The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at r = 0 and r → ∞ are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus). In the limit that the cutoff is reduced to zero, we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at r = 0 and r → ∞ are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus). In the limit that the cutoff is reduced to zero, we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.

arXiv:1612.06706American Journal of PhysicsApr 18, 20176 citations

Modeling atom–atom interactions at low energy by Jost–Kohn potentials

Subhanka Mal, Kingshuk Adhikary, Dibyendu Sardar +2 more

Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.

arXiv:1902.02177Journal of Physics B Atomic Molecular and Optical PhysicsMay 21, 20193 citations

Remarks on the quantum potential

Robert Carroll

To appear in the proceedings ofJan 1, 20060 citations

On the Quantum Potential

Springer eBooksJan 1, 20062 citations

Application of a quantum wave impedance method for zero-range singular potentials

Orest Hryhorchak

An application of a quantum wave impedance method for a study of quantum-mechanical systems which con\-tain singular zero-range potentials is considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance. As a result both the scattering and bound states problems are solved for systems of single $δ$ , double $δ$ and single $δ-δ'$ potentials. The formalization of solving systems with an arbitrary combination of a piesewise constant potential and a $δ$ -potentials with the help of a quantum wave impedance approach is described.An application of a quantum wave impedance method for a study of quantum-mechanical systems which con\-tain singular zero-range potentials is considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance. As a result both the scattering and bound states problems are solved for systems of single $δ$, double $δ$ and single $δ-δ'$ potentials. The formalization of solving systems with an arbitrary combination of a piesewise constant potential and a $δ$-potentials with the help of a quantum wave impedance approach is described.

arXiv:2010.06930Oct 14, 20206 citations

δ

, double

δ

and single

δ-δ'

potentials. The formalization of solving systems with an arbitrary combination of a piesewise constant potential and a

δ

-potentials with the help of a quantum wave impedance approach is described.