q-Deformed quantum mechanics related to the Tamm-Dancoff oscillator algebra and some physical applications

Abstract In this paper, we consider a system of the q -deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q -calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q -derivative, q -integral and q -exponential function are introduced. Using these properties, we construct a new formalism for the q -deformed quantum mechanics, which accordingly involve the q -adjoint operator and the q -Hermitian operator properties. We then derive the q -deformed Heisenberg relation, and develop the q -Hermitian momentum operator. The q -deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q -deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.