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, ∫ ^xB _xAk(x)dx-∫ ^xB _xAk(x)dx = nπ, where k(x)=√2M[E-V(x)]/h{stroke}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 Φ(x)or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral ∫ ^xB _xAk(x)dx 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.