# Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians

## Abstract

Based on recent results on quasi-exactly solvable Schrodinger equations, we review a new phenomenological potential class lately reported. In the present paper, we consider the quantum differential equations resulting from position-dependent mass (PDM) particles. We first focus on the PDM version of the hyperbolic potential V(x) = asech{sup 2}x + bsech{sup 4}x, which we address analytically with no restrictioon the parameters and the energies. This is the celebrated Manning potential, a double-well widely used in molecular physics, until now not investigated for PDM. We also evaluate the PDM version of the sixth power hyperbolic potential V(x) = asech{sup 6}x + bsech{sup 4}x for which we could find exact expressions under some special settings. Finally, we address a triple-well case V(x) = asech{sup 6}x + bsech{sup 4}x + csech{sup 2}x of particular interest for its connection to the new trends in atomtronics. The PDM Schrodinger equations studied in the present paper yield analytical eigenfunctions in terms of local Heun functions in its confluents forms. In all the cases PDM particles are more likely tunneling than ordinary ones. In addition, it is observed a merging of eigenstates when the mass becomes nonuniform.

- Authors:

- Grupo de Física Teórica, State University of Ceara (UECE), Av. Paranjana 1700, 60740-903 Fortaleza - CE (Brazil)
- (Brazil)

- Publication Date:

- OSTI Identifier:
- 22306031

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENFUNCTIONS; EIGENSTATES; HAMILTONIANS; POTENTIALS; SCHROEDINGER EQUATION; TUNNEL EFFECT

### Citation Formats

```
Christiansen, H. R., State University Vale do Acaraú, Av. da Universidade 850, 62040-370 Sobral - CE, and Cunha, M. S.
```*Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians*. United States: N. p., 2014.
Web. doi:10.1063/1.4894056.

```
Christiansen, H. R., State University Vale do Acaraú, Av. da Universidade 850, 62040-370 Sobral - CE, & Cunha, M. S.
```*Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians*. United States. doi:10.1063/1.4894056.

```
Christiansen, H. R., State University Vale do Acaraú, Av. da Universidade 850, 62040-370 Sobral - CE, and Cunha, M. S. Mon .
"Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians". United States. doi:10.1063/1.4894056.
```

```
@article{osti_22306031,
```

title = {Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians},

author = {Christiansen, H. R. and State University Vale do Acaraú, Av. da Universidade 850, 62040-370 Sobral - CE and Cunha, M. S.},

abstractNote = {Based on recent results on quasi-exactly solvable Schrodinger equations, we review a new phenomenological potential class lately reported. In the present paper, we consider the quantum differential equations resulting from position-dependent mass (PDM) particles. We first focus on the PDM version of the hyperbolic potential V(x) = asech{sup 2}x + bsech{sup 4}x, which we address analytically with no restrictioon the parameters and the energies. This is the celebrated Manning potential, a double-well widely used in molecular physics, until now not investigated for PDM. We also evaluate the PDM version of the sixth power hyperbolic potential V(x) = asech{sup 6}x + bsech{sup 4}x for which we could find exact expressions under some special settings. Finally, we address a triple-well case V(x) = asech{sup 6}x + bsech{sup 4}x + csech{sup 2}x of particular interest for its connection to the new trends in atomtronics. The PDM Schrodinger equations studied in the present paper yield analytical eigenfunctions in terms of local Heun functions in its confluents forms. In all the cases PDM particles are more likely tunneling than ordinary ones. In addition, it is observed a merging of eigenstates when the mass becomes nonuniform.},

doi = {10.1063/1.4894056},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 9,

volume = 55,

place = {United States},

year = {2014},

month = {9}

}