Weakly q-deformed Heisenberg algebra and non-Hermitian Hamiltonians: Application in statistical physics

Abstract

We have investigated a weakly q-deformed algebra, which leads to non-Hermitian operators. It has been explicitly shown that the harmonic oscillator Hamiltonian is quasi-Hermitian, and that the corresponding physical Hilbert-space metric Θ differs from that obtained in Ref. [1], by hermitizing the position operator. The reality of the Hamiltonian eigenvalues has been illustrated by analytically computing the first order correction to the energy spectrum of the harmonic oscillator (HO). Furthermore, we have shown that this q-deformation leads to a GUP with minimal uncertainties in both position and momentum measurements. Moreover, the physical implications of this q-deformation, have been investigated by studying the thermostatistics of a system of HOs and an ideal gas. For both systems, we computed the partition function, and then we derived some thermodynamic functions. In addition, for the ideal gas, a modified equation of state and a generalized Mayer's equation, consistent with the real behavior of gases, have been established. The obtained results show that the impact of this model would be significant at high temperatures. However, unlike earlier research, we observe that this q-deformation induced the similar corrections regardless of the system under study.