8 research outputs found
New Eaxactly Solvable Hamiltonians: Shape Invariance and Self-Similarity
We discuss in some detail the self-similar potentials of Shabat and
Spiridonov which are reflectionless and have an infinite number of bound
states. We demonstrate that these self-similar potentials are in fact shape
invariant potentials within the formalism of supersymmetric quantum mechanics.
In particular, using a scaling ansatz for the change of parameters, we obtain a
large class of new, reflectionless, shape invariant potentials of which the
Shabat-Spiridonov ones are a special case. These new potentials can be viewed
as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for the energy eigenvalues,
eigenfunctions and transmission coefficients for these potentials are obtained.
We show that these potentials can also be obtained numerically. Included as an
intriguing case is a shape invariant double well potential whose supersymmetric
partner potential is only a single well. Our class of exactly solvable
Hamiltonians is further enlarged by examining two new directions: (i) changes
of parameters which are different from the previously studied cases of
translation and scaling; (ii) extending the usual concept of shape invariance
in one step to a multi-step situation. These extensions can be viewed as
q-deformations of the harmonic oscillator or multi-soliton solutions
corresponding to the Rosen-Morse potential.Comment: 26 pages, plain tex, request figures by e-mai
A Group-Theoretical Method for Natanzon Potentials in Position-Dependent Mass Background
A new manner for deriving the exact potentials is presented. By making use of
conformal mappings, the general expression of the effective potentials deduced
under su(1,1) algebra can be brought back to the general Natanzon
hypergeometric potentials
Exact solutions of the radial Schrodinger equation for some physical potentials
By using an ansatz for the eigenfunction, we have obtained the exact
analytical solutions of the radial Schrodinger equation for the pseudoharmonic
and Kratzer potentials in two dimensions. The energy levels of all the bound
states are easily calculated from this eigenfunction ansatz. The normalized
wavefunctions are also obtained.Comment: 13 page