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

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.