Solutions of the nonrelativistic wave equation with position-dependent effective mass
TL;DR: In this article, the energy spectrum of the bound states and their wave functions are explicitly written down and mapped the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation.
Abstract: Given a spatially dependent mass distribution, we obtain potential functions for exactly solvable nonrelativistic problems. The energy spectrum of the bound states and their wave functions are written down explicitly. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation. The Oscillator, Coulomb, and Morse class of potentials are considered.
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TL;DR: In this article, the authors show that there exist intimate connections between three unconventional Schrodinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space.
Abstract: We show that there exist some intimate connections between three unconventional Schrodinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.
257 citations
TL;DR: In this article, the shape-invariant superpotential of the Schrodinger equation is taken as effective potentials in a position-dependent effective mass (PDEM) one.
Abstract: Known shape-invariant potentials for the constant-mass Schrodinger equation are taken as effective potentials in a position-dependent effective mass (PDEM) one. The corresponding shape-invariance condition turns out to be deformed. Its solvability imposes the form of both the deformed superpotential and the PDEM. A lot of new exactly solvable potentials associated with a PDEM background are generated in this way. A novel and important condition restricting the existence of bound states whenever the PDEM vanishes at an end point of the interval is identified. In some cases, the bound-state spectrum results from a smooth deformation of that of the conventional shape-invariant potential used in the construction. In others, one observes a generation or suppression of bound states, depending on the mass-parameter values. The corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials.
216 citations
TL;DR: In this article, the analytical series solutions of the Schrodinger equation with position-dependent mass for the Morse potential are obtained by the series expansion method, and the position dependent mass themselves are expanded in the series about the origin.
196 citations
TL;DR: In this article, the authors show that there exist intimate connections between three unconventional Schrodinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively.
Abstract: We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.
189 citations
TL;DR: In this paper, the authors studied the Schrodinger equation with spatial mass distributions for a class of exactly solvable potentials and obtained 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 precisely solvable.
186 citations
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TL;DR: In this paper, an exact solution for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others, was obtained.
Abstract: An exact solution is obtained for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others. The allowed vibrational energy levels are found to be given by the formula $E(n)={E}_{e}+h{\ensuremath{\omega}}_{0}(n+\frac{1}{2})\ensuremath{-}h{\ensuremath{\omega}}_{0}x{(n+\frac{1}{2})}^{2}$, which is known to express the experimental values quite accurately. The empirical law relating the normal molecular separation ${r}_{0}$ and the classical vibration frequency ${\ensuremath{\omega}}_{0}$ is shown to be ${{r}_{0}}^{3}{\ensuremath{\omega}}_{0}=K$ to within a probable error of 4 percent, where $K$ is the same constant for all diatomic molecules and for all electronic levels. By means of this law, and by means of the solution above, the experimental data for many of the electronic levels of various molecules are analyzed and a table of constants is obtained from which the potential energy curves can be plotted. The changes in the above mentioned vibrational levels due to molecular rotation are found to agree with the Kratzer formula to the first approximation.
3,299 citations
TL;DR: In this article, general conditions for dynamical supersymmetry breaking are discussed and examples are given (in 0 + 1 and 2 + 1 dimensions) in which such a program in four dimensions is possible.
3,270 citations
Book•
01 Jan 1966
3,150 citations
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.
2,688 citations