Abstract
Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.
Highlights
In material science, the electron effective mass m* is the mass, it appears to have when probed in the periodic potential of a crystal lattice
We show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass
We have shown that the effect of position dependence in the effective mass in this oscillator model is the induction of a sort of thermodynamic resonance during which the entropy of the system becomes infinitely large
Summary
The electron effective mass m* is the mass, it appears to have when probed in the periodic potential of a crystal lattice. Notwithstanding the Hermiticity necessity, an all around characterized quantum kinetic energy operator is relied upon to satisfy various exigencies, for example, Be Galilee Invariant, not prompt infringement of Heisenberg uncertainty rule [6] and guarantee continuity of the envelope function at abrupt interfaces [7] Ordering these two operators cannot be done in a unique way for a consistent quantum theory with variable mass, reason why one finds a good number of proposals in the literature. In an attempt to resolve this issue, Dutra and Almeida [13] used two exactly solved models to propose that exact solutions could be used as a guide towards restricting the possible number of kinetic energy operators at hand They iterated in a concluding note: “...
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