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On the deformed oscillator and the deformedderivative 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) CampusTharamani, Chennai 600113, India

$Department of Mathematics and Sciences

College of Arts and Applied Sciences (CAAS), Dhofar UniversityPost Box No. 2509, Postal Code: 211, Salalah, Oman

Abstract

The Tsallis q-exponential function eq(x) = (1 + (1 − q)x)1

1−q isfound to be associated with the deformed oscillator defined by therelations

[

N, a†]

= a†, [N, a] = −a, and[

a, a†]

= φT (N +1)−φT (N),with φT (N) = N/(1+(q−1)(N −1)). In a Bargmann-like representa-tion of this deformed oscillator the annihilation operator a correspondsto a deformed derivative with the Tsallis q-exponential functions asits eigenfunctions, and the Tsallis q-exponential functions become thecoherent states of the deformed oscillator. When q = 2 these de-formed oscillator coherent states correspond to states known variouslyas 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 bosonand the excited energy levels lie in a band of finite width, and whenq −→ 2 it becomes a two-level system with a nondegenerate groundstate and an infinitely degenerate excited state.

Keywords: Nonextensive statistical mechanics, Tsallis q-exponential, de-formed exponentials, deformed oscillators, deformed numbers, deformedderivatives, nonlinear coherent states, f -oscillators, f -coherent states, phasecoherent states.

§Retired Faculty (Physics); email: jagan@imsc.res.inORCID ID: https://orcid.org/0000-0003-2968-2044

¶email: rohelakhan@yahoo.com

ORCID ID: https://orcid.org/0000-0003-1264-2302

http://arxiv.org/abs/1911.02428v3https://orcid.org/0000-0003-2968-2044https://orcid.org/0000-0003-1264-2302

1 Introduction

A posteriori knowledge gained from special relativity and quantum mechanicssuggests that it may be possible to build models in physics actually realizablein Nature by mathematically deforming existing theories (see, e.g., [1] andreferences therein). We may look at the attempts to build q-deformed modelsin physics using q-deformations of the mathematical structures associatedwith the existing models from this point of view. There have been twototally different q-deformation schemes in physics running in parallel since1980s: One emerged, essentially, from studies on quantum algebras (see,e.g., [2] and references therein) and the other stemmed from the pioneeringpaper of Tsallis [3] on nonextensive statistical mechanics.

Studies on the deformation of quantum commutation relations startedin 1970s from different points of view (see [4–6]; see [7] for an account ofthe early history of deformed quantum commutation relations). With theadvent of quantum algebras in 1980s, representation theory of quantum al-gebras led to the q-deformed quantum harmonic oscillator, or the so calledq-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 relatedmathematical structures have been used in building models in several areas ofphysics: nuclear structure physics using models like shell model, interactingboson model, etc., vibrational-rotational molecular spectroscopy, statisticsof deformed oscillators, statistics interpolating between the Bose statisticsand 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], andreferences therein). In the discussion of vibrational-rotational spectroscopy,besides the q-deformation of the quantum harmonic oscillator, the quantumrigid rotator is q-deformed based on the q-deformation of the Lie algebra ofangular momentum operators.

The other q-deformation scheme follows from the nonextensive entropy

Sq = k

(

1−∑Wi=1 pqiq − 1

)

, withW∑

i=1

pi = 1, (1)

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

lnq x =x1−q − 11− q , (2)

the Tsallis q-logarithm, we get

Sq = kW∑

i=1

pi lnq(

p−1i)

, (3)

such that

limq−→1

Sq = −kW∑

i=1

pi ln pi, (4)

the Boltzmann-Gibbs-Shannon entropy. From (2) we have

x = (1 + (1− q) lnq x)1

1−q . (5)

Thus, if we define

eq(x) = (1 + (1− q)x)1

1−q , (6)

the Tsallis q-exponential function, then

eq(lnq x) = x. (7)

Note that

limq−→1

lnq x = ln x, limq−→1

eq(x) = ex. (8)

It is to be observed that in the definition of the ordinary exponential function,

ex = limN−→∞

(

1 +x

N

)N

, (9)

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 othervalues of q, < 1 or > 1. In general, for q 6= 1 the definition is

eq(x) =

{

(1 + (1− q)x) 11−q if 1 + (1− q)x ≥ 0,0 otherwise.

(10)

3

It follows that eq(0) = 1, 0 ≤ eq(x) ≤ ∞, anddeq(x)

dx= (eq(x))

q . (11)

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 applicationsin 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, neuralnetworks, economics, stock markets, etc. (see, e.g., [28–32] and referencestherein, and the extensive bibliography in [33]).

It should be noted that in nonextensive statistical mechanics and its ap-plications one defines

eq(kx) = (1 + (1− q)kx)1

1−q , (12)

and eq(kx) 6= eq(x)k. For example, to define the q-Gaussian function onetakes eq (−βx2) = (1− (1− q)βx2)1/(1−q). Carlitz introduced a deformedanalogue of the exponential while defining the socalled degenerate Bernoullinumbers and polynomials (see [34, 35]). The deformed exponential functionof Carlitz is defined by expλ(x) = (1 + λx)

1/λ and expλ(tx) = (expλ(x))t =

(1+λx)t/λ . This difference between the deformed exponential of Carlitz andthe Tsallis q-exponential is crucial in applications (see, e.g., [36]).

In this paper we are concerned only with the mathematical structures ofq-deformations of the quantum harmonic oscillator. The canonical harmonicoscillator, 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 Fockrepresentation 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, . . . ,

(14)with {|n〉|n = 0, 1, 2, . . .} being an orthonormal basis. Starting with theground state |0〉 we can build the higher states as

|n〉 = b†n

√n!|0〉. (15)

4

The Hamiltonian of the boson oscillator is

H =

(

b†b+1

2

)

=1

2

(

bb† + b†b)

, (16)

with the energy spectrum

En = n+1

2, n = 0, 1, 2, . . . , (17)

such thatH|n〉 = En|n〉. (18)

Coherent states are eigenfunctions of b:

b|α〉 = α|α〉, (19)

where α is any complex number. The solution for |α〉 is

|α〉 = 1√e|α|2

∞∑

n=0

αn√n!|n〉, (20)

with the normalization 〈α|α〉 = 1. We can write

|α〉 = 1√e|α|2

eαb† |0〉. (21)

In the Bargmann-like representation

b† = x, b = D =d

dx, N = xD, (22)

and the monomials

ξn(x) =xn√n!, n = 0, 1, 2, . . . , (23)

carry the Fock representation (14) as given by

b†ξn =√n+ 1 ξn+1, bξn =

√n ξn−1, Nξn = nξn, n = 0, 1, 2, . . . .

(24)The monomials {ξn(x) |n = 0, 1, 2, . . .} are seen to form an orthonormal basiswith respect to the inner product

〈f |g〉 = [f ∗(D)g(x)]|x=0 . (25)

5

Note that in this representation eαx is an eigenfunction of b for any complexnumber α. Under the inner product (25) it is seen that 〈eαx |eαx〉 = e|α|2.Thus, in this representation, the coherent states are given by

ψα(x) =1√e|α|2

eαx =1√e|α|2

∞∑

n=0

αn√n!ξn(x), (26)

which are normalized as 〈ψα |ψα〉 = 1.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)

where a†, a, and N are the deformed oscillator creation, annihilation, andnumber operators, respectively. The real function φ(n), sometimes calledthe deformation, or structure, function, characterizes the deformed oscillator.Note that the first two relations of (27) imply that a†a and aa† commute withN . In the case of the canonical boson oscillator (13) φ(N) = N .

The Fock representation of the algebra (27) can be constructed easily asfollows:

a†|n〉 =√

φ(n+ 1) |n+ 1〉, a|n〉 =√

φ(n) |n− 1〉,N |n〉 = n|n〉, n = 0, 1, 2, . . . , (28)

Note that a†a|n〉 = φ(n)|n〉. Since a|0〉 = 0, we must have φ(0) = 0. Now,defining

φ(0)! = 1, φ(n)! =

n∏

j=1

φ(j), n = 1, 2, . . . , (29)

we can write

|n〉 = a†n

√

φ(n)!|0〉. (30)

The Hamiltonian of the deformed oscillator is taken to be

Hφ =1

2

(

aa† + a†a)

, (31)

generalizing (16). The energy spectrum of the deformed oscillator becomes

En,φ =1

2(φ(n+ 1) + φ(n)), (32)

6

generalizing (17), andHφ|n〉 = En,φ|n〉. (33)

Associated to any φ(n) let us define the deformed exponential function

exφ =

∞∑

n=0

xn

φ(n)!, (34)

which we shall refer to as the φ-exponential function. It can be verified thatthe normalized coherent states of the deformed oscillator (27), eigenstates ofa, are given by

|α〉φ =1

√

e|α|2

φ

∞∑

n=0

αn√

φ(n)!|n〉 = 1√

e|α|2

φ

eαa†

φ |0〉, (35)

in which the complex number α is such that e|α|2

φ

The monomials {ξn,φ(x)|n = 0, 1, 2, . . .} form an orthonormal basis with re-spect to the inner product

〈f |g〉φ = [f ∗ (Dφ) g(x)]|x=0 . (42)Note that in this representation eαxφ is an eigenfunction of a for the complexnumber α. Under the inner product (42)

〈

eαxφ∣

∣eαxφ〉

= e|α|2

φ . (43)

Thus, in this representation, the coherent states are given by

ψα,φ(x) =1

√

e|α|2

φ

eαxφ =1

√

e|α|2

φ

∞∑

n=0

αn√

φ(n)!ξn,φ(x), (44)

with the normalization 〈ψα,φ |ψα,φ〉 = 1, for any α such that e|α|2

φ

was studied much before the advent of quantum groups (see [4–7]). Since

[N + 1]q − q[N ]q = 1, (46)

the relation between a and a† is written as

aa† − qa†a = 1. (47)

The Fock representation is

a†|n〉 =√

[n+ 1]q |n+ 1〉, a|n〉 =√

[n]q |n− 1〉,N |n〉 = n|n〉, n = 0, 1, 2, . . . , (48)

where

[n]q =1− qn1− q (49)

is Heine’s basic number, or q-number (see, e.g., [38, 39]). Note that

[0]q = 0, [1]q = 1, limq−→1

[n]q = n. (50)

Now, with the definitions

[0]q! = 1, [n]q! =n∏

j=1

[j]q, n = 1, 2, . . . , (51)

we have

|n〉 = a†n

√

[n]q!|0〉. (52)

Defining the original q-exponential function

exq =

∞∑

n=0

xn

[n]q!, (53)

the normalized coherent states of the q-oscillator (45) are given by

|α〉q =1

√

e|α|2q

∞∑

n=0

αn√

[n]q!|n〉 = 1√

e|α|2q

eαa†

q |0〉. (54)

9

In the Bargmann-like representation

a† = x, a = Dq =1

x[xD]q =

1

x

(

1− qxD1− q

)

, N = xD, (55)

where Dq is the Jackson derivative, or q-derivative, operator (see, e.g., [38,39]) such that

DqF (x) =F (x)− F (qx)

(1− q)x . (56)

Note that

Dqxn = [n]qx

n−1. (57)

In the Bargmann-like representation the monomials

ξn,q(x) =xn

√

[n]q!, n = 0, 1, 2, . . . , (58)

carry the Fock representation of the q-oscillator algebra (45) as given by

a†ξn,q =√

[n + 1]q ξn+1,q, aξn,q =√

[n]q ξn−1,q,

Nξn,q = nξn,q, n = 0, 1, 2, . . . . (59)

The monomials {ξn,q(x)|n = 0, 1, 2, . . .} form an orthonormal basis with re-spect to the inner product

〈f |g〉q = [f ∗ (Dq) g(x)]|x=0 . (60)

It follows from (57) that the q-exponential function

eαxq =

∞∑

n=0

αnxn

[n]q!, (61)

for any complex number α, becomes an eigenfunction of a, and the normalizedcoherent states of the q-oscillator are given by

ψα,q(x) =1

√

e|α|2q

eαxq =1

√

e|α|2q

∞∑

n=0

αn√

[n]q!ξn,q(x). (62)

10

The development of quantum groups and the representation theory ofassociated quantum algebras led first to the (q−1, q)-oscillator [8,9] (see also[10, 11]):

[

N, a†]

= a†, [N, a] = −a,[

a, a†]

= φ(N + 1)− φ(N),

φ(N) =q−N − qNq−1 − q = [N ](q−1,q). (63)

With the relation

[N + 1](q−1,q) − q−1[N ](q−1,q) = qN , (64)

the relation between a and a† becomes

aa† − q−1a†a = qN . (65)

Note the q ←→ q−1 symmetry. The Fock representation is

a†|n〉 =√

[n+ 1](q−1,q) |n+ 1〉, a|n〉 =√

[n](q−1,q) |n− 1〉,N |n〉 = n|n〉, n = 0, 1, 2, . . . , (66)

where

[n](q−1,q) =q−n − qnq−1 − q (67)

defines the (q−1, q)-basic number. Note that

[0](q−1,q) = 0, [1](q−1,q) = 1, limq−→1

[n](q−1,q) = n. (68)

Now, with the definitions

[0](q−1,q)! = 1, [n](q−1,q)! =n∏

j=1

[j](q−1,q), n = 1, 2, . . . , (69)

we have

|n〉 = a†n

√

[n](q−1,q)!|0〉. (70)

Defining the (q−1, q)-exponential function as

ex(q−1,q) =∞∑

n=0

xn

[n](q−1,q)!, (71)

11

the normalized coherent states of the (q−1, q)-oscillator (63) are given by

|α〉(q−1,q) =1

√

e|α|2

(q−1,q)

∞∑

n=0

αn√

[n](q−1,q)!|n〉 = 1√

e|α|2

(q−1,q)

eαa†

(q−1,q)|0〉. (72)

The Bargmann-like representation is given by

a† = x, a = D(q−1,q) =1

x

(

q−xD − qxDq−1 − q

)

, N = xD. (73)

Note that

D(q−1,q)F (x) =F (q−1x)− F (qx)

(q−1 − q)x , D(q−1,q)xn = [n](q−1,q)x

n−1. (74)

Then, the monomials

ξn,(q−1,q)(x) =xn

√

[n](q−1,q)!, n = 0, 1, 2, . . . , (75)

carry the Fock representation of the (q−1, q)-oscillator algebra (63) as givenby

a†ξn,(q−1,q) =√

[n+ 1](q−1,q) ξn+1,(q−1,q),

aξn,(q−1,q) =√

[n](q−1,q) ξn−1,(q−1,q),

Nξn,(q−1,q) = nξn,(q−1,q), n = 0, 1, 2, . . . . (76)

The monomials {ξn,(q−1,q)(x)|n = 0, 1, 2, . . .} form an orthonormal basis withrespect to the inner product

〈f |g〉(q−1,q) =[

f ∗(

D(q−1,q))

g(x)]∣

∣

x=0. (77)

In this Bargmann-like representation, the (q−1, q)-exponential function

eαx(q−1,q) =

∞∑

n=0

αnxn

[n](q−1,q)!, (78)

for any complex number α, is an eigenfunction of a, and

ψα,(q−1,q)(x) =1

√

e|α|2

(q−1,q)

eαx(q−1,q) =1

√

e|α|2

(q−1,q)

∞∑

n=0

αn√

[n](q−1,q)!ξn,(q−1,q)(x)

(79)

12

is a normalized coherent state of the (q−1, q)-oscillator (63).Representation theory of two-parameter quantum algebras led to the

generalization of the (q−1, q)-oscillator to the (p, q)-oscillator [40] (see also[41, 42]):

[

N, a†]

= a†, [N, a] = −a,[

a, a†]

= φ(N + 1)− φ(N),

φ(N) =pN − qNp− q = [N ](p,q). (80)

Since[N + 1](p,q) − p[N ](p,q) = qN , (81)

we can writeaa† − pa†a = qN . (82)

Note the p←→ q symmetry. The Fock representation is

a†|n〉 =√

[n+ 1](p,q) |n+ 1〉, a|n〉 =√

[n](p,q) |n− 1〉,N |n〉 = n|n〉, n = 0, 1, 2, . . . , (83)

where

[n](p,q) =pn − qnp− q (84)

defines the (p, q)-basic number, or the twin-basic number [43]. Note that

[0](p,q) = 0, [1](p,q) = 1, limp,q−→1

[n](p,q) = n. (85)

Now, with the definitions

[0](p,q)! = 1, [n](p,q)! =n∏

j=1

[j](p,q), n = 1, 2, . . . , (86)

we have

|n〉 = a†n

√

[n](p,q)!|0〉. (87)

Defining the (p, q)-exponential function as

ex(p,q) =∞∑

n=0

xn

[n](p,q)!, (88)

13

the normalized coherent states of the (p, q)-oscillator (80) are given by

|α〉(p,q) =1

√

e|α|2

(p,q)

∞∑

n=0

αn√

[n](p,q)!|n〉 = 1√

e|α|2

(p,q)

eαa†

(p,q)|0〉. (89)

The Bargmann-like representation of the (p, q)-oscillator algebra (80) is givenby

a† = x, a = D(p,q) =1

x

(

pxD − qxDp− q

)

, N = xD. (90)

The (p, q)-derivative D(p,q) is such that

D(p,q)F (x) =F (px)− F (qx)

(p− q)x , D(p,q)xn = [n](p,q)x

n−1. (91)

The monomials

ξn,(p,q)(x) =xn

√

[n](p,q)!, n = 0, 1, 2, . . . , (92)

carry the Fock representation:

a†ξn,(p,q) =√

[n + 1](p,q) ξn+1,(p,q), aξn,(p,q) =√

[n](p,q) ξn−1,(p,q),

Nξn,(p,q) = nξn,(p,q), n = 0, 1, 2, . . . . (93)

The monomials {ξn,(p,q)(x)|n = 0, 1, 2, . . .} form an orthonormal basis withrespect to the inner product

〈f |g〉(p,q) =[

f ∗(

D(p,q))

g(x)]∣

∣

x=0. (94)

Note that, in this Bargmann-like representation, the (p, q)-exponential func-tion

eαx(p,q) =∞∑

n=0

αnxn

[n](p,q)!, (95)

for any complex number α, is an eigenfunction of a and a normalized coherentstate of the (p, q)-oscillator is given by

ψα,(p,q)(x) =1

√

e|α|2

(p,q)

eαx(p,q) =1

√

e|α|2

(p,q)

∞∑

n=0

αn√

[n](p,q)!ξn,(p,q)(x). (96)

14

Note that the canonical boson oscillator (13), the q-oscillator (45), andthe (q−1, q)-oscillator (63), are special cases of the (p, q)-oscillator (80) corre-sponding to (p = 1, q = 1), p = 1, and p = q−1, respectively. The canonicalfermion oscillator corresponds to (p = −1, q = 1). The q-fermion oscillator( [44, 45]) and the Tamm-Dancoff oscillator ( [46, 47]) are also special casesof the (p, q)-oscillator corresponding to p = −q−1 and p = q, respectively.Actually, one can choose p as any function of q in building models. Thus,starting with the (p, q)-oscillator a plethora of interesting types of q-deformedoscillators have been found with their energy spectra having accidental pair-wise energy level degeneracies [48–50]. Further, there exist several otherdeformed oscillator structures with single and multiple deformation param-eters (see, e.g., [51–55]). It may also be noted that (p, q)-calculus based onthe (p, q)-number (84) and the (p, q)-derivative (91), sometimes called thepost-quantum calculus following the name quantum calculus given to thecalculus based on the q-number (49) and the q-derivative (56) (see [39]), is avery active area of research in applied mathematics with applications to ap-proximation theory, computer-aided geometric design, etc. (see, e.g., [56–58]and references therein).

It is known since the early days of deformed oscillators that the creationand annihilation operators of the deformed oscillator (27) can be realized interms of the canonical boson creation and annihilation operators as

a† =

√

φ(N)

Nb†, a = b

√

φ(N)

N=

√

φ(N + 1)

N + 1b, N = b†b. (97)

Writing φ(N) = f 2(N)N , the deformed oscillator (27) has been presented as

a† = f(N)b†, a = bf(N) = f(N + 1)b, N = b†b,[

N, a†]

= a†, [N, a] = −a,[

a, a†]

= f 2(N + 1)(N + 1)− f 2(N)N, (98)called an f -oscillator, and a general theory of f -oscillators has been developed[59] (see also [16, 17]) with many applications to nonlinear physics includingnonlinear coherent states, or f -coherent states, relevant for quantum optics(see, e.g., [60]; see also [61–63]). The function f(n), called the nonlinearityfunction, characterizes the f -oscillator (98). In terms of f(n) the deformedexponential function (34) can be rewritten as

exφ =∞∑

n=0

xn

f 2(n)!n!= exf , (99)

15

where

f 2(0)! = 1, f 2(n)! =

n∏

j=1

f 2(j), for n ≥ 1. (100)

Then, the normalized coherent states of the f -oscillator (98), eigenstates ofa = f(N + 1)b, are given by

|α〉f =1

√

e|α|2

f

∞∑

n=0

αn

f(n)!√n!|n〉, (101)

where f(0)! = 1, f(n)! =∏n

j=1 f(j) for n ≥ 1, and α is such that e|α|2

f

and comparing this expression with (103) one identifies eq(x) with exf corre-

sponding to

f(0)! = 1, f(n)! =1√Qn−1

, n = 1, 2, . . . . (105)

In [69] this choice of f(n)! has been substituted in (101) to get the normalizedstates

|α〉 = N (α){

|0〉+∞∑

n=1

√

Qn−1αn√n!|n〉}

,

N (α) = 1√1 +

∑∞n=1Qn−1

|α|2

n!

, (106)

called the f -coherent states attached to the Tsallis q-exponential function.This class of f -coherent states and their nonclassical properties and otherphysical aspects have been studied in detail for 1 < q ≤ 2 in [69].

Noting that in (103)

Qn−1 = (1+(q−1)(n−1))! =n∏

j=1

(1+(q−1)(j−1)), n = 1, 2, , . . . , (107)

the Taylor series expression for eq(x) has been identified in [70] with a de-formed exponential function as

eq(x) =

∞∑

n=0

xn

[n](q−1)!, (108)

where

[n](q−1) =n

1 + (q − 1)(n− 1) , n = 0, 1, 2, . . . ,

[0]q−1! = 1, [n](q−1)! =

n∏

j=1

[j](q−1) =n!

(1 + (q − 1)(n− 1))! , for n ≥ 1.

(109)

Note that[0](q−1) = 0, [1](q−1) = 1, lim

q−→1[n](q−1) = n. (110)

17

Regarding [n](q−1) as a deformed number, a scheme of q-deformation of non-linear maps was proposed in [70] and this proposal has been found to havemany applications (see, e.g., [71–76]). Equation (108) shows that we canwrite eq(x) as a φ-exponential function:

eq(x) =∞∑

n=0

xn

φT (n)!= exφT , with φT (n) = [n](q−1). (111)

Let us close this section with an interesting observation. Note that wecan write

eq(x) = eln

(

(1+(1−q)x)1

1−q

)

= e∑∞

n=1(q−1)n−1

nxn, (112)

showing explicitly limq−→1 eq(x) = ex. The expression (112) for eq(x) as an

ordinary exponential could be useful for approximations in applications. Inparticular, for the Tsallis q-Gaussian function we can write, for q ≈ 1,

eq(

−βx2)

≈ e−βx2+ q−12 β2x4− (q−1)2

3β3x6. (113)

In [77] (see also [78]) it has been shown that if

h(x) =

∞∑

n=1

anxn, (114)

then

eh(x) =

∞∑

n=0

cnxn, (115)

where c0 = 1, c1 = a1, and

cn = an +1

n

n−1∑

j=1

jcn−jaj, n = 2, 3, . . . . (116)

Using this result the q-exponential function exq in (53) has been expressedas the exponential of an infinite series in [79]. For the Tsallis q-exponentialfunction we have

h(x) =

∞∑

n=1

(q − 1)n−1n

xn,

eh(x) = eq(x) =

∞∑

n=0

xn

[n](q−1)!= 1 + x+

∞∑

n=2

(1 + (q − 1)(n− 1))!n!

xn.

(117)

18

We find c0 = 1 and c1 = a1 = 1 as should be. Now, substituting

an =(q − 1)n−1

n, cn =

(1 + (q − 1)(n− 1))!n!

, for n ≥ 2, (118)

in (116) we get an identity: for n ≥ 2,

(1 + (n− 1)(q − 1))!n!

=(q − 1)n−1

n+

1

n

n−1∑

j=1

(q − 1)j−1(1 + (n− j − 1)(q − 1))!(n− j)! . (119)

We can verify this identity directly. Let us take (q − 1) = 1/τ . Then theidentity (119) becomes

(τ)n = (n− 1)!τn−1∑

j=0

(τ)jj!

, (120)

where (τ)n = τ(τ + 1)(τ + 2) . . . (τ + (n − 1)) is the rising factorial, or thePochhammer symbol, with (τ)0 = 1. Now, observe that

(τ)n+1 = n!τ

n∑

j=0

(τ)jj!

= n

[

(n− 1)!τ(

(τ)nn!

+

n−1∑

j=0

(τ)jj!

)]

= (τ + n)(τ)n,

(121)showing that (τ)n given by the right hand side of (120) satisfies the definingrecurrence relation for (τ)n, namely, (τ)n+1 = (τ + n)(τ)n with (τ)0 = 1.

As observed in [80], we can represent the Tsallis q-exponential functionas a hypergeometric series (see, e.g., [81] for details of the theory of hyper-geometric series) as follows:

eq(x) = 1F0

(

1

q − 1;−; (q − 1)x)

, (122)

where

1F0(a;−; z) =∞∑

n=0

(a)nzn

n!. (123)

19

This relation can be verified as follows:

eq(x) =

∞∑

n=0

xn

[n]q−1!= 1 +

∞∑

n=1

(

n∏

j=1

(1 + (q − 1)(j − 1)))

xn

n!

= 1 +∞∑

n=1

(

n∏

j=1

(

1

q − 1 + (j − 1))

)

((q − 1)x)nn!

=

∞∑

n=0

(

1

q − 1

)

n

((q − 1)x)nn!

= 1F0

(

1

q − 1;−; (q − 1)x)

. (124)

One may also identify eq(x) with 2F1

(

1q−1

, b; b; (q − 1)x)

, where b is an arbi-

trary nonzero parameter, since 1F0(a;−; x) = 2F1(a, b; b; x), where

2F1(a, b; c; z) =

∞∑

n=0

(a)n(b)n(c)n

zn

n!(125)

is the Gauss hypergeometric function.For the Tsallis q-logarithm we get

lnq(1 + x) = x 2F1(q, 1; 2;−x). (126)The proof of this relation is as follows:

lnq(1 + x) =(1 + x)1−q − 1

1− q =1

1− q

(

∞∑

n=0

(

1− qn

)

xn − 1)

= x

∞∑

n=0

(q)n(−x)n(n + 1)!

= x

∞∑

n=0

(q)n(1)n(2)n

(−x)nn!

= x 2F1(q, 1; 2;−x). (127)Using the well known Euler integral representation of 2F1(a, b; c; z) it can beverified that

x 2F1(q, 1; 2;−x) =∫ 1

0

dtx

(1 + xt)q=

(1 + x)1−q − 11− q = lnq(1 + x). (128)

Note that when q −→ 1 the relation (126) becomes the well known relationln(1 + x) = x 2F1(1, 1; 2;−x). We hope these identifications would help fur-ther analysis and generalizations of the Tsallis q-exponential and q-logarithmfunctions.

20

4 The deformed oscillator and the deformed

derivative associated with the Tsallis q-

exponential function

Let us now consider the deformed oscillator with the commutation relations

[

N, a†]

= a†, [N, a] = −a,[

a, a†]

= φT (N + 1)− φT (N),

φT (N) = [N ](q−1) =N

1 + (q − 1)(N − 1) . (129)

The energy spectrum of this deformed oscillator is given by

En,T =1

2(φT (n+ 1) + φT (n))

=1

2

(

2(q − 1)n2 + 2n+ 2− q(q − 1)2n2 + (3− q)(q − 1)n+ 2− q

)

,

n = 0, 1, 2, . . . , (130)

as follows from (32). For 1 ≤ q ≤ 2 we find that

En,T (q = 1) = n+1

2, n = 0, 1, 2, . . . ,

E0,T =1

2, for 1 ≤ q < 2, E0,T =

1

2, when q −→ 2,

En,T = 1, for all n > 0, when q −→ 2,lim

n−→∞En,T =

1

q − 1 , for 1 ≤ q ≤ 2,

En+1,T −En,T =2− q

(q − 1)2n2 + 2(q − 1)n+ q(2− q) ≥ 0,

for 1 ≤ q ≤ 2,limq−→2

(En+1,T −En,T ) = 0, for all n > 0. (131)

From these equations we can conclude as follows. The deformed oscillator(129) becomes a canonical boson oscillator when q = 1. For 1 < q ≤ 2it has a ground state with energy E0,T = 1/2, same as a boson, and aninfinity of excited states starting with E1,T = (1/2)+ (1/q) and increasing indiminishing steps up to the maximum energy E∞,T = 1/(q−1)

states, E∞,T − E1,T = (1/(q(q − 1))) − (1/2), decreases and when q −→ 2the energy band shrinks to a single energy level with infinite degeneracy(En,T = 1 for all n > 0). Thus, when q −→ 2 this oscillator becomes a two-level system with a nondegenerate ground state with energy E0,T = 1/2 andan infinitely degenerate excited state with energy eigenvalues En,T = 1 forall n > 0. It should be interesting to investigate the physical applications ofsuch deformed oscillators related to the Tsallis q-exponential function. Thedeformed oscillator algebra (129) has been derived in [82] in a completelydifferent context, unrelated to the Tsallis q-exponential function, and thethermodynamical properties of a gas of photons obeying such an algebrahave been evaluated (see also [20] for a discussion of the algebra (129)).

It is seen that the µ-oscillator, introduced in [83], for which φ(N) =N/(1+µN), with µ > 0, is only slightly different from (129). The µ-oscillatoris the first deformed oscillator found with its energies lying within a band offinite width exactly like in the case of the deformed oscillator (129). For µ = 0the µ-oscillator becomes a canonical boson oscillator and for any µ > 0 it hasa spectrum bounded from above, but it does not share the other interestingproperties of the deformed oscillator (129). This can be seen by comparing(131) with the energy spectrum of the µ-oscillator:

En,µ =1

2

(

n+ 1

1 + µ(n+ 1)+

n

1 + µn

)

=1

2

(

2µn2 + 2(µ+ 1)n+ 1

µ2n2 + 2µn+ µ2 + µ+ 1

)

,

En,µ=0 = n +1

2, n = 0, 1, 2, . . . ,

E0,µ =1

2 (µ2 + µ+ 1), for µ ≥ 0,

limn−→∞

En,µ =1

µ, for µ > 0,

En+1,µ −En,µ =1

µ2n2 + 2µ(µ+ 1)n+ 2µ+ 1, for µ ≥ 0. (132)

In particular, note that there is no finite value of µ for which En,µ becomesindependent of n for all n > 0. It is this property which makes the deformedoscillator (129) become a two-level system when q −→ 2.

The deformed oscillator (129) can be presented as an f -oscillator as fol-

22

lows:

a† = fT (N)b†, a = bfT (N) = fT (N + 1)b, N = b

†b,

fT (N) =

√

φT (N)

N=

√

[N ](q−1)N

=1

√

(1 + (q − 1)(N − 1))). (133)

It has been found in [69] that the states in (106) are the coherent states ofthis f -oscillator corresponding to the eigenfunctions of a = fT (N + 1)b andare related to the Tsallis q-exponential function. To understand this let usproceed as follows. The Fock representation of (129) is given by

a†|n〉 =√

[n+ 1](q−1) |n+ 1〉, a|n〉 =√

[n](q−1) |n− 1〉,N |n〉 = n|n〉, n = 0, 1, 2, . . . . (134)

It follows that

|n〉 = a†n

√

[n](q−1)!|0〉. (135)

With the Tsallis q-exponential function expressed as in (108), the normalizedcoherent states of the deformed oscillator (129) are seen to be given by

|α〉T =1

√

eq (|α|2)

∞∑

n=0

αn√

[n](q−1)!|n〉 = eq

(

αa†)

|0〉. (136)

Noting that

|n〉 = a†n

√

[n](q−1)!|0〉 =

(

fT (N)b†)n

√

[n](q−1)!|0〉 = fT (n)!b

†n

√

[n](q−1)!|0〉, (137)

and using (99), (101), and (111), it is seen that

|α〉T =1

√

eq (|α|2)

∞∑

n=0

fT (n)!√n!αn

[n](q−1)!|n〉 = 1√

e|α|2

fT

∞∑

n=0

αn

fT (n)!√n!|n〉, (138)

is an f -coherent state of the boson oscillator associated with the Tsallis q-exponential function as identified in [69].

To get the Bargmann-like representation of the deformed oscillator (129)we must have a deformed derivative D(T,q) such that

D(T,q)xn = [n](q−1)x

n−1. (139)

23

Following (39), as has been done in ( [82, 84, 85]), one may choose the for-mal operator (1/x)φ(xD) = (1/(1 + (q − 1)xD))D as the required deformedderivative which satisfies (139) and has eq(kx) as its eigenfunction for any k.However, in this case it is only a formal operator, not suitable to operate onan arbitrary function without a series expansion. Another suggestion for therequired deformed derivative is (1 + (1− q)x)D for which eq(x) is an eigen-function (see [86,87]). But, it is also not suitable for the purpose since eq(kx)is not its eigenfunction for arbitrary k unless one defines eq(kx) = (eq(x))

k asin [34, 35, 80]. As we have seen earlier, for the Tsallis q-exponential functioneq(kx) = (1+(1−q)kx)1/(1−q) 6= eq(x)k. Further, the derivative (1+(1−q)x)Ddoes not satisfy the requirement in (139).

To derive the required deformed derivative we follow [88] where the de-formed derivative corresponding to the µ-oscillator [83] has been obtained.Slightly modifying the suggestion in [88] we get the desired deformed deriva-tive as

D(T,q)F (x) =

∫ 1

0

dt t1−qDF(

tq−1x)

. (140)

It can be verified that for F (x) = xn

D(T,q)xn =

∫ 1

0

dt t1−qD(

t(q−1)nxn)

=

∫ 1

0

dt t(q−1)(n−1)Dxn

= nxn−1∫ 1

0

dt t(q−1)(n−1) =nxn−1

1 + (q − 1)(n− 1)= [n](q−1)x

n−1, (141)

as required. Then, it follows that

D(T,q)eq(kx) = D(T,q)

(

∞∑

n=0

knxn

[n](q−1)!

)

= keq(kx). (142)

It can also be verified directly that

D(T,q)eq(kx) = keq(kx). (143)

This is seen as follows. From (140) we have

D(T,q)eq(kx) =

∫ 1

0

dt t1−qD[

(

1 + (1− q)ktq−1x)

11−q

]

24

= k

∫ 1

0

dt(

1 + (1− q)ktq−1x)

q

1−q

= k

∫ 1

0

dt t−q(

t1−q + (1− q)kx)

q

1−q

=

[

k

1− q

∫

d(t1−q)(

t1−q + (1− q)kx)

q

1−q

]∣

∣

∣

∣

t=1

t=0

=[

kt(

1 + (1− q)ktq−1x)

11−q

]∣

∣

∣

t=1

t=0

=[

kteq(

ktq−1x)]∣

∣

t=1

t=0= keq(kx), (144)

proving (143). The inverse of the deformed derivative D(T,q), the deformedintegral, is seen to be given by

∫

d(T,q)x f(x) =

[

d

dt

(∫

dx tf(

t(q−1)x)

)]∣

∣

∣

∣

t=1

t=0

, (145)

such that if∫

d(T,q)x f(x) = F (x) then DT,qF (x) = f(x). It can be verifiedthat

∫

d(T,q)x xn =

[

d

dt

(∫

dx t(q−1)n+1xn)]∣

∣

∣

∣

t=1

t=0

=xn+1

[n+ 1](q−1), (146)

as should be, since D(T,q)xn+1 = [n+ 1](q−1)x

n.Now, the Bargmann-like representation of the deformed oscillator (129)

is given bya† = x, a = D(T,q), N = xD. (147)

Under the inner product

〈f |g〉T =[

f ∗(

D(T,q))

g(x)]∣

∣

x=0, (148)

the monomials

ξn,T (x) =xn

√

[n](q−1)!, n = 0, 1, 2, . . . , (149)

form an orthonormal basis and carry the Fock representation

a†ξn,T (x) =√

[n + 1](q−1) ξn+1,T (x), aξn,T (x) =√

[n](q−1) ξn−1,T (x),

Nξn,T (x) = nξn,T (x), n = 0, 1, 2, . . . . (150)

25

In the Bargmann-like representation, with

〈eq(αx) |eq(αx)〉T = eq(

|α|2)

, (151)

the normalized coherent states of the deformed oscillator (129), eigenfunc-tions of a = D(T,q), are given by

ψα,T (x) =1

√

eq (|α|2)eq(αx) =

1√

eq (|α|2)

∞∑

n=0

αnxn

[n](q−1)!

=1

√

eq (|α|2)

∞∑

n=0

αn√

[n](q−1)!ξn,T (x). (152)

The case q = 2 is interesting. In this case we have fT (n) = 1/√n and the

state |α〉T becomes the harmonious state (102) as observed in [69]. Now, forq = 2 we have

[0](1) = 0, [n](1) = 1, for all n. (153)

Thus, for q = 2 the deformed oscillator (129) has the Fock representation

a†|n〉 = |n+ 1〉, n = 0, 1, 2, . . . ,a|0〉 = 0, a|n〉 = |n− 1〉, n = 1, 2, . . . ,N |n〉 = n|n〉, n = 0, 1, 2, . . . , (154)

and can be identified with the algebra of the Susskind-Glogower exponen-tial phase operators (see, e.g., [59]). The coherent states of this deformedoscillator are

|α〉(T,q=2) =√

1− |α|2∞∑

n=0

αn|n〉, |α| < 1, (155)

where√

1− |α|2 = 1/√

e2 (|α|2) in which e2 (|α|2) is a Tsallis q-exponentialfunction corresponding to q = 2.

It is instructive to look at the deformed derivative and integral operatorsexplicitly for q = 2. Corresponding to q = 2 the deformed derivative becomes,as seen from (140),

D(T,q=2)F (x) =

∫ 1

0

dt t−1DF (tx). (156)

26

For F (x) = xn

D(T,q=2)xn =

∫ 1

0

dt ntn−1xn−1 = xn−1, (157)

as required. To verify that

e2(kx) =1

1− kx, (158)

is an eigenfunction of D(T,q=2), satisfying

D(T,q=2)e2(kx) = ke2(kx), (159)

we follow (144) to get

D(T,q=2)e2(kx) =

∫ 1

0

dt t−1D[

(1− ktx)−1]

= k

∫ 1

0

dt (1− ktx)−2

= k

∫ 1

0

dt t−2(

t−1 − kx)−2

=

[

−k∫

d(t−1)(

t−1 − kx)−2]∣

∣

∣

∣

t=1

t=0

=[

kt (1− ktx)−1]∣

∣

t=1

t=0

= [kte2(ktx)]|t=1t=0 = ke2(kx). (160)

In this case the deformed integral is, as seen from (145),

∫

d(T,q=2)x f(x) =

[

d

dt

(∫

dx tf(tx)

)]∣

∣

∣

∣

t=1

t=0

. (161)

For f(x) = xn

∫

d(T,q=2)x xn =

[

d

dt

(∫

dx tn+1xn)]∣

∣

∣

∣

t=1

t=0

= xn+1, (162)

as should be, since D(T,q=2)xn+1 = xn.

27

5 Conclusion

In fine, it is found that the deformed oscillator defined by the commutationrelations

[

N, a†]

= a†, [N, a] = −a,[

a, a†]

= φT (N + 1)− φT (N),

with φT (N) =N

1 + (q − 1)(N − 1) , (163)

is associated with the Tsallis q-exponential function

eq(x) = (1 + (1− q)x)1

1−q , (164)

representable as

eq(x) =

∞∑

n=0

xn

φT (n)!, with φT (n)! = φT (n)φT (n− 1) . . . φT (2)φT (1). (165)

In a Bargmann-like representation the annihilation operator a correspondsto a deformed derivative defined by

D(T,q)F (x) =

∫ 1

0

dt t1−qDF(

tq−1x)

, (166)

with the Tsallis q-exponential functions as its eigenfunctions. Thus the Tsal-lis q-exponential functions are the coherent states of the deformed oscillator(129). When q = 2 these deformed oscillator coherent states correspond tostates known variously as phase coherent states, harmonious states, or pseu-dothermal states. Further, when q = 1 this deformed oscillator is a canonicalboson oscillator, when 1 < q < 2 it has an energy spectrum bounded fromabove and with the ground state same as a boson, and when q −→ 2 it be-comes a two-level system with a nondegenerate ground state and an infinitelydegenerate excited state. It should be worthwhile to study the physical appli-cations of such deformed oscillators. The expression in (166) for the deformedderivative D(T,q) for which the Tsallis q-exponential functions are eigenfunc-tions should lead to interesting applications, particularly, in nonextensivestatistical mechanics which is essentially based on the Tsallis q-exponentialfunction and its inverse function, namely, the Tsallis q-logarithm function. A

28

remark in [88], made in the context of the µ-oscillator, points to the existenceof possibilities for extending the deformed derivative (166) and the Tsallisq-exponential function with more deformation parameters. For example, fol-lowing the remark in [88], if we substitute the ordinary derivative D in (166)by the two-parameter derivative D(p,r) (D(p,q) in (91) with q replaced by r) wewould get a (p, q, r)-deformed derivative and its eigenfunctions would be theTsallis (p, q, r)-exponential functions (see [89] for the existing two-parametergeneralization of the Tsallis q-logarithm and q-exponential function).

References

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[2] Curtright, T., Fairlie, D., and Zachos, C. (Eds.), Quantum Groups:Proceedings of the Argonne Workshop, (World Scientific, 1991).https://www.worldscientific.com/worldscibooks/10.1142/1206.

[3] Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 479-487 (1988).https://doi.org/10.1007/BF01016429.

[4] Arik, M. and Coon, D. D., Hilbert spaces of analytic functionsand generalized coherent states, J. Math. Phys. 17 524-527 (1976).https://doi.org/10.1063/1.522937.

[5] Kuryshkin, V. V., Generalized quantum operators of creationand annihilation, Ann. Fond. L. de Broglie 5 111-125 (1980).https://inis.iaea.org/search/searchsinglerecord.aspx?recordsFor=SingleRecord&RN=12600560.

[6] Jannussis, A., Brodimas, G., Sourlas, D., and Zisis, V., Remarkson the q-quantization, Lett. Nuovo Cimento 30 123-127 (1981).https://doi.org/10.1007/BF02817324.

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1 Introduction2 Deformed oscillators: Some examples3 Tsallis q-exponential function as a -exponential function4 The deformed oscillator and the deformed derivative associated with the Tsallis q-exponential function5 Conclusion