https://doi.org/10.1140/epjp/i2019-12492-6
Regular Article
Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics
Athens Institute for Education and Research, Mathematics and Physics Divisions, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece
* e-mail: nabulsiahmadrami@yahoo.fr
Received:
11
October
2018
Accepted:
3
January
2019
Published online:
7
May
2019
In this study, we have constructed a new fractional Schrödinger Hamiltonian for a quantum system characterized by a position-dependent mass and a modified fractional spatial derivative operator. This approach is characterized by a modified commutation relation and a modified Heisenberg’s uncertainty relation which both led to a modified Schrödinger operator typically used to describe quantum mechanics in fractional dimensions. By selecting an effective mass involved in semiconductor heterostructures, we have discussed two independent cases: the confinement of a particle in a one-dimensional infinite well and the motion of a particle in a linear potential used to describe the falling of a particle under gravity. Several features were obtained, yet the main outcome of the present manuscript concerns the fact that, for some specific fractional dimensions, a particle fallen under gravity may have a quantized energy.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2019
