TITLE:
Planck’s Oscillators at Low Temperatures and Haken’s Perturbation Approach to the Quantum Oscillators Reconsidered
AUTHORS:
Stanisław Olszewski
KEYWORDS:
Planck’s Quantum Oscillators at Low Temperatures, Calculation of the Extremal Specific Heat of the Oscillator Energy and Extremal Entropy, Simplified Haken’s Time-Dependent Approach to the Perturbation Energy of a Non-Degenerate Quantum State
JOURNAL NAME:
Journal of Modern Physics,
Vol.12 No.12,
October
28,
2021
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 Schr?dinger 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.