On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential Ramaswamy Jagannathan∗§ and Sameen Ahmed Khan$¶ ∗The Institute of Mathematical Sciences 4th Cross Street, Central Institutes of Technology (CIT) Campus Tharamani, Chennai 600113, India $Department of Mathematics and Sciences College of Arts and Applied Sciences (CAAS), Dhofar University Post Box No. 2509, Postal Code: 211, Salalah, Oman Abstract 1 The Tsallis q-exponential function e (x) = (1 + (1 q)x) 1−q is q − found to be associated with the deformed oscillator defined by the relations N, a† = a†, [N, a]= a, and a, a† = φ (N + 1) φ (N), − T − T with φ (N)= N/(1+(q 1)(N 1)). In a Bargmann-like representa- T − − tion 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 de- formed 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 os- cillator, 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. arXiv:1911.02428v3 [math-ph] 4 Jul 2020 Keywords: Nonextensive statistical mechanics, Tsallis q-exponential, de- formed exponentials, deformed oscillators, deformed numbers, deformed derivatives, nonlinear coherent states, f-oscillators, f-coherent states, phase coherent states. §Retired Faculty (Physics); email: [email protected] ORCID ID: https://orcid.org/0000-0003-2968-2044 ¶email: [email protected] ORCID ID: https://orcid.org/0000-0003-1264-2302 1 Introduction A posteriori knowledge gained from special relativity and quantum mechanics suggests that it may be possible to build models in physics actually realizable in Nature by mathematically deforming existing theories (see, e.g., [1] and references therein). We may look at the attempts to build q-deformed models in physics using q-deformations of the mathematical structures associated with the existing models from this point of view. There have been two totally different q-deformation schemes in physics running in parallel since 1980s: One emerged, essentially, from studies on quantum algebras (see, e.g., [2] and references therein) and the other stemmed from the pioneering paper of Tsallis [3] on nonextensive statistical mechanics. Studies on the deformation of quantum commutation relations started in 1970s from different points of view (see [4–6]; see [7] for an account of the early history of deformed quantum commutation relations). With the advent of quantum algebras in 1980s, representation theory of quantum al- gebras led to the q-deformed quantum harmonic oscillator, or the so called q-oscillator [8–11]. Harmonic oscillator being a basic paradigm playing a cen- tral role in modeling various physical phenomena q-oscillators and their gen- eralizations have been studied extensively. Deformed oscillators and related mathematical structures have been used in building models in several areas of physics: nuclear structure physics using models like shell model, interacting boson model, etc., vibrational-rotational molecular spectroscopy, statistics of deformed oscillators, statistics interpolating between the Bose statistics and the Fermi statistics, deformed thermodynamics and applications to con- densed matter physics, nonclassical states in quantum optics, noncommu- tative probability theory, information theory, etc. (see, e.g., [12–27], and references therein). In the discussion of vibrational-rotational spectroscopy, besides the q-deformation of the quantum harmonic oscillator, the quantum rigid rotator is q-deformed based on the q-deformation of the Lie algebra of angular momentum operators. The other q-deformation scheme follows from the nonextensive entropy 1 W pq W S = k − i=1 i , with p =1, (1) q q 1 i ! i=1 P− X introduced by Tsallis in [3] where W is the total number of possible con- figurations of the system under consideration, pi is the probability that the 2 system is in the ith configuration, and q is a real parameter. Defining x1−q 1 ln x = − , (2) q 1 q − the Tsallis q-logarithm, we get W −1 Sq = k pi lnq pi , (3) i=1 X such that W lim Sq = k pi ln pi, (4) q−→1 − i=1 X the Boltzmann-Gibbs-Shannon entropy. From (2) we have 1 x = (1+(1 q) ln x) 1−q . (5) − q Thus, if we define 1 e (x)=(1+(1 q)x) 1−q , (6) q − the Tsallis q-exponential function, then eq(lnq x)= x. (7) Note that x lim lnq x = ln x, lim eq(x)= e . (8) q−→1 q−→1 It is to be observed that in the definition of the ordinary exponential function, x N ex = lim 1+ , (9) N−→∞ N if we replace the inifinitesimally small 1/N corresponding to large N by 1 q, with 0 < 1 q = ǫ 0, then we get the Tsallis q-exponential function− (6). − ≈ Having understood eq(x) in this way its definition can be extended to other values of q, < 1 or > 1. In general, for q = 1 the definition is 6 1 (1+(1 q)x) 1−q if 1 + (1 q)x 0, eq(x)= − − ≥ (10) 0 otherwise. 3 It follows that e (0) =1, 0 e (x) , and q ≤ q ≤∞ de (x) q =(e (x))q . (11) dx q It is not surprising that being a natural deformation of ex the Tsallis q- exponential function eq(x) and its inverse function lnq(x) find applications in several areas of physics and other sciences like nonequilibrium processes, anomalous diffusion, turbulence, spin-glasses, nonlinear dynamics, high en- ergy particle collisions, dissipative optical lattices, atmospheric physics, as- trophysics, biological systems, medical imaging, earthquake studies, neural networks, economics, stock markets, etc. (see, e.g., [28–32] and references therein, and the extensive bibliography in [33]). It should be noted that in nonextensive statistical mechanics and its ap- plications one defines 1 e (kx)=(1+(1 q)kx) 1−q , (12) q − k and eq(kx) = eq(x) . For example, to define the q-Gaussian function one 6 2 2 1/(1−q) takes eq ( βx ) = (1 (1 q)βx ) . Carlitz introduced a deformed analogue− of the exponential− − while defining the socalled degenerate Bernoulli numbers and polynomials (see [34, 35]). The deformed exponential function 1/λ t of Carlitz is defined by expλ(x)=(1+ λx) and expλ(tx) = (expλ(x)) = (1+λx)t/λ . This difference between the deformed exponential of Carlitz and the Tsallis q-exponential is crucial in applications (see, e.g., [36]). In this paper we are concerned only with the mathematical structures of q-deformations of the quantum harmonic oscillator. The canonical harmonic oscillator, or the boson, is defined by the commutation relations N, b† = b†, [N, b]= b, b, b† =1, (13) − where b†, b and N are the boson creation, annihilation and number operators, respectively. These relations imply that b†b = N and bb† = N + 1. The Fock representation of the boson algebra (13) is given by b† n = √n +1 n+1 , b n = √n n 1 , N n = n n , n =0, 1, 2,..., | i | i | i | − i | i | i (14) with n n = 0, 1, 2,... being an orthonormal basis. Starting with the {| i| } ground state 0 we can build the higher states as | i b†n n = 0 . (15) | i √n! | i 4 The Hamiltonian of the boson oscillator is 1 1 H = b†b + = bb† + b†b , (16) 2 2 with the energy spectrum 1 E = n + , n =0, 1, 2,..., (17) n 2 such that H n = E n . (18) | i n| i Coherent states are eigenfunctions of b: b α = α α , (19) | i | i where α is any complex number. The solution for α is | i 1 ∞ αn α = n , (20) | i √ |α|2 √ | i e n=0 n! X with the normalization α α = 1. We can write h | i 1 † α = eαb 0 . (21) | i √e|α|2 | i In the Bargmann-like representation d b† = x, b = D = , N = xD, (22) dx and the monomials xn ξn(x)= , n =0, 1, 2,..., (23) √n! carry the Fock representation (14) as given by † b ξn = √n +1 ξn+1, bξn = √nξn−1, Nξn = nξn, n =0, 1, 2,.... (24) The monomials ξ (x) n =0, 1, 2,... are seen to form an orthonormal basis { n | } with respect to the inner product f g = [f ∗(D)g(x)] . (25) h | i |x=0 5 Note that in this representation eαx is an eigenfunction of b for any complex number α. Under the inner product (25) it is seen that eαx eαx = e|α|2 . h | i Thus, in this representation, the coherent states are given by ∞ n 1 αx 1 α ψα(x)= e = ξn(x), (26) √ |α|2 √ |α|2 √ e e n=0 n! X which are normalized as ψ ψ = 1. h α | αi In general, a deformed oscillator algebra is prescribed by the relations N, a† = a†, [N, a]= a, − a, a† = φ(N + 1) φ(N), or aa† = φ(N + 1), a†a = φ(N), (27) − wherea†, a, and N are the deformed oscillator creation, annihilation, and number operators, respectively. The real function φ(n), sometimes called the deformation, or structure, function, characterizes the deformed oscillator. Note that the first two relations of (27) imply that a†a and aa† commute with N. In the case of the canonical boson oscillator (13) φ(N)= N.