Effective Mass Schrödinger Equation via Point Canonical Transformation

Exact solutions of the effective radial Schrödinger equation are obtained for some inverse potentials by using the point canonical transformation. The energy eigenvalues and the corresponding wave functions are calculated by using a set of mass distributions.     

A class of exactly solvable Schrödinger equation with moving boundary condition

Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary condition.     

Exactly solvable associated Lamé potentials and supersymmetric transformations

A systematic procedure to derive exact solutions of the associated Lamé equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.     

Intertwining operator method and supersymmetry for effective mass Schrödinger equations

By application of the intertwining operator method to Schrödinger equations with position-dependent (effective) mass, we construct Darboux transformations, establish the supersymmetry factorization technique and show equivalence of both formalisms. Our findings prove equivalence of the intertwining technique and the method of point transformations.     

Two-dimensional Schrödinger Hamiltonians with effective mass in SUSY approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties—symmetry, irreducibility, and separation of variables—are investigated.     

Generalized Morse and Pöschl-Teller potentials: The connection via Schrödinger equation

A systematic and unified treatment to connect the Schrödinger equation for generalized Morse and Pöschl-Teller potentials, generated by supersymmetry quantum mechanics, is used. An algebraic treatment of bound-state problems is presented.     

Schrödinger equation with delta potential in superspace

A superspace version of the Schrödinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is smaller than or equal to 1. In the case when there is one commuting and 2n anti-commuting variables also the wave function is given explicitly.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

Approximate Solutions of the Schrödinger Equation for the Eckart Potential and Its Parity-Time-Symmetric Version Including Centrifugal Term

The energy eigenvalues and eigenfunctions of the Schrödinger equation for Eckart potential as well as the parity-time-symmetric version of the potential in three dimensions with the centrifugal term are investigated approximately by using the Nikiforov-Uvarov method. To show the accuracy of our results, we calculate the energy eigenvalues for various values of n and l. It is found that the results are in good agreement with the numerical solutions for short-range potential (large a). For the case of 1/a i/a, the potential is also studied briefly.     

Perturbed Coulomb potentials in the Klein-Gordon equation via the asymptotic iteration method

The asymptotic iteration method is used to construct the exact energy eigenvalues for a Lorentz vector or a Lorentz scalar, and an equally mixed Lorentz vector and Lorentz scalar Coulombic potentials. Highly accurate and rapidly converging ground-state energies for Lorentz vector Coulomb with a Lorentz vector or a Lorentz scalar linear potential, , respectively, are obtained.     

Exactly solvable energy-dependent potentials

We introduce a method for constructing exactly-solvable Schrödinger equations with energy-dependent potentials. Our method is based on converting a general linear differential equation of second order into a Schrödinger equation with energy-dependent potential. Particular examples presented here include harmonic oscillator, Coulomb and Morse potentials with various types of energy dependence.     

Parameter differentiation and quantum state decomposition for time varying Schrödinger equations

For the unitary operator, solution of the Schrödinger equation corresponding to a time-varying Hamiltonian, the relation between the Magnus and the product of exponentials expansions can be expressed in terms of a system of first-order differential equations in the parameters of the two expansions. A method is proposed to compute such differential equations explicitly and in a closed form.     

Trapping neutral fermions with a PT-symmetric trigonometric potential in the presence of position-dependent mass

The relativistic problem of neutral fermions subject to PT-symmetric trigonometric potential (∼iαtanαx)$(\sim \mathrm{i}\alpha \tan \alpha x)$ in 1+1$1+1$ dimensions is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Dirac equation with the vector coupling scheme and obtain the bound state solutions in closed form. The behavior of the energy spectra is discussed in detail.     

Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass

We present a new method to construct the exactly solvable PT-symmetric potentials within the framework of the position-dependent effective mass Dirac equation with the vector potential coupling scheme in 1 + 1 dimensions. In order to illustrate the procedure, we produce three PT-symmetric potentials as examples, which are PT-symmetric harmonic oscillator-like potential, PT-symmetric potential with the form of a linear potential plus an inversely linear potential, and PT-symmetric kink-like potential, respectively. The real relativistic energy levels and corresponding spinor components for the bound states are obtained by using the basic concepts of the supersymmetric quantum mechanics formalism and function analysis method.     

Exact solution of the Schrödinger equation for a particle in a regular N-simplex

We present the exact solution for the Schrödinger equation for a particle inside an N-dimensional regular simplex shaped enclosure. This result extends and unifies the earlier results for equilateral triangle and K-tetrahedron billiards.     

Solutions to the Modified Pöschl-Teller Potential in D-Dimensions

An approximate solution of the D-dimensional Schrödinger equation with the modified Pöschl-Teller potential is obtained with an approximation of the centrifugal term. Solution to the corresponding hyper-radial equation is given using the conventional Nikiforov-Uvarov method. The normalization constants for the Pöschl-Teller potential are also computed. The expectation values of -2_> and are also obtained using the Feynman-Hellmann theorem.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

A time-dependent closed-form formulation of the linear unitary transformation for harmonic-oscillator annihilation and creation operators is presented in the Schrödinger picture using the Lie algebraic approach. The time evolution of the quantum mechanical system described by a general time-dependent quadratic Hamiltonian is investigated by combining this formulation with the time evolution equation of the system. The analytic expressions of the evolution operator and propagator are found. The motion of a charged particle with variable mass in the time-dependent electric field is considered as an illustrative example of the formalism. The exact time evolution wave function starting from a Gaussian wave packet and the operator expectation values with respect to the complicated evolution wave function are obtained readily.     

Exactly complete solutions of the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential

A spherically harmonic oscillatory ring-shaped potential is proposed and its exactly complete solutions are presented by the Nikiforov-Uvarov method. The effect of the angle-dependent part on the radial solutions is discussed.