Planck’s Oscillators at Low Temperatures and Haken’s Perturbation Approach to the Quantum Oscillators Reconsidered

Stanisław Olszewski

doi:10.4236/jmp.2021.1212100

Abstract

In the first step the extremal values of the vibrational specific heat and entropy represented by the Planck oscillators at the low temperatures could be calculated. The positions of the extrema are defined by the dimensionless ratios between the quanta of the vibrational energy and products of the actual temperature multiplied by the Boltzmann constant. It became evident that position of a local maximum obtained for the Planck’s average energy of a vibration mode and position of a local maximum of entropy are the same. In the next step the Haken’s time-dependent perturbation approach to the pair of quantum non-degenerate Schrödinger eigenstates of energy is re-examined. An averaging process done on the time variable leads to a very simple formula for the coefficients entering the perturbation terms.

Highlights

Planck has developed—a time ago—a quantum approach to the oscillator ensembles for which both the vibrational energy and entropy are considered [1]
In the first step the extremal values of the vibrational specific heat and entropy represented by the Planck oscillators at the low temperatures could be calculated
The positions of the extrema are defined by the dimensionless ratios between the quanta of the vibrational energy and products of the actual temperature multiplied by the Boltzmann constant

Summary

Introduction

Planck has developed—a time ago—a quantum approach to the oscillator ensembles for which both the vibrational energy and entropy are considered [1]. Beyond of the general formulae for E and S presented in (1) and (2), Planck examined a limiting situation when T becomes very high. In this case we obtain ehν kT ≅ 1+ hν. The problem of the Planck oscillators for energy and entropy could be connected with the Haken’s time-dependent perturbation calculation of a single nondegenerate quantum state. Since an averaging process makes the time-dependent terms equal to zero, the coefficients entering the final solution approach the terms characteristic for the harmonic oscillator

Properties of Planck’s E and S Characteristic for Small Temperature T

Average Energy for the Planck’s Vibration Mode and Its Properties

Summary