Thermodynamic properties of quantum models based on the complex unitary Cayley-Klein groups

In this work, the goal is to reach the thermodynamics properties produced by a family of the integrable Hamiltonians of the unitary groups of rank one in the Cayley-Klein space. In fact, this makes it possible through the general commutator relations between their generators of the desired Cayley-Klein group under investigation. On the other hand, by converting the coordinates to the Euler parameterization, the metric obtained from the Casimir is rewritten in terms of these coordinates, which is equal to the same the Lagrangian of the geodesic motion of the corresponding space. By matching coordinates, Lagrangian is written in terms of canonical momenta, and so the Hamiltonian can be obtained in two forms: one for a free particle and another for an involved particle with the integral potential. Hence, by solving the Hamiltonian of each of the two states in a different way(NU and SUSY QM methods), the quantum model for its corresponding system is shown in terms of the contraction parameter (κ1). Finally, with finding eigenvalues for each of these two systems, their thermodynamic properties are discussed with changing two parameters λ and δ, respectively. In our study, one can only see a phase transition in the heat capacity diagram at critical point τ0. In other words, it can be stated that a phase transition of the second type possible occurs for free particle in high temperature. Also, other results are reported in the end.