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arXiv:1906.02274

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

Yu. G. Rudoy, Enock Oluwole Oladimeji

Published Mar 13, 2019• 2 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1906.02274

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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.
In this paper the detailed investigation of one of the most interesting models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is continued; the starting point of dynamic analysis for potentials, wave functions and energy spectrum was carried on by the same authors in the paper. The generalization of these results on the case of nonzero temperature includes the approximate calculation of the partition function of the model and then also the Massieu Planck thermodynamic potential and its most important derivatives, i.e. internal energy, heat capacity and pressure. The analysis of the results is carried on separately for low and high temperature regions because of the lack of exact unifying expressions of the potential at all temperatures.
Relativistic, scalar particles are considered, contained in a box with periodic boundary conditions. Although interactions are not expected to be a fundamental problem, we concentrate on free particles. By considering them to be harmonic oscillators, it is found that their dynamical variables can be replaced by a completely ontological set, which means that, here, quantum mechanics does not deviate from a purely geometric, ontological particle system. The effects of the mass terms are included. Locality holds for the quantum theory, and seems to be fully obeyed also by the classical treatment, although further discussion will be needed. Quantised interactions are briefly speculated on, but mostly postponed to later. We do discuss extensively the distinction between the quantum treatment and the classical one, even though they produce exactly the same equations mathematically. We briefly explain how this result can be squared with the usual quantum no-go theorems. It is suggested to apply this theory for real time quantum model simulations.
A critical study of the wave mechanics of a particle trapped in a 1-D box having infinite potential walls and small flexibility in its size reveals its several important and hither to unknown aspects which could be relevant for better understanding of systems like quantum -dot/wire/well. Since most of these aspects arise from the zero-point force coming into operation when the particle occupies its ground state in the box, they are expected to have great significance at low temperatures. To demonstrate this we briefly analyze some important aspects of an electron bubble in liquid helium and its nano-droplets.
In this chapter we shall discuss some elementary examples of quantum dynamics. In Sec. 4.1 we shall go back to the problem of a particle in a box, this time with finite potential wells. In Sec. 4.2 we shall analyze the effects of a potential barrier on a moving particle. In Sec. 4.3 we shall consider another quantum effect which has no analogue in the classical domain: a quantum particle can tunnel in a classically forbidden region. In Sec. 4.4 perhaps the most important dynamical typology (with a wide range of applications) is considered: the harmonic oscillator. Finally, in Sec. 4.5 several types of elementary fields are considered. Finite potential wells In Sec. 3.4 we have considered what is perhaps the simplest example of quantum dynamics, that is a free particle moving in a box with infinite potential walls. Consider the motion of, say, a one-dimensional particle in a rectangular potential well with finite steps. In Fig. 4.1 we show two of such potentials, symmetric in (a) and asymmetric in (b). Let us consider the case pictured in Fig. 4.1(a) and indicate with V 0 the energy of the potential well. We may therefore distinguish three regions on the x -axis: region I ( x < 0), where the potential energy is equal to V 0 ; region II (0 ≤ x ≤ a ), where the particle is free; and region III ( x > a ), where the potential energy is again equal to V 0 .