Using operators of the invariance group of their time-dependent Schr\"odinger equations it is shown that the bound states of Morse oscillators are bases for discrete nonunitary irreducible representations (non-UIRs) of the groups $\mathrm{Sp}(2, R)$ , SU(1,1), and SO(2,1). Hermiticity requirements and the dependence of the group generators on the dynamical variables determine the operator of an isometry which connects these non-UIRs of the noncompact invariance groups to UIRs of the compact groups SU(2) and SO(3) that have been used as spectrum-generating groups for Morse systems.Using operators of the invariance group of their time-dependent Schr\"odinger equations it is shown that the bound states of Morse oscillators are bases for discrete nonunitary irreducible representations (non-UIRs) of the groups $\mathrm{Sp}(2, R)$, SU(1,1), and SO(2,1). Hermiticity requirements and the dependence of the group generators on the dynamical variables determine the operator of an isometry which connects these non-UIRs of the noncompact invariance groups to UIRs of the compact groups SU(2) and SO(3) that have been used as spectrum-generating groups for Morse systems.

Physical Review AAug 1, 19963 citations

Bound State Solutions of the Schrödinger Equation

Ajoy Kumar Ghatak, S. Lokanathan

Springer eBooksJan 1, 20040 citations

POTENTIAL REPRESENTATION METHOD FOR THE SCHRÖDINGER EQUATION

Arvydas Juozapas Janavičius, Donatas Jurgaitis, Sigita Turskienė

A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.

Mathematical Modelling and AnalysisAug 24, 20111 citation

Comments on a bound state model for a two body system

L. Micu

We show that in classical mechanics, as well as in nonrelativistic quantum mechanics the equation of the relative motion for a two-body bound system at rest can be replaced by individual dynamical equations of the same kind as the first one, but with different parameters. We assume that in relativistic quantum mechanics the individual equations are Dirac equations with modified parameters in agreement with the individual Schrödinger equations. We find that products of solutions to the individual equations with correlated arguments are the quantum analogues of the classical representation of a bound system and represent suitable models for the bound state wave functions. As validity test for the new representation of bound states we suggest to use some observable differences between this one and the representation in terms of relative coordinates.We show that in classical mechanics, as well as in nonrelativistic quantum mechanics the equation of the relative motion for a two-body bound system at rest can be replaced by individual dynamical equations of the same kind as the first one, but with different parameters. We assume that in relativistic quantum mechanics the individual equations are Dirac equations with modified parameters in agreement with the individual Schrödinger equations. We find that products of solutions to the individual equations with correlated arguments are the quantum analogues of the classical representation of a bound system and represent suitable models for the bound state wave functions. As validity test for the new representation of bound states we suggest to use some observable differences between this one and the representation in terms of relative coordinates.

arXiv:quant-ph/0602104Feb 13, 20060 citations

\mathrm{Sp}(2, R)

, SU(1,1), and SO(2,1). Hermiticity requirements and the dependence of the group generators on the dynamical variables determine the operator of an isometry which connects these non-UIRs of the noncompact invariance groups to UIRs of the compact groups SU(2) and SO(3) that have been used as spectrum-generating groups for Morse systems.