states for the Goldman and Krivchenkov oscillator, which are the eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed within the framework of supersymmetric quantum mechanics. The constructed ladder operators can be a useful tool in quantum chemistry computations of non-trivial matrix elements. In particular, they can be employed in molecular vibrational–rotational spectroscopy of diatomic molecules to compute transition energies and dipole matrix elements.     

Minimum-uncertainty coherent states of the hyperbolic and trigonometric Rosen–Morse oscillators

A mixed supersymmetric-algebraic approach is employed to generate the minimum uncertainty coherent states of the hyperbolic and trigonometric Rosen–Morse oscillators. The method proposed produces the superpotentials, ground state eigenfunctions and associated eigenvalues as well as the Schrödinger equation in the factorized form amenable to direct treatment in the algebraic or supersymmetric scheme. In the standard approach the superpotentials are calculated by solution of the Riccati equation for the given form of potential energy function or by differentiation of the ground state eigenfunction. The procedure applied is general and permits derivation the exact analytical solutions and coherent states for the most important model oscillators employed in molecular quantum chemistry, coherent spectroscopy (femtochemistry) and coherent nonlinear optics.

     

Molecular translational-rovibronic hamiltonian. I. Non-linear molecules

The translational-rovibronic hamiltonian for a non-linear polyatomic molecule is derived by using the Schrödinger equation in tensor form and employing the Eckart conditions (determining the nuclear-framework rotational variables). The present derivation is a unified comprehensive one by a quantum-mechanical pathway and contrasts with fragmentary previous derivations via a classical-intermediate path. The method presented affords a firm conceptual picture of the nature of the transformation and the origin of coupling terms, and avoids mathematical complexities with their residue of obscurity. The correct form of the total angular momentum operators is also derived quantum mechanically.     

On the quantum momentum method for the exact solution of separale multiple well bound state and scattering problems

A simple method is suggested for the convenient application of Milne's method of solution of the one-dimensional Schrödinger equation to multiple well potentials. The method involves the introduction of “quantum momentum” functions which resemble the classical momentum over each “well”. The wavefunction is matched at intermediate point between the “wells”, and the quantization requirements are determined. The double oscillator and scattering by spherically symmetric potentials are cited as applications.     

Anharmonicity in rotational and vibrational excitation of H2 by Li+ collisions

Integral cross sections for pure rotational and vibrational-rotational excitation of H₂(X¹Σ⁺_g) by Li⁺(¹S) impact are computed by close-coupling methods at 0.2, 0.6, and 1.2 eV in the c.m. system using vibrational functions that are numerical solutions of the one-dimensional radial Schrödinger equation for harmonic, Morse, and adiabatically corrected Kolos-Wolniewicz (KW) potential functions. Comparison of results employing KW and Morse functions shows excellent agreement for all transitions studied. Findings using harmonic oscillator functions, however, differ noticeably from KW and Morse values for vibrational (0 → 1) and very large rotational (Δj = 10) transitions, but are satisfactory for lower order (0 → 2, 4, 6, 8) rotational transitions.     

Analytical solution of the Schrödinger equation for a double minimum morse potential and application to intramolecular inversion

An exact solution is obtained for the one-dimensional time-independent Schrödinger equation with a symmetric double minimum potential constructed from two Morse potentials. This model potential is used to describe the inversion motions in NH₃, ND₃, and NT₃, and its adequacy is discussed.     

Analytical solutions to the Hulthén and the Morse potentials by using the asymptotic iteration method

We present the exact analytical solution of the radial Schrödinger equation for the deformed Hulthén and the Morse potentials within the framework of the asymptotic iteration method. The bound state energy eigenvalues and corresponding wave functions are obtained explicitly. Our results are in excellent agreement with the findings of the other methods.     

New analytic approximations based on the Magnus expansion

The Magnus expansion is a frequently used tool to get approximate analytic solutions of time-dependent linear ordinary differential equations and in particular the Schrödinger equation in quantum mechanics. However, the complexity of the expansion restricts its use in practice only to the first terms. Here we introduce new and more accurate analytic approximations based on the Magnus expansion involving only univariate integrals which also shares with the exact solution its main qualitative and geometric properties.