Thermodynamic Potentials Generalized from a đ-Deformed Exponential Function

Abstract - In the transformation of the effective Hamiltonian for the case of a spatially dependent mass, into its canonical form, a generalized (đ-deformed) Hermitian linear momentum operator p_Î»is deduced. Added to this, a spatial canonical transformation x_Î»to solve the resulting differential equation leads to a đ-deformed quantum mechanics.In this scenario it is possible to propose a đ-deformed exponential function ăexpă_Î» (x)=expâĄ(x_Î») and its correspondingđ-deformed logarithmic partnerălnă_Î» (x).From these results, and the definition of the internal energy U=-k â/âÎČ ln(Z), it is straightforward to get the generalized internal energyU_Î»=-k â/âÎČ ălnă_Î» (Z_Î» ) which enables a natural generalization of others thermodynamic functions such as the EntropyS_Î», the Helmholtz free energy F_Î» and the heat capacity C_Î».Some particular mass distributionsm(Î»;x) are used as example to illustrate the proposal. Besides, in the case of m(Î»;x)=m_0 ă(1+Î»x)ă^(-2) our proposal leads straightforwardly to the Tsallis results obtained by solving a variational problem with suitable Lagrange parameters. Keywords - SchrĂ¶dinger Equation, Position-dependent Mass, Generalized Exponential Function, Internal Energy, Thermodynamic Functions.