q-deformed and c-deformed harmonic oscillators

Hamilton functions of classical deformed oscillators (c-deformed oscillators) are derived from Hamiltonians of q-deformed oscillators of the Macfarlane and Dubna types. A new scale parameter, lq, with the dimension of length, is introduced to relate a dimensionless parameter characterizing the deformation with the natural length of the harmonic oscillator. Contraction from q-deformed oscillators to c-deformed oscillators is accomplished by keeping lq finite while taking the limit h → 0. The c-deformed Hamilton functions for both types of oscillators are found to be invariant under discrete translations: the step of the translation for the Dubna oscillator is half of that for the Macfarlane oscillator. The c-deformed oscillator of the Macfarlane type has propagating solutions in addition to localized ones. Reinvestigation of the q-deformed oscillator carried out in the light of these findings for the c-deformed systems proves that the q-deformed systems are invariant under the same translation symmetries as the c-deformed systems and have propagating waves of the Bloch type. (author)