Dual quantum oscillator: Effective Hamiltonian, propagator, quantum impedance, and secular equation

Abstract

In previous articles, the author proposed a conceptual model of the dual quantum oscillator (DQO). Research and development of this model are ongoing. This paper presents an effective Hamiltonian for this model. The effective Hamiltonian is constructed based on known eigenvalues and has a quantum-statistical structure that incorporates all the characteristics of the DQO. The Hamiltonian matrix is two-dimensional (2×2) and diagonalized. This is not a mathematical reduction of a multi-level system to a two-level system, but a real physical representation of an innovative structure. The effective Hamiltonian is statistically generalized. Through its determinant, it yields an effective statistical sum, which, depending on the conditions, is divided into two components - one for Bose-Einstein statistics and one for Fermi-Dirac statistics. These statistical sums, in turn, turned out to be integral functions of the spectral densities derived in the previous article. This means that the DQO consists of two open subsystems - a bosonic and a fermionic one. Energy is exchanged between them, and for the material fermionic subsystem, the bosonic field acts as a thermostat. The vacuum enters the system as the dynamic first cause and the medium of this process, so that the system is closed as a whole. This closed nature determines a convergent probability density and specific energies - the trace of the matrix yields the average energy, the determinant yields the statistical sum, and the discriminant yields the quantum scale of energy splitting. It also serves as a natural factor in suppressing divergences. Globally, such internal isolation of individual oscillators is ensured by the thermodynamic equilibrium of their ensemble, in which they are distributed by frequency. Unique characteristics have been calculated: the internal energy as a generalization of the mean-field value of the Bose and Fermi subsystems together with the temperature shift of the bosonic field, the propagator, and the quantum dynamic impedance as fundamental measures of the system’s response to fluctuations. The secular (centennial) equation of the Hamiltonian accurately reflects the general nature of DQO statistics. Its free term acquires particular physical significance and status, becoming the determinant of the Hamiltonian and a metric criterion for the transition between Bose and Fermi statistics. And the corresponding characteristic function of the equation is the determinant of a special quantum impedance matrix. It is expected that this fundamental model will truly broaden the horizons of quantum physics and pave the way for new, effective approaches and methods for applying DQO in science and technology.