About one interesting and important model in quantum mechanics I. Dynamic decription

TL;DR

This paper provides a detailed analysis of a fundamental quantum model introduced in 1933, exploring its special cases, wave functions, spectra, and pressure properties, with implications for nanotechnology applications.

Contribution

It offers an extensive, elementary exposition of the model, its special cases, and introduces a pressure operator analysis, extending understanding of quantum confinement and oscillators.

Findings

Pressure absent in the Bloch oscillator due to infinite box width

Unified treatment of confinement and unconfined models

Derivation of baroenergetic equation of state for all models

Abstract

In this paper the detailed investigation of one of the most interested models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is presented. This model includes as particular cases two most popular and valuable models: the quasi free particle in the box with impenetrable hard walls (i.e., the model with confinement) and Bloch quantum harmonic oscillator, which is unconfined in space; both models are frequently and effectively exploited in modern nanotechnology e.g., in quantum dots and magnetic traps. We give the extensive and elementary exposition of the potentials, wave functions and energetic spectra of all these interconnected models. Moreover, the pressure operator is defined following the lines of G. Helmann and R. Feynman which were the first who introduced this idea in the late 30ies in quantum chemistry. By these means the baroenergetic equation of state is obtained and analyzed for all three models; in particular, it is shown the absence of the pressure for the Bloch oscillator due to the infinite width of the box. The generalization of these results on the case of nonzero temperature will be given later.