Vibrational Entropy and Complexity Measures in Modified Pöschl–Teller Plus Woods–Saxon potential

Abstract

The explicit expression of the vibrational partition function for the modified Pöschl–Teller plus Woods–Saxon potential has been presented in a closed-form. The analytical expression for the vibrational mean energy have also been calculated were other thermodynamic functions like the vibrational specific heat, free energy, and the entropy for the gallium nitride wurtzite crystal structure have been determined in details. The dependence of these functions on the potential parameters has also been discussed in detail. By using the ground state energy and the probability density, the behaviours of some theoretic quantities (Shannon entropy and Fisher information entropy) have also been calculated and analyzed graphically as a function of the potential parameters.

Appendix

$$\begin{aligned} \int \limits _{-1}^1 {\left( {\frac{1-y}{2}} \right) }^{a}\left( {\frac{1+y}{2}} \right) ^{b}\left[ {P_{n}^{(a,b)} (x)} \right] ^{2}dx= & {} \frac{2\Gamma (a+n+1)\Gamma (b+n+1)}{n!a\Gamma (a+b+2n+1)\Gamma (a+b+n+1)}. \end{aligned}$$

(A.1)

$$\begin{aligned} P_{n}^{(a,b)} (1-2x)= & {} \frac{\Gamma (a+n+1)}{n!\Gamma (a+1)}{ }_{2}F_{1} (-n,n+a+b+1;a+1;x). \end{aligned}$$

(A.2)

where $P_{n} (z)$ is Jacobi polynomials.

$$\begin{aligned} { }_{2}F_{1} (a,b;c;y)=\frac{\Gamma (c)}{\Gamma (a)\Gamma (b)}\sum \limits _{n=0}^\infty {\frac{\Gamma (a+n)\Gamma (b+n)}{\Gamma (c+n)}} \frac{y^{n}}{n!}. \end{aligned}$$

(A.3)