$\kappa$-Deformed quantum and classical mechanics for a system with position-dependent effective mass

We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $\kappa$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.


I. INTRODUCTION
Minimum length scales are of crucial importance in several areas of physics like quantum gravity, string theory, relativity, fundamentally due to the techniques developed for removing divergences in field theories maintaining the parameters lengths as universal constants of the theory in question (for a review see for instance Ref. 1). In this sense, the seek for these minimum lengths in quantum mechanics has been translated into generalizations of the standard commutation relationship between position and momentum. 2 Further studies in noncommuting quantum spaces led to a Schrödinger equation with a position-dependent effective mass (PDM). 3 Along the last decades the PDM systems have attracted attention because of their wide range of applicability in semiconductor theory, 4-7 nonlinear optics, 8 quantum liquids, 9,10 inversion potential for NH 3 in density functional theory, 11 particle physics, 12 many body theory, 13 molecular physics, 14 Wigner functions, 15 relativistic quantum mechanics, 16 superintegrable systems, 17 nuclear physics, 18 magnetic monopoles, 19,20 astrophysics, 21 nonlinear oscillations, [22][23][24][25][26][27][28][29][30][31] factorization methods and supersymmetry, 32-36 coherent states, [37][38][39] etc.
In this work we employ the κ-algebra for generalizing classical and quantum mechanics with the aim of studying the properties of the resulting noncommuting space originated by the deformation. Between these properties we found that the κ-deformed space, classical and quantum, allows to characterize a PDM system with the mass being univocally determined by the κ-algebra. The work is organized as follows. In Section II, we review the properties of the κ-algebra that are used in the forthcoming sections. Next, we present in Section III the dynamics resulting from a generic PDM and then we specialize with the mass function m(x) associated to the κ-algebra. Here we obtain the Schrödinger equation associated to the κ-derivative and we show that all the standard properties remain to be valid in the deformed structure such as the continuity equation, the wave-function normalization, the classical limit, etc. In Section IV we illustrate our proposal with a particle in an infinite potential well. In Section V, we use the κ-deformed formalism to revisit the problem of the Mathews-Lakshmanan oscillator. [22][23][24][25][26][27][28][29][30][31] Finally, in Section VI we draw some conclusions and outline future perspectives.

II. REVIEW OF THE κ-ALGEBRA
The κ-statistics emerges from a generalization of the Boltzmann-Gibbs entropy derived by means of a kinetic interaction principle, that allows to characterize nonlinear kinetics in particle systems (see, for instance, Ref. 59 for more details). In the last two decades several theoretical developments have shown that the κ-formalism preserve features as Legendre transform in thermodynamics, 62 H-theorem, 63 Lesche stability, 64 composition law of the κ-entropy, 65 among others. The mathematical background of the κ-deformed formalism is based on generalizations of the standard exponential and logarithm functions, from which it is possible to introduce deformed versions of algebraic operators and calculus, 59-61 trigonometric and hyperbolic functions, 66,67 Fourier transform, 68 Gaussian law of error, 69 Stirling approximation and Gamma function, 70 Cantor set, 71 Lambert W function 72 , information geometry, 73 and other possible exponential and logarithm functions, 74 etc.
More specifically, the so-called κ-exponential is a deformation of the ordinary exponential function, defined by 59-61 The inverse function of the κ-exponential is the κ-logarithm, given by In the limit κ → 0, the ordinary exponential and logarithmic functions are recovered, i.e.
exp 0 x = exp x and ln 0 x = ln x. These functions satisfy the properties exp κ (a) exp A κ-deformed calculus has been introduced in Ref. 59 from the deformed differential The definition of a deformed variable u κ (also named deformed κ-number) is implies d κ u = du κ , i.e., the deformed differential of an ordinary variable u can be rewritten as with the ordinary differential of a deformed variable u κ . In this way, one defines the κ-derivative operator by with the κ-exponential an eigenfunction of D κ , D κ exp κ u = exp κ u. Similarly, the dual κ-derivative is defined by which satisfies D κ ln κ u = 1/u. These operators obey D κ x(y) = [D κ y(x)] −1 . In particular, (5) and (6) we see that the standard derivative is recovered as κ → 0. The deformed derivative operator (5) can be seen as the variation of the function f (u) with respect to a nonlinear variation of the independent variable u, i.e., D κ f (u) = df (u)/du κ . On the other hand, the dual deformed derivative operator (6) is the rate of change of a nonlinear variation of the function f (u) with respect to the standard variation of the independent variable u, D κ f (u) = d κ f (u)/du. The deformed second derivatives satisfy and These rules can be extended to deformed derivatives of higher order.

POSITION-DEPENDENT MASS
A. κ-Deformed classical formalism Let us first consider the problem of a particle with a position-dependent mass (PDM) m(x) in 1D for the classical formalism. The Hamiltonian of the system is whose the linear momentum is p = m(x)ẋ, leads to the equation of motion with F (x) = −dV /dx the force acting on the particle, whereẋ = dx/dt,ẍ = d 2 x/dt 2 and m ′ (x) = dm/dx give velocity, acceleration and mass gradient, respectively. The point maps the Hamiltonian (9) of a particle with PDM m(x) in the usual phase space (x, p) into another Hamiltonian of a particle with a constant mass m 0 represented in the deformed phase space (η, Π), with U(η) = V (x(η)) the potential expressed in the deformed space-coordinate η. When m(x) = m 0 , both representations coincide.
Let us consider in particular the mass function where the parameter κ has units of inverse length and controls the dependence of the mass with position, where κ = 0 corresponds to the standard case. Thus equation of motion (10) becomes This equation can be compactly rewritten in the form of a deformed Newton's second law Moreover, for the mass function (13) the κ-deformed spatial coordinate and its conjugated linear momentum are and with Poisson brackets {x κ , Π κ } x,p = 1. The deformed displacement d κ x of a particle with the non-constant mass m(x), given in Eq. (13), is mapped into the usual displacement dx κ in a deformed space x κ provided with a constant mass m 0 : up to first order. The time evolution of the system is governed by the dual derivative, i.e.

B. κ-Deformed quantum formalism
In the quantization of a PDM system an ordering ambiguity arises for defining the kinetic energy operator in terms of the mass operator m(x) and the linear momentump. There are several ways to define a Hermitian kinetic energy operator, and a general two-parameter form is given bŷ For more details see the discussions, for instance, of von Roos, 4 Lévy-Leblond, 40 and others.
Among many particular cases in the literature, we point out the proposals by Ben Daniel and Duke (α = β = 0), 41 Gora and Williams (α = 1, β = 0), 42 Zhu and Kroemer (α = β = 1 2 ), 43 Li and Kuhn (α = 1 2 , β = 0). 44 Morrow and Brownstein 45 have shown that only the case α = β satisfies the conditions of continuity of the wave-function at the boundaries of a heterojunction in crystals. In particular, Mustafa and Mazharimousavi 46 have shown that the case α = β = 1 4 allows the mapping of a quantum Hamiltonian with PDM into a Hamiltonian with constant mass by means a PCT. More precisely, considering the quantum with Ψ(x, t) = ψ(x)e −iEt/ and E the eigenvalue corresponding to the eigenfunction ψ(x) of where the probability current is Equation (19) can be conveniently rewritten by means of the transformation Ψ(x, t) = Let us consider in particular the mass function (13). The modified wave-function Φ(x, t) = with which is the analog of the κ-derivative operator (5). Using Eq. (8), Equation (23) is indeed equivalent to a Schrödinger-like equation for Φ(x, t) with the non- wherep κ ≡ −i D κ = √ 1 + κ 2x2p stands for a κ-deformed non-Hermitian momentum operator, and obeys the commutation relation This leads to generalized uncertainty principle ∆x∆p κ ≥ 2 √ 1 + κx 2 . We notice that if the Besides, we obtain the κ-deformed continuity equation It is worth noting that there is an equivalence between the Schrödinger equation for the Hermitian system (18) with the mass function m(x) given by (13) and the non-Hermitian one (25) expressed in terms of a κ-deformed momentum operator, where Ψ(x, t) must be Moreover, we see that in the description of quantum systems with the mass function (13) in terms of the modified wave-function Φ(x, t), the usual derivative and integral with respect to the variable x are replaced by their corresponding κ-deformed versions. Analogous features apply in the classical formalism, with the motion equation expressed in terms of the dual κ-derivative (see Eq. (15)).
Using the change of variable ) a modified potential in terms of the original one V and the inverse transformation x = x(x κ ). Therefore, the wave-equation for The quantum Hamiltonian associated with the Schrödinger wave-equation , that can be obtained by applying the point canonical transformation (x,p) → (x κ ,Π κ ) on the quantum Hamiltonian (18) wherê with [x κ ,Π κ ] = i 1 . Also, we have thatΠ κ is in accordance with the definition of a PDM pseudo-momentum operator introduced in Ref. 46. Thus, the dynamical variables (11) are the classical counterparts of the Hermitian operators (31).
From the eigenvalue equationΠ κ |k = k|k , the eigenfunctions in the representation where C is a constant. As in the non deformed case (κ = 0), the function ψ k (x) is not normalizable. Even though, a deformed wave-packet can be defined from the κ-deformed where g(k) is the distribution function of the wave-vectors k. It is verified straightforwardly that the corresponding wave-packet of the operatorp κ is ϕ( From the Plancherel theorem, we have

IV. PARTICLE IN AN INFINITE POTENTIAL WELL
In Secs. IV and V we illustrate the quantum and classical κ-deformed formalism with two paradigmatic examples.

A. Classical case
First we consider the problem of a particle confined in an infinite square potential well between x = 0 and x = L. If H(x, p) = E is the energy of the classical particle, then the For v(0) = v 0 and 0 < x < L, the position as a function of time is . Hence, the classical probability density ρ classic (x)dx ∝ dx/v to find the from which the uniform distribution ρ classic (x) = 1/L is recovered when κ → 0. The first and the second moments of the position and the linear momentum for the classical distribution (35) are We can verify that lim κ→0 x = L/2, lim κ→0 x 2 = L 2 /3 and lim κ→0 p 2 = 2m 0 E. From the change of variable x → x κ the PDM particle confined in an interval [0, L] is mapped into a particle with constant mass in [0, L κ ], where L κ = arcsinh(κL)/κ corresponds to the length of the box in the deformed space.

B. Quantum case
Let us now analyze the problem in the κ-deformed quantum formalism. Considering Φ(x, t) = ϕ(x)e −iEt/ , this leads to the time-independent Schrödinger-like equation , whose eigenfunctions are given by for 0 ≤ x ≤ L, and ϕ n (x) = 0 elsewhere, with C 2 κ = 2/L κ and k κ,n = nπ/L κ , where n is an integer number and L κ = κ −1 arcsinh(κL). The energy levels corresponding to these eigenfunctions are with ε 0 = 2 π 2 /(2m 0 L 2 ). The effect of the deformation parameter κ corresponds to a contraction of the space (L κ < L for κ = 0), and consequently this leads to an increase of the energy levels of the particle. In Fig. 1 we illustrate the energy levels of the particle as a function of the quantum number for different values of κ. The probability densities of the stationary states in position space are Substituting Eq. (37) into the inverse Fourier transform (34), we obtain the eigenfunctions for the particle confined in a box in momentum space k g n (k) = n L κ 2 Consequently, its associated probability density results Interestingly, the eigenfunctions (40) and the probability densities (41) have the same form as in the case of a particle with constant mass, but with L κ instead of L. In Fig. 2 we plot the eigenfunctions ψ n (x) and their probability densities in the coordinate and momentum spaces, ρ n (x) and γ n (k), for the three states of lower energy and for some values of the deformation parameter κ. We can see that as κ increases, ρ n (x) becomes more asymmetric and γ n (k) more spread along its domain. In Fig. 3 we show that the average value of the quantum probability density ρ n (x) approaches to the classical probability density ρ classic (x) (illustrated here for n = 20) in accordance with the correspondence principle. The distribution γ n (k) is also shown for the same state n = 20. product L 0 ϕ n (x)ϕ n ′ (x)d κ x = δ n,n ′ , so that any continuous function in the interval [0, L] can be written as a linear combination with the coefficients c n of the series given by Concerning the Sturm-Liouville problem, Braga et al. 54 Similarly as was done in Ref. 54, we consider as a quantitative measure of the error the Fig. 4 we show that when N becomes large, the partial sum f N (x) converges to f (x) = 1, as well as R(N) goes to zero.
Expected values ofx andΠ κ for stationary states can be obtained from usual internal products of the eigenfunctions ψ n (x) or, equivalently, from the deformed internal products of the modified eigenfunctions ϕ(x), i.e., which l is a positive integer. The expectation values x , x 2 , p , and p 2 for the eigenstates of the particle in a one dimensional infinite potential well are respectively with I j,l (z) = 2 z 0 sech 2j (λ κ u)tanh 2l (λ κ u) sin 2 (nπu)du and λ κ = κL κ . The analytical form of the functions I j,l (z) is expressed by means of the Appell hypergeometric function of two variables (http://functions.wolfram.com/ElementaryFunctions/Sech/21/01/14/01/10/01/0001/) and due to its complicated expression, it becomes convenient to write the expectation value (45d) in terms of I j,l (z).
with κ a continuous parameter that controls the anharmonicity of the potential. In Fig. 6 we plot the phase spaces (x, p) and (x κ , Π κ ) for different values of κA 0 . The bounded motion in the interval −A κ < x < A κ of the standard space turns out into the interval −x κ,max < x κ < x κ,max = κ −1 atanh(κA 0 ) in the deformed space. Besides, the unbounded motion has the interval of the linear momentum 0 < |p| < m 0 ω 0 A 0 turned into m 0 ω 0 A 0 1 − 1 As the dimensionless parameter κA 0 increases from 0 to 1.1 within the interval [0.9, 1.1] it is observed that the horizontal axe of the ellipses become infinite, thus giving place to an unbounded motion. By means of the WKB approximation we can obtain the energy levels of the corresponding quantum system. Using this method, we have with n = 0, 1, 2, . . ..
which corresponds to the energy levels of an anharmonic oscillator.
From Eq. (50), the classical density probability of finding the particle between x and κ −x 2 . The first and second moments of the position and the linear momentum in terms of the amplitude or the energy for the deformed oscillator are The mean values of the kinetic and potential energies satisfy the relationship we have that the virial theorem (V = T ) is satisfied only for κA 0 = 0, which implies κ = 0.

B. κ-Deformed quantum oscillator
The corresponding κ-deformed time-independent Schrödinger equation for the PDM oscillator is 23 Making the change of variable x → x κ = κ −1 arcsinh(κx) (see Eq. (4)) we obtain a particle with constant mass m 0 subjected to the Pöschl-Teller potential with ǫ = E − ω 0 /2κ 2 a 2 0 , ν(ν + 1) = 1/κ 4 a 4 0 and a 2 0 = /m 0 ω 0 . The solutions of the Eq. (57) are where µ = ν − n, n is an integer and P µ ν are the associated Legendre polynomials. Then, the eigenfunctions for the κ-deformed oscillator in the space representation x are The energy levels are given by with ω κ = ω 0 1 +  From the Legendre differential equation the identities (see Eqs. (2) and (3)  κ-deformed oscillator, along with the uncertainties ∆x and ∆p of x and p, for the ground state and the first three excited ones. As expected, while ∆x increases as the dimensionless deformation parameter κa 0 varies within the interval [−1, 1], ∆p decreases and viceversa. In turn, this implies a generalized κ-uncertainty inequality (Fig. 9 (c)) which is an increasing function of the quantum number n and it also grows fast as κa 0 varies. The symmetry exhibited around the axis κa 0 = 0 in the curves of the Fig. 9 are a consequence of the invariance of the mass function (and then of the Hamiltonian too) given by Eq. (13) against the transformation κ → −κ.

VI. CONCLUSIONS
We have presented the quantum and the classical mechanics that results from assuming a position-dependent mass related to the κ-algebra, which is the mathematical background underlying κ-statistics. Indeed, we have characterized both the quantum and classical formalism of a particle with a PDM determined univocally by the κ-algebra. The consistency of the κ-deformed formalism is manifested in the following arguments.
The κ-deformed Schrödinger equation turns out to be equivalent to a Schrödinger-like equation for a deformed wave-function provided with a κ-deformed non-Hermitian momentum operator. Within the κ-formalism one can define deformed versions of the continuity equation, the Fourier transform, etc. In particular, a deformed Newton's second law in terms of the deformed dual κ-derivative (Eq. (15)) follows in the classical limit.
We have illustrated the approach with the problems of a particle confined in an infinite potential well and a κ-deformed oscillator which is equivalent to the Mathews-Lakshmanan oscillator (in the standard space) or to the Pösch-Teller potential problem (in the κ-deformed space), provided with the change of variable x → x κ . We have obtained the distributions for the classical case as well as the eigenfunctions and eigenenergies for the quantum case.
Although we have applied the mapping approach to a κ-deformed space in order to study the quantum Mathews-Lakshmanan oscillator, it is important to mention that other equivalent approaches can be found in the literature. For instance, factorization methods, supersymmetry and coherent states have also been investigated for this nonlinear oscillator (see [34][35][36][37][38] and references therein). in the usual coordinates (x, p) and in the deformed ones (x κ , Π κ ). It is verified that for a certain range of values of κ the motion is unbounded (Fig. 6).
We consider that the techniques employed in this work could stimulate the seek of other generalizations of classical and quantum mechanical aspects, as has been reported in recent researches by means of the q-algebra.

DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.