q-Deformed Dynamics and Virial Theorem

In the framework of the q-deformed Heisenberg algebra the investigation of qdeformation of Virial theorem explores that q-deformed quantum mechanics possesses better dynamical property. It is clarified that in the case of the zero potential the theoretical framework for the q-deformed Virial theorem is self-consistent. In the selfadjoint states the q-deformed uncertainty relation essentially deviates from the Heisenberg one. § E-mail address: [email protected] [email protected] In hunting new physics at extremely high energy scale clarifying modification of the ordinary quantum mechanics at extremely small space scale is important. The ordinary quantum mechanics is based on the Heisenberg commutation relation. From the space scale 10 cm down to, according to the present test of quantum electrodynamics, at least 10 cm every test confirms that the Heisenberg commutation relation is correct. There is a possibility that the Heisenberg commutation relation at extremely short distances much smaller than 10 cm may need to be modified. In search for such possibility q-deformed quantum mechanics is a candidate. In literature different frameworks of qdeformed quantum mechanics were established [1–20]. The framework of the q-deformed Heisenberg algebra developed in Refs. [2, 4] shows interesting physical content and dynamical properties. Its relation to the corresponding q-deformed boson commutation relations and the limiting process of q-deformed harmonic oscillator to the undeformed one are clear. The q-deformed uncertainty relation undercuts the Heisenberg minimal uncertainty relation [15, 16, 18]. The non-perturbation energy spectrum of the q-deformed Schrödinger equation exhibits an exponential structure [3, 4, 17], corresponding to new degrees of freedom and new quantum numbers [17]. The perturbation expansion of the q-deformed Hamiltonian possesses a complex structure, which amounts to some additional momentum-dependent interaction [2, 4, 17, 19, 20]. A reliable foundation for the perturbation calculations in q-deformed dynamics is established [19, 20]. In this paper we study the q-deformation of Virial theorem to demonstrate that qdeformed quantum mechanics possesses better dynamical property. In the ordinary quantum mechanics there is a delicate point in the theoretical treatment of Virial theorem for the case of the zero potential. We clarified that in the case of the zero potential the theoretical framework for q-deformed Virial theorem is self-consistent. The demonstration of such self-consistency of q-deformed Virial theorem is equivalent to the problem of finding a selfadjoint extension for the representation of the q-deformed Heisenberg algebra. Furthermore, we find that in the selfadjoint states the q-deformed uncertainty relation essentially deviates from the Heisenberg one. In terms of q-deformed phase space variables the position operator X and the momen-