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arXiv:1912.11697 · Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

Pressure Operator for the Pöeschl-Teller Oscillator

Yu. G. Rudoy, Enock Oluwole Oladimeji

Published Jan 1, 2017• 3 citations

Abstract

The quantum-mechanical properties of the strongly non-linear quantum oscillator in the Pöeschl-Teller model are considered. In the first place, the energy spectrum and its dependence upon the confinement parameter (i.e., the width of the “box”) are studied. Moreover, on the grounds of the Hellman-Feynman theorem the pressure operator in this model is obtained and (along with the energy spectrum) is studied in two main approximations: the “particle in the box” and “linear (harmonic) oscillator” for large and low values of the main quantum number; the critical value is also evaluated. Semi-classical approximation as well as perturbation theory for the Pöeschl-Teller are also considered. The results obtained here are intended for future thermodynamic calculations: first of all, for the generalization of the well-known Bloch result for the linear harmonic oscillator in the thermostat. To this end, the density matrix for the Pöeschl-Teller oscillator will be calculated and the full Carnot cycle conducted.

DOI: 10.22363/2312-9735-2017-25-3-276-282

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The quantum-mechanical properties of the strongly non-linear quantum oscillator in the Pöeschl-Teller model are considered. In the first place, the energy spectrum and its dependence upon the confinement parameter (i.e., the width of the “box”) are studied. Moreover, on the grounds of the Hellman-Feynman theorem the pressure operator in this model is obtained and (along with the energy spectrum) is studied in two main approximations: the “particle in the box” and “linear (harmonic) oscillator” for large and low values of the main quantum number; the critical value is also evaluated. Semi-classical approximation as well as perturbation theory for the Pöeschl-Teller are also considered. The results obtained here are intended for future thermodynamic calculations: first of all, for the generalization of the well-known Bloch result for the linear harmonic oscillator in the thermostat. To this end, the density matrix for the Pöeschl-Teller oscillator will be calculated and the full Carnot cycle conducted.

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.

The first five energy levels and the oscillator strengths for transitions involving the first three states have been numerically computed for a bounded linear harmonic oscillator for various values of the boundary parameter. It is found that for l>3l0, where l is the length of the box within which the oscillator is confined and l0 is the classical amplitude of the oscillator when it has energy hv, the bounded oscillator behaves more or less like a free oscillator (in the first few energy levels), while for l≪l0 it has properties closely approaching those of a free particle enclosed in a box.

In a previous note (1) we argue that the entropy for a pure quantum bound state is given by S=Sx+Sp where Sx = -k{ Integral dx Wn(x)Wn(x)ln(WnWn) + Integral dp an(p)an(p) ln(an an) }. It was shown that for any energy level pure bound state for the case of a particle in a box with an infinite potential or for an oscillator S does not depend on the parameters L (length of box) or k (spring constant) or m (mass). Thus a quantum adiabatic change of the parameter leaving the particle in the same quantum state leads to no change in entropy. In (2) a two level quantum Carnot cycle is considered for a particle in a box with an infinite potential. Energy levels n=0 and n=1 are considered.. The wavefunction is a(1)W1(x) + a(2)W2(x). Shannon’s entropy is used namely S=-k Sum over n an(an) ln{|an an|} ((1)) and an expression is found linking an*an to L and T (temperature) for a particle in a box with infinite potential walls. A Carnot cycle with two isotherms and two adiabatic dS=0 parts is considered. No consideration is given to the internal entropy of a bound state eigenstate. In this note we try to show how the overall entropy changes if the internal bound energy eigenfuntion entropy ((1)) is considered and also show how an*an(T,L) changes. We find that a1*a1 changes as do the overall values of Q (heat) and W(work) for the Carnot cycle, but that the efficiency does not.

In (1) a thermodynamical-statistical mechanical approach is presented for a two energy level quantum system for a particle in a box with infinite potential walls. It is shown that classical statistical mechanical efficiency of a Carnot cycle 1- Tc/Th holds for this quantum system. In this note we argue that the Carnot cycle efficiency holds in general if one uses delta E = delta Q - delta Work with delta Q/T = delta S if E and S are state variables as they are. One does not need to perform the two energy level specific integrals presented in (1) because the Carnot efficiency result is guaranteed for any number of energy levels.