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, integral(xB)(xA) k(x)dx - integral(x0B)(x0A) k(0)(x)dx = n pi, where k(x) = root 2M[E - V(x)]/(h) over bar 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 phi(x) or the number of the nodes of the wave function psi(x) increases by one, the momentum integral integral(xB)(xA) k(x)dx will increase by pi. 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 beta 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 lambda and then decreases with it.