On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential

Abstract

The Tsallis $q$-exponential function $e_q(x)=(1+(1-q)x{)}^{\frac{1}{1-q}}$ is found to be associated with the deformed oscillator defined by the relations $\left[N,a^{†}\right]=a^{†}$, $[N,a]=-a$, and $\left[a,a^{†}\right]={\varphi }_T(N+1)-{\varphi }_T(N)$, with ${\varphi }_T(N)=N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed oscillator the annihilation operator $a$ corresponds to a deformed derivative with the Tsallis $q$-exponential functions as its eigenfunctions, and the Tsallis $q$-exponential functions become the coherent states of the deformed oscillator. When $q=2$ these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when $q=1$ this deformed oscillator is a canonical boson oscillator, when $1<q<2$ its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when $q⟶2$ it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state.

Third version in which three more references have been added. To appear in International Journal of Theoretical Physics

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Cited by in corpus (2)

• Non-Debye impedance and relaxation models for dissipative electrochemical capacitors
• Coherent phase states in the coordinate and Wigner representations