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Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics

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Abstract.

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.

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Correspondence to Rami Ahmad El-Nabulsi.

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El-Nabulsi, R.A. Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics. Eur. Phys. J. Plus 134, 192 (2019). https://doi.org/10.1140/epjp/i2019-12492-6

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