Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

We apply our recently proposed proper quantization rule, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_{x_A}^{x_B}k(x) dx -\int_{x_{0A}}^{x_{0B}}k_0(x) dx=n\pi}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k(x)=\sqrt{2 M [E-V(x) ]}/\hbar}$$\end{document} to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning—whenever the number of the nodes of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi(x)}$$\end{document} or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_{x_A}^{x_B} k(x)dx}$$\end{document} will increase by π. Based on this new approach, we present a vibrational high temperature partition function in order to study thermodynamic functions such as the vibrational mean energy U, specific heat C, free energy F and entropy S. It is surprising to note that the specific heat C (k = 1) first increases with β and arrives to the maximum value and then decreases with it. However, it is shown that the entropy S (k = 1) first increases with the deepness of potential well λ and then decreases with it.