Second-Order Supersymmetric Partners
of the Trigonometric Rosen–Morse
Potential
Rosa Reyes, D. J. Fernández, and H. Gasperín
Abstract The second-order supersymmetric partners of the trigonometric Rosen–
Morse potential are studied. The stationary Schrödinger equation for this potential
is solved in such a way that the general solution supplies straightforwardly the
eigenstates of the Hamiltonian while the non-physical solutions turn out to be conveniently expressed for characterizing its global properties. This allows to implement
in a simple and systematic way the second-order supersymmetry transformations.
Keywords Supersymmetric quantum mechanics · Trigonometric Rosen–Morse
potential
1 Introduction
The trigonometric Rosen–Morse (TRM) potentials belong to the exactly solvable
class of potentials, i.e., there exist explicit analytic expressions for their energy
eigenstates and eigenvalues [1, 2]. These potentials are interesting in physics mainly
for two reasons: the first one is their possible use for describing the quark-gluon
interaction in quantum chromodynamics [3]; the second one is their intrinsic
properties, making them ideal as a toy model for studying nonlinear algebras and
supersymmetric quantum mechanics (SUSY QM) [4–6]. For example, they have a
relatively simple dependence of the x-coordinate in a finite domain. In addition,
they have an infinite discrete energy spectrum, with a nonlinear dependence of the
energy levels on the index labeling them, making these potentials a clear case study
for nonlinear algebras.
On the other hand, SUSY QM is a powerful tool for generating, from an exactly
solvable initial Hamiltonian, new families of exactly solvable Hamiltonians whose
spectra are quite similar to the initial one [7–11]. In this work we will apply the
R. Reyes () · D. J. Fernández · H. Gasperín
Departamento de Física, Cinvestav, AP, Ciudad de México, Mexico
e-mail: rmreyes@fis.cinvestav.mx
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_22
235
of the Trigonometric Rosen–Morse
Potential
Rosa Reyes, D. J. Fernández, and H. Gasperín
Abstract The second-order supersymmetric partners of the trigonometric Rosen–
Morse potential are studied. The stationary Schrödinger equation for this potential
is solved in such a way that the general solution supplies straightforwardly the
eigenstates of the Hamiltonian while the non-physical solutions turn out to be conveniently expressed for characterizing its global properties. This allows to implement
in a simple and systematic way the second-order supersymmetry transformations.
Keywords Supersymmetric quantum mechanics · Trigonometric Rosen–Morse
potential
1 Introduction
The trigonometric Rosen–Morse (TRM) potentials belong to the exactly solvable
class of potentials, i.e., there exist explicit analytic expressions for their energy
eigenstates and eigenvalues [1, 2]. These potentials are interesting in physics mainly
for two reasons: the first one is their possible use for describing the quark-gluon
interaction in quantum chromodynamics [3]; the second one is their intrinsic
properties, making them ideal as a toy model for studying nonlinear algebras and
supersymmetric quantum mechanics (SUSY QM) [4–6]. For example, they have a
relatively simple dependence of the x-coordinate in a finite domain. In addition,
they have an infinite discrete energy spectrum, with a nonlinear dependence of the
energy levels on the index labeling them, making these potentials a clear case study
for nonlinear algebras.
On the other hand, SUSY QM is a powerful tool for generating, from an exactly
solvable initial Hamiltonian, new families of exactly solvable Hamiltonians whose
spectra are quite similar to the initial one [7–11]. In this work we will apply the
R. Reyes () · D. J. Fernández · H. Gasperín
Departamento de Física, Cinvestav, AP, Ciudad de México, Mexico
e-mail: rmreyes@fis.cinvestav.mx
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_22
235