Algebraic structures and deformed Schr\"{o}dinger equations from groups entropies

TL;DR

This paper develops a generalized algebraic framework inspired by group entropy theory, leading to a G-deformed Schrödinger equation that encompasses Tsallis and Kappa statistics, with applications to quantum systems like the infinite potential well.

Contribution

It introduces the G-algebra and G-differential calculus, unifying various statistical mechanics frameworks and deriving a generalized Schrödinger equation with novel eigenfunction properties.

Findings

G-algebra generalizes real number algebra based on group entropies.

The G-deformed Schrödinger equation includes Tsallis and Kappa cases as special instances.

Eigenfunctions exhibit non-uniform zeros spacing described by G-algebra sums.

Abstract

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schr\"{o}dinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schr\"{o}dinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.

Tables

Table 1. Table 1: Several structures derived from the group entropy approach, by means of the formal group law, are listed. The application to the quantum mechanics implies the canonical transformation { x ^ , p ^ } ↦ { x ^ G , p ^ G } maps-to ^ 𝑥 ^ 𝑝 subscript ^ 𝑥 𝐺 subscript ^ 𝑝 𝐺 \{\hat{x},\hat{p}\}\mapsto\{\hat{x}_{G},\hat{p}_{G}\} where the form of the G 𝐺 G -Schrödinger equation remains invariant.

Structure	Group entropy structure
algebra of real numbers	$G$-algebra
integers $n\in \mathbb{Z}$	$G$-integers: $n_G=G\left(nG\left(1\right)\right)$
real number $x$	$G$-real number: $x_G=G^{-1}\left(x\right)$
differential $dx$	$G$-differential: $d{x}_G=\frac{dx}{G^{\prime }\left(G^{-1}\left(x\right)\right)}$
derivative $\frac{\partial }{{\partial }_x}$	$G$-derivative $D_G$: $G^{\prime }\left(G^{-1}\left(x\right)\right)\frac{\partial }{{\partial }_x}$
integral $\int f\left(x\right)𝑑x$	$G$-integral: ${\int }_{\left(G\right)}f\left(x_G\right)𝑑x_G$
	$G$-deformed operators ${\hat{x}}_G,{\hat{p}}_G$
conjugated operators $\hat{x},\hat{p}$	${\hat{x}}_G=G^{-1}\left(\hat{x}\right)$
	${\hat{p}}_G=\frac{1}{2}\{G^{\prime }\left(G^{-1}\left(\hat{x}\right)\right),\hat{p}\}$
$[\hat{x},\hat{p}]=i\hslash \hat{1}$	$[{\hat{x}}_G,{\hat{p}}_G]=i\hslash \hat{1}$
Schrödinger equation	$G$-Schrödinger equation
$i\hslash \frac{\partial \Psi }{\partial t}=-\frac{{\hslash }^2}{2m_0}\frac{{\partial }^2\Psi }{\partial x^2}+V\Psi$	$i\hslash \frac{\partial {\Phi }_G}{\partial t}=-\frac{{\hslash }^2}{2m_0}{D}_G^2{\Phi }_G+V{\Phi }_G$
wavefunction	$G$-wavefunction:
$\Psi (x,t)$	${\Phi }_G(x,t)=\Psi (x,t)\sqrt{G^{\prime }\left(G^{-1}\left(x\right)\right)}$

Equations

$$\Phi (x,y)=G(G^{-1}(x)+G^{-1}(y))=x{\oplus }_{G}y$$

$$G(t)=t+\sum _{k=1}^{\infty }{a}_k\frac{t^{k+1}}{k+1}$$

$$F(s)=G^{-1}(s)=t+\sum _{i=1}^{\infty }{b}_i\frac{s^{i+1}}{i+1}$$

$${\log }_G(x)=G(\log x)\text{(group logarithm)}$$

$${\exp }_G(x)=\exp (G^{-1}(x))\text{(group exponential)}.$$

$${\log }_G(x{\otimes }_{G}y)={\log }_G(x)+{\log }_G(y)$$

$$x{\otimes }_{G}y={\exp }_G({\log }_G(x)+{\log }_G(y)).$$

$${\exp }_G(x+y)={\exp }_G(x){\otimes }_G{\exp }_G(y),$$

$${\exp }_G(x){\exp }_G(y)={\exp }_G(x{\oplus }_{G}y),$$

$${\log }_G(xy)={\log }_G(x){\oplus }_G{\log }_G(y).$$

$$G_{\mathcal{B}\mathcal{G}}(t)=t\text{(BG-class)}$$

$$G_q(t)=\frac{\exp ((1-q)t)-1}{1-q}\text{(Tsallis class)}$$

$${\log }_{q}x=\frac{x^{1-q}-1}{1-q}$$

$${\exp }_q^x=[1+(1-q)x{]}_{+}^{\frac{1}{1-q}},(x\in [-1/(1-q),\infty ))$$

$$G_{\mathcal{K}}(t)=\frac{\exp (\sigma t)-\exp (-\sigma t)}{2\sigma }\text{(Kaniadakis class)}$$

$$G_{a,b}(t)=\frac{\exp (at)-\exp (bt)}{a-b}\text{(Abe class)}$$

$$(n+m{)}_G=n_G{\oplus }_G{m}_G$$

$$x_q=\log ({\exp }_q(x))=G_q^{-1}(x)(q-\text{deformation of}x)$$

$$x_G=\log ({\exp }_G(x))=G^{-1}(x)(G-\text{deformation of}x)$$

$$(x{\oplus }_{G}y{)}_G$$

$$\widetilde{x_G}={\log }_G(\exp (x))=G(x)(G-\text{dual deformation of}x)$$

$$\widetilde{(x{\oplus }_{G}y{)}_G}$$

$$(\widetilde{x_G}{)}_G=\widetilde{(x_G{)}_G}=x.$$

$$({\mathbb{R}}_G,+)\simeq (\mathbb{R},{\oplus }_G)\simeq (\widetilde{{\mathbb{R}}_G},+)$$

$$dx=G^{\prime }(G^{-1}(x))d{x}_G$$

$$dx=\frac{1}{G^{\prime }(x)}d\widetilde{x_G}$$

$$\frac{d\widetilde{x_G}}{d{x}_G}=G^{\prime }(G^{-1}(x))G^{\prime }(x)$$

$$D_{G}f(x)=\frac{df}{d{x}_G}=G^{\prime }(G^{-1}(x))\frac{df}{dx}$$

$$\widetilde{D_G}f(x)=\frac{df}{d\widetilde{x_G}}=\frac{1}{G^{\prime }(x)}\frac{df}{dx}$$

$$\frac{D_{G}f(x)}{\widetilde{D_G}f(x)}=G^{\prime }(G^{-1}(x))G^{\prime }(x).$$

Full text

1 Introduction

The achievement of nonextensive statistics in describing multiple phenomena (where the application of the traditional Boltzmann-Gibbs (BG) statistical mechanics have shown some problems) as anomalous diffusion [1], long-range interactions [2], plasmas [3], quantum tunneling [4], cold atoms [5], onset of chaos [6], quantum chaos scales [7], and measure theory [8] among others, motivated the construction of several approaches of generalized statistical mechanics from which the foundations of the theory have been extended [9, 10]. In these advances, many mathematical frameworks associated with generalized statistics have been proposed [11, 12, 13, 14, 15]. Concerning with the nonextensive Tsallis and Kaniadakis entropies, their associated algebraic structures, called the $q$-algebra [14, 15] and the $\kappa$-algebra [12, 13], were formulated. In particular, the $q$-algebra has been employed to characterize position-dependent mass systems [16, 17, 18, 19] from a framework doted with a deformed space and a deformed derivative, where the functional form of the effective mass is related with the deformation inherited from the algebraic structure [20, 21, 22, 23, 24].

To provide a unified scenario for generalized entropies in nonextensive statistics from a group theory viewpoint, the group entropy approach has been developed [25, 26, 27, 28, 29, 30]. In this framework, the entropy functional is proposed in function of the class of interactions that are considered and it defines a group entropy that satisfies the three Shannon-Kinchin (SK) axioms and the composability action. Each group entropy is expressed in terms of a formal group element, being the formal group law a generalized additivity. The group entropies have been recently studied as generalized information measures in information geometry [30]. From group entropy approach the generalized logarithm and its inverse are derived: the group logarithm and the group exponential. When the group entropy corresponds to the BG class, all the standard definitions are recovered.

Inspired by the $q$-algebra and the $\kappa$-algebra, the goal of this letter is to propose a generalized algebraic structure, that we called $G$-algebra, based on the formal group law, the group logarithm ($G$-logarithm) and the group exponential ($G$-exponential). Thus, the present proposal allows to obtain several deformed algebraic structures in a unified way, in terms of the group entropy considered. The $q$-algebra and the $\kappa$-algebra are obtained as special cases for the Tsallis and Kaniadakis classes. From the $G$-algebra we construct its associated differential calculus, that we called $G$-calculus, provided with the $G$-differential, the $G$-derivative, and the $G$-integral.

The letter is structured as follows. In Section 2 we review the concept of group entropy, along with the Lazard formal group and some properties. With the help of the formal group law, the $G$-logarithm and the $G$-exponential in Section 3 we present the $G$-algebra; where the operations of the real numbers (sum, substraction, product and division) are generalized. Next, we present the $G$-differential calculus that can be derived from the $G$-algebra, where the $G$-differential, the $G$-derivative and the $G$-integral are developed. We show that the usual algebra, the $q$-algebra and the $\kappa$-algebra are obtained when the BG, the Tsallis and the Kaniadakis classes are considered, respectively. Then, we present a $G$-deformed Schrödinger equation and we study the corresponding versions to the Tsallis and the Kaniadakis classes. We solve them for the unidimensional infinite potential well and we obtain the corresponding effective masses, that are related with their algebraic structures according to [22, 23]. Finally, in Section 4 some conclusions and perspectives are outlined.

2 Review of group entropy theory

In this section we give the notions and concepts necessary for the development of the paper. We follow the exposition given in the Refs. [25, 26, 27, 28, 29, 30].

The group entropy approach is a construction inspired from generalizations of the notion of entropy that are compatible with the SK axioms, by making use of the formal group theory, thus giving a connection between entropic measures and formal group theory. Given a entropy $S_{n}$ ($n\in\mathbb{N}$) defined over the set of probability distributions $\{p=(p_{1},\ldots,p_{n}):\sum_{i=1}^{n}p_{i}=1\ ,\ p_{j}\geq 0\}$ the SK axioms establishes:

• (I)

the continuity of the functional $S_{n}[p]$,

• (II)

the maximum of $S_{n}$ is attained by the uniform distribution,

• (III)

$S_{n}$ is expandable to $S_{n+1}$ by adding events of zero probability,

• (IV)

Composability Axiom: $S_{nm}[pq]=S_{n}[p]+S_{m}[q|p]$ where $S_{m}[q|p]=\sum_{k=1}^{n}p_{k}S_{m}[q]$, $(pq)_{ik}=p_{k}q_{i|k}$ for all $i=1,\ldots,m$ and $k=1,\ldots,n$.

It can be shown that the Boltzmann-Gibbs entropy $S[p]=\sum_{i=1}^{n}p_{i}\log(p_{i}^{-1})$ is the only one that satisfies the SK axioms. In this axiomatical characterization of the Boltzmann-Gibbs entropy by means of first principles the composability axiom turns to be essential.

With the aim to generalize the composability axiom for deriving more entropic forms, the notion of group entropy was introduced. A group entropy is a function satisfying the (I)–(III) SK axioms along with following generalized composable notion. If there exist an entropy $S$ and a continuous real function $\Phi(x,y)$ such that

• (C1)

Composability: If $A$ and $B$ are two statistically independent systems provided with probability distribution $\{p_{i}\}_{i=1}^{n}$ and $\{q_{j}\}_{j=1}^{m}$ then $S(A\cup B)=\Phi(S(A),S(B))$.

• (C2)

Symmetry: $\Phi(x,y)=\Phi(y,x)$.

• (C3)

Associativity: $\Phi(x,\Phi(y,z))=\Phi(\Phi(x,y),z)$.

• (C4)

Null-composability: $\Phi(x,0)=x$.

then we say that $S$ is strongly composable. The importance of this definition is that C1–C4 generalize the (IV) SK axiom in such a way that a uniqueness theorem for an entropic form (which depends on $\Phi$) satisfying the (I)–(III) SK axioms and the C1–C4 conditions, can be guaranteed [27, 29].

A formal group law defined over the real numbers $\mathbb{R}$ is any power series $\Phi(x,y)$ with real coefficients satisfying C2–C4. The polynomial $\Phi(x,y)$ can be expressed by the Lazard formal group law

[TABLE]

where: $\oplus_{G}$ denotes the generalized sum associated with the formal group structure,

[TABLE]

is the formal group exponential and

[TABLE]

is the formal group logarithm, which satisfy $F(G(t))=t$, $G(F(s))=s$, $b_{i}\in\mathbb{Z}$, and $a_{1}=-b_{1},a_{2}=\frac{3}{2}b_{1}^{2}-b_{2},\ldots$ Using these expressions, the group logarithm $\log_{G}(x)$ and the group exponential $\exp_{G}(x)$ are defined as

[TABLE]

and

[TABLE]

Postulating the reasonable requirement for the group logarithm

[TABLE]

that constitutes the definition of the product $\otimes_{G}$ derived from the formal group structure, and using that $\log_{G}(x)$ and $\exp_{G}(x)$ are one the inverse of the other we can obtain

[TABLE]

From this equation and the definitions of the group logarithm and the group exponential it follows that

[TABLE]

[TABLE]

and

[TABLE]

One of the main advantages of the group entropy approach is that it allows to express the additivity property in a unified way. For instance, if we choose $\Phi_{\mathcal{BG}}(x,y)=x+y$ and $\Phi_{q}(x,y)=x+y+(1-q)xy$ (with $q$ a real continuous parameter) we recover the usual additivity and the non-additivity of the $q$-Tsallis entropies, which correspond to the group entropies

[TABLE]

and

[TABLE]

In such cases the group logarithm and the group exponential become into the standard ones for the Boltzmann-Gibbs class, and into the $q$-logarithm

[TABLE]

along with the $q$-exponential

[TABLE]

for the Tsallis class. Analogously, the deformed logarithm and exponential functions associated to the Kaniadakis class and the Abe class can be obtained from their group entropies

[TABLE]

and

[TABLE]

with $\sigma,a,b\in\mathbb{R}$.

3 Algebraic structure and calculus associated to

the groups entropies

In order to complete the algebraic structure corresponding to the Lazard formal group law, from Eqns. (1) and (7) (that we call from now on $G$-sum and $G$-product) we need to define their neutral elements, along with the $G$-substraction, the $G$-division and their inverse elements. Given $x,y\in\mathbb{R}$ it is not difficult to see that the “$G$-algebra” desired is given by the following generalized operations in terms of the group entropy $G(t)$:

• (a)

G-sum: $x\oplus_{G}y=G(G^{-1}(x)+G^{-1}(y))$

• (b)

G-substraction: $x\ominus_{G}y=G(G^{-1}(x)-G^{-1}(y))$

• (c)

G-product: $x\otimes_{G}y=\exp_{G}(\log_{G}(x)+\log_{G}(y))$

• (d)

G-division: $x\oslash_{G}y=\exp_{G}(\log_{G}(x)-\log_{G}(y))$

together with the neutral and inverse elements:

• (e)

neutral element of G-sum: $x\oplus_{G}0=0\oplus_{G}x=x$

• (f)

inverse of G-sum: $\ominus_{G}x=G(-G^{-1}(x))$

• (g)

neutral element of G-product: $x\otimes_{G}1=1\otimes_{G}x=x$

• (h)

inverse of G-product: $\oslash_{G}x=\exp_{G}(-\log_{G}(y))$

which naturally become into the standard operations for the BG-class $G(t)=t$. From the list (a)-(h) we have that the $G$-sum and the $G$-product are associative, i.e. $x\oplus_{G}(y\oplus_{G}z)=(x\oplus_{G}y)\oplus_{G}z$ and $x\otimes_{G}(y\otimes_{G}z)=(x\otimes_{G}y)\otimes_{G}z$ but the $G$-sum is not distributive neither in relation to the usual product ($a(x\oplus_{G}y)\neq ax\oplus_{G}ay$) nor to the $G$-product ($a\otimes_{G}(x\oplus_{G}y)\neq(a\otimes_{G}x)\oplus_{G}(a\otimes_{G}y)$) (see for instance [15]). The $G$-sum and the $G$-product allow to generalize the notion of integer and power of a number in the following way:

• (i)

$G$-integers: $n_{G}=G(nG^{-1}(1))$ for all $n\in\mathbb{Z}$.

• (j)

$G$-powers of $x$: $x^{{G}_{n}}=\exp_{G}(n\log_{G}(x))$ for all $n\in\mathbb{Z}$.

The $G$-integers (that we denote by $\mathbb{Z}_{G}$) satisfy

[TABLE]

for all $n,m\in\mathbb{Z}$. Then, the map $n\mapsto n_{G}$ is a group isomorphism between the integers with the usual sum $(\mathbb{Z},+)$ and the $G$-integers with the $G$-sum $(\mathbb{Z}_{G},\oplus_{G})$, i.e. $(\mathbb{Z},+)\simeq(\mathbb{Z}_{G},\oplus_{G})$.

Considering the Tsallis class $G_{q}(t)$, the $q$-logarithm and the $q$-exponential have motivated the $q$-deformation of a number $x$ [9, 14, 15], given by

[TABLE]

Following this recipe, we define the $G$-deformation of x as

[TABLE]

From this definition we have the property:

[TABLE]

for all $x,y\in\mathbb{R}$. The property (20) defines an isomorphism between the groups $(\mathbb{R},\oplus_{G})$ and $(\mathbb{R}_{G},+)$ where $\mathbb{R}_{G}$ stands for the $G$-real numbers, i.e. $\mathbb{R}_{G}:=\{x_{G}=G^{-1}(x):x\in\mathbb{R}\}$. Other deformation by means of the group entropy $G$ can be proposed as follows, that we called $G$-dual deformation

[TABLE]

from which we have

[TABLE]

for all $x,y\in\mathbb{R}$. This defines a group isomorphism between $(\mathbb{R},\oplus_{G})$ and $(\widetilde{\mathbb{R}_{G}},+)$ where $\widetilde{\mathbb{R}_{G}}:=\{\widetilde{x_{G}}=G(x):x\in\mathbb{R}\}$ stands for the $G$-dual real numbers. Naturally, for all $x\in\mathbb{R}$ we also have

[TABLE]

Thus, the following isomorphisms result

[TABLE]

which will play an important role for constructing a $G$-deformed calculus.

3.1 $G$-calculus and its dual structure

From the definitions of the $G$-deformation and its dual one we can derive their corresponding differentials by means of the identity $G(G^{-1}(x))=G^{-1}(G(x))=x$. We obtain

[TABLE]

and

[TABLE]

from which follows

[TABLE]

that establishes the relationship between the differential structures of $(\mathbb{R}_{G},+)$ and $(\widetilde{\mathbb{R}_{G}},+)$. Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$, from Eqns. (25)–(27) we define the $G$-derivative

[TABLE]

and the corresponding one to the $G$-dual deformation

[TABLE]

which implies

[TABLE]

Thus, the associated $G$-integral (noted by $I_{G}=\int_{(G)}dx_{G}$) is

[TABLE]

along with its dual version

[TABLE]

Naturally, we have

[TABLE]

which is an expression of the fundamental calculus theorem within the $G$-deformed structure.

3.2 Some algebraic structures as special cases

We begin illustrating the relevance of the formalism presented by studying the algebra and calculus of two particular classes: the Tsallis and the Kaniadakis group entropies (Eqns. (12) and (15)). We omit the Abe class since the inverse of the group entropy (16) has not an explicit formula.

3.2.1 $q$-Algebra and calculus

Considering the Tsallis class and replacing the group entropy (12), its inverse together with the $q$-exponential and the $q$-logarithm in the list (a)-(h) we obtain the full structure of the $q$-algebra [14, 15]:

• $(a_{q})$

$q$-sum: $x\oplus_{q}y=x+y+(1-q)xy$

• $(b_{q})$

$q$-substraction: $x\ominus_{q}y=\frac{x-y}{1+(1-q)y}$  $(y\neq\frac{-1}{1-q})$

• $(c_{q})$

$q$-product: $x\otimes_{q}y=\left[x^{1-q}+y^{1-q}-1\right]_{+}^{\frac{1}{1-q}}$

• $(d_{q})$

$q$-division: $x\oslash_{q}y=\left[x^{1-q}-y^{1-q}+1\right]_{+}^{\frac{1}{1-q}}$

• $(e_{q})$

neutral element of $q$-sum: $x\oplus_{q}0=0\oplus_{q}x=x$

• $(f_{q})$

inverse of $q$-sum: $\ominus_{q}x=\frac{-x}{1+(1-q)x}$  $(x\neq\frac{-1}{1-q})$

• $(g_{q})$

neutral element of $q$-product: $x\otimes_{q}1=1\otimes_{q}x=x$

• $(h_{q})$

inverse of $q$-product: $1\oslash_{q}x=\left[2-x^{1-q}\right]_{+}^{\frac{1}{1-q}}$

along with the $q$-integers and the $q$-powers for all $n\in\mathbb{Z}$

• $(i_{q})$

$q$-integers: $n_{q}=\frac{1}{1-q}([2-q]^{n}-1)$

• $(j_{q})$

$q$-powers of $n$: $x^{\otimes_{q}^{n}}=\left[nx^{1-q}-(n-1)\right]_{+}^{\frac{1}{1-q}}$.

Moreover, the $G$-derivative and the $G$-integral become into the $q$-derivative

[TABLE]

and the $q$-integral

[TABLE]

We also obtain the $q$-derivative and the $q$-integral associated to the dual $q$-deformation:

[TABLE]

and

[TABLE]

3.2.2 $\kappa$-algebra and calculus

For the case of the Kaniadakis class (Eq. (15)), by letting $\kappa=\sigma$, the definitions (a)-(h) provide the full structure of the $\kappa$-algebra [12, 13]:

• $(a_{\mathcal{K}})$

$\kappa$-sum: $x\oplus_{\kappa}y=x\sqrt{1+\kappa^{2}y^{2}}+y\sqrt{1+\kappa^{2}x^{2}}$

• $(b_{\mathcal{K}})$

$\kappa$-substraction: $x\ominus_{\kappa}y=x\sqrt{1+\kappa^{2}y^{2}}-y\sqrt{1+\kappa^{2}x^{2}}$

• $(c_{\mathcal{K}})$

$\kappa$-product:

$x\otimes_{\kappa}y=$ $\exp\left(\frac{1}{\kappa}(\textrm{arcsinh}((x^{\kappa}+y^{\kappa}-x^{-\kappa}-y^{-\kappa})/2)\right)$

• $(d_{\mathcal{K}})$

$\kappa$-division:

$x\oslash_{\kappa}y=$ $\exp\left(\frac{1}{\kappa}(\textrm{arcsinh}((x^{\kappa}-y^{\kappa}-x^{-\kappa}+y^{-\kappa})/2)\right)$

• $(e_{\mathcal{K}})$

neutral element of $\kappa$-sum: $x\oplus_{\kappa}0=0\oplus_{\kappa}x=x$

• $(f_{\mathcal{K}})$

inverse of $\kappa$-sum: $\ominus_{\kappa}x=\frac{-x}{1+(1-q)x}$  $(x\neq\frac{-1}{1-q})$

• $(g_{\mathcal{K}})$

neutral element of $\kappa$-product: $x\otimes_{\kappa}1=1\otimes_{\kappa}x=x$

• $(h_{\mathcal{K}})$

inverse of $q$-product: $1\oslash_{\kappa}x=x^{-1}$

where we have used that the $\kappa$-logarithm is

[TABLE]

and the $\kappa$-exponential is

[TABLE]

For all $n\in\mathbb{Z}$ the ${\mathcal{K}}$-integers and the ${\mathcal{K}}$-powers are given by

• $(i_{\mathcal{K}})$

$\kappa$-integers: $n_{\kappa}=\frac{1}{\kappa}\sinh\left(n\textrm{arcsinh}(\kappa x)\right)$

• $(j_{\mathcal{K}})$

$\kappa$-powers of $n$:

$x^{\otimes_{\kappa}^{n}}=\left[n\sinh(\kappa\log x)+\sqrt{n^{2}\sinh^{2}(\kappa\log x)+1}\right]_{+}^{\frac{1}{\kappa}}$.

The ${\mathcal{K}}$-derivative and the ${\mathcal{K}}$-integral result

[TABLE]

and the $\kappa$-integral

[TABLE]

We also have the ${\mathcal{K}}$-derivative and the ${\mathcal{K}}$-integral associated to the dual Kaniadakis deformation, expressed by

[TABLE]

and

[TABLE]

3.3 $G$-Deformed version of the

Schrödinger equation

With the aim to formulate a nonrelativistic Schrödinger in the context of group entropy theory, we propose a generalized translation operator, given by

[TABLE]

where the nonlinearity of the infinitesimal displacement $\varepsilon$ depends on the group entropy $G$. Analogously to the standard case $G(t)=t$, it can be shown that the operator $\hat{\mathcal{T}}_{G}(\varepsilon)$ has the following associated position-dependent momentum

[TABLE]

being $\hat{p}=-i\hbar\partial/\partial x$ the standard momentum operator. With the aim of guaranteeing classical analogues, real eigenvalues and a orthonormal basis, we redefine $\hat{p}_{G}$ as

[TABLE]

that is Hermitian. From the well-known commutator $[\hat{x},\hat{p}]=i\hbar\hat{1}$ (being $\hat{1}$ the identity operator) and using that for all analytical function $f(x)$ is $[f(\hat{x}),\hat{p}]=i\hbar f^{\prime}(\hat{x})$, we have that

[TABLE]

satisfy

[TABLE]

The relationship (50) shows that $\{\hat{x},\hat{p}\}\mapsto\{\hat{x}_{G},\hat{p}_{G}\}$ is a canonical transformation and constitutes a “natural” generalization of the conjugated operators $\hat{x},\hat{p}$ in the context of group entropy approach. Now we explore some consequences about this by considering the $G$-deformed Hamiltonian $\hat{H}_{G}$

[TABLE]

of a single particle of mass $m_{0}$ having a $G$-deformed $\hat{p}_{G}$ and subjected to a potential $V(\hat{x})$, being $A(z)=G^{\prime}(G^{-1}(z))$. We also have the relationship

[TABLE]

From this we have

[TABLE]

where $\{\hat{C},\hat{D}\}=\hat{C}\hat{D}+\hat{D}\hat{C}$ denotes the anti-commutator between $\hat{C}$ and $\hat{D}$. Then, for an arbitrary state $|\Psi\rangle$ we obtain

[TABLE]

Using that $\langle x^{\prime}|\hat{p}^{n}|\Psi\rangle=(-i\hbar)^{n}\frac{\partial^{n}}{\partial x^{n}}\langle x^{\prime}|\Psi\rangle$ for all $n\in\mathbb{N}$ and $\langle x|\Psi\rangle=\Psi(x)$ we can recast the Eq. (3.3) to obtain

[TABLE]

From (51), (55) and $i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\hat{H}_{G}|\Psi(t)\rangle$ we arrive to the time-independent $G$-deformed Schrödinger equation for the wave-function $\Psi(x,t)$:

[TABLE]

with $A(x)=G^{\prime}(G^{-1}(x))$ depending on the group entropy $G$. Several remarks deserve to be mentioned about the equation (3.3). First, the space deformation $x_{G}=G^{-1}(x)=\int^{x}A(s)^{-1}ds$ implies that the particle has a position-dependent mass (that we call $G$-deformed mass $m(x)_{G}$)

[TABLE]

whose functional form depends exclusively of the group entropy. Second, the group entropy introduces a new term of field $-\frac{1}{8m_{0}}\hbar^{2}[A^{\prime}(x)^{2}+2A(x)A^{\prime\prime}(x)]\Psi(x,t)$ depending on the derivatives of the deformation, which physically can be interpreted as a field whose force changes point by point due to the deformation. And third, it is straightforwardly to show that for the Tsallis class we have $A(x)=G^{\prime}(G^{-1}(x))=(1+\gamma_{q}x)$ (with $\gamma_{q}=1-q$) and then the $G$-deformed Schrödinger equation (3.3) results

[TABLE]

which is the $q$-deformed Schrödinger equation corresponding to a position-dependent mass $m(x)=m_{0}/(1+\gamma_{q}x)^{2}$, presented previously in [20, 22, 23, 24].

Next step is to show that the $G$-deformed Schrödinger equation admits a rewrite in terms of the $G$-derivative and having a constant mass. For accomplish this, we note that the normalization condition for $\Psi(x,t)$ expressed with the $G$-integral, i.e.

[TABLE]

allows to identify the $G$-deformed wave-function

[TABLE]

The $G$-deformed derivative $D_{G}=A(x)\frac{\partial}{\partial_{x}}$ applied two times to $\Phi_{G}(x,t)$ gives

[TABLE]

with

[TABLE]

and then

[TABLE]

Thus, using Eqns. (62) and (3.3) we can recast (3.3) as

[TABLE]

Therefore, multiplying the $G$-deformed Schrödinger equation (3.3) by $\sqrt{A(x)}$, using (3.3) and $x=x(x_{G})$ we obtain

[TABLE]

which is entirely written in the $G$-deformed position coordinate $x_{G}=G^{-1}(x)$ and provided with $G$-deformed derivatives and a constant mass $m_{0}$. It is clear that when there is no deformation ($G^{-1}(x)=x$), from Eq. (65) we recover the standard Schrödinger equation, and for $G^{-1}(x)=x_{q}$ we obtain the $q$-deformed Schrödinger equation (3.3) written in the $q$-deformed space. In order to solve the Eq. (65) we use the following properties of the $G$-exponential:

[TABLE]

where we have also

[TABLE]

for all differential function $f(x)$. From the group exponential $\exp_{G}(x)=\exp(x_{G})$ we define the group cosine and group sine as

[TABLE]

and

[TABLE]

Using Eqns. (68), (69) we obtain a $G$-deformed version of the De Moivre formula

[TABLE]

Thus, the solution of the differential equation

[TABLE]

is $\phi(x_{G})=A_{1}\cos(\alpha x_{G})+A_{2}\sin(\alpha x_{G})$ where $A_{1},A_{2}$ are fixed numbers determined by the initial conditions $\phi(x_{G})|_{x_{G}=x_{G}^{(0)}},D_{G}\phi(x_{G})|_{x_{G}=x_{G}^{(0)}}$.

3.4 Infinite potential well in the $G$-deformed space

For illustrating the physical consequences of the $G$-deformed Schrödinger equation we consider a particle provided with a position-dependent mass $m(x)_{G}=m_{0}/[G^{\prime}(G^{-1}(x))]$ in a one-dimensional infinite potential well $V(x)$ of width $L$, given by

[TABLE]

Now if we assume that the deformation $x_{G}=G^{-1}(x)$ is an increasing function of $x$ (this is the case of the Tsallis and the Kaniadakis classes) then the potential well has the following expression in the $G$-deformed space

[TABLE]

since $G(0)=0$ for all group entropy $G$ and $L_{G}=G^{-1}(L)$. Physically, this means that the problem of a particle of a non-constant mass $m_{G}(x)=m_{0}/[G^{\prime}(G^{-1}(x))]^{2}$ in an infinite potential well of length $L$ is equivalent to the problem of a particle of constant mass $m_{0}$ in the $G$-deformed space subjected to the same potential but of width $L_{G}$. For obtaining the autofunctions $\Phi_{G}(x_{G})$ of this problem we need to solve the $G$-deformed eigenvalue equation

[TABLE]

with the boundary conditions $\Phi_{G}(x_{G}=0)=\Phi_{G}(x_{G}=L_{G})=0$. Basing us on the standard case we have

[TABLE]

where the $G$-deformed eigenenergies $E_{G}^{(n)}$ are

[TABLE]

being $(A_{G}^{(n)})^{2}=2/L_{G}$ the $G$-deformed normalization constant. Now that we have solved the deformed Schrödinger equation (3.3) in the $G$-deformed space, by means of the transformation $x_{G}\mapsto x$ we can obtain the corresponding autofunctions $\Psi_{n}(x)=\Phi_{G}^{(n)}(x)/\sqrt{G^{\prime}(G^{-1}(x))}$ (see Eq. (3.3)) in the standard position space $x$:

[TABLE]

We can see how the formal group law $\oplus_{G}$ underlies the zeros spacing of $\Psi_{n}$. From (77) for all $n\geq 1$ the $n+1$ zeros of $\Psi_{n}$ are

[TABLE]

which obey precisely the formal group law ($G$-sum):

[TABLE]

It is instructive to illustrate the non-uniform zeros spacing for the Tsallis and the Kaniadakis classes. If we call

[TABLE]

the $m$-th spacing111Which naturally reduces to the uniform distribution of zeros $\Delta^{(m)}=L/m$ for the standard case $G(t)=t$. (with $1\leq m\leq n$) then we have

[TABLE]

for the Tsallis case, and

[TABLE]

for the Kaniadakis case.

In Fig. 1 we illustrate the probability distributions of some autofunctions (77) for the Boltzmann (standard case), Tsallis and Kaniadakis classes. For comparing we set the $\kappa$ and $q$ parameters such that $\kappa=1-q$. As the deformation increases a rapidly breaking of the symmetry of the wavefunctions takes place and the probability density tends to accumulate in the zones where the mass takes big values, i.e. around $x=0$. Thus, when the deformation is appreciable the particle has a big mass around $x=0$ and then the probability to find it near to $x=L$ is vanishingly small. Moreover, we observe that the deformation of the probability distributions is slightly more pronounced for the Kaniadakis case than the Tsallis one.

In the top panel of Fig. 2 we illustrate the classical limit ($n=50$) of the wavefunction probability distributions for the standard case and for the Tsallis and Kappa classes. In both classes (Tsallis and Kappa ones) the classical probability distribution tends to a delta probability distribution centered at $x=0$ due to the particular form of the group entropies $G_{q}$ and $G_{\kappa}$. Regarding the parameter $1-q$, the asymmetry of the deformed probability distribution of the Tsallis class is depicted in the bottom panel of Fig. 2. We observe that as the parameter $q$ tends to 2 the deformed distribution tends to accumulate probability around $x=1$ due to the argument $(1+(1-q)x)$ makes to diverge the distribution for $1-q=-1$. By last, we can see that the similarity between the functional form of the Tsallis mass

[TABLE]

and the Kaniadakis mass

[TABLE]

implies that they are essentially the same for $\kappa=1-q\gg 1$ because $m_{q}(x)/m_{\kappa}(x)\sim(\kappa x)^{2}/(1-q)x^{2}=1$. This is reflected in the fact of not observing a behavior substantially different between their corresponding autofunctions, for the example and the range of parameters studied.

4 Conclusions

We have presented the algebraic structure (that we called $G$-algebra) along with its associated differential calculus, which is derived from the formal Lazard group law and the exponential and logarithm groups. From the $G$-algebra and its $G$-calculus we have explored some of its consequences, that are summarized in Table 1.

We point out the following consequences of our contribution.

First, we have obtained the $q$-algebra and the $\kappa$-algebra as special cases for the Tsallis and Kaniadakis group entropies respectively. Also, their associated calculus, constituted by the deformed derivative and integral, have been consistently derived. Thus, for each choice of the group entropy one can obtain its corresponding algebra and calculus by means of the $G$-algebra and $G$-calculus formulas.

Second, we have given some clues in the connection of the non-relativistic quantum mechanics and the entropy group approach, by proposing a $G$-deformed Schrödinger equation that can derived from the $G$-calculus in a deformed position space. We have shown that the $G$-deformed position and momentum operators constitute a canonical transformation of the standard ones and therefore, the motion equations have as those of standard (non-deformed) quantum mechanics. As a consequence, we have obtained the Schrödinger equation derived from the $G$-deformed position and momentum operators (that we called $G$-deformed Schrödinger equation), which is physically identical to the corresponding of a particle having a position-dependent mass, where the functional form of the mass is univocally determined by the derivatives of the group entropy. Furthermore, from the rewriting of the $G$-deformed Schrödinger in terms of the $G$-derivative and position and provided with a constant mass, we have shown the equivalence between the problem of a particle having an arbitrary position-dependent mass and the corresponding to a particle of constant mass within a $G$-deformed space. This connection constitutes an extension of the investigated in previous works, but for any arbitrary form of the functional mass.

Third, by studying the Tsallis and Kaniadakis classes applied to the unidimensional motion of a particle in a infinite potential well, we have characterized the effect of the deformation of both group entropies. As a consequence of the deformation, the probability distributions of the Tsallis and Kappa classes present an asymmetry which is more pronounced as the parameter $\kappa=1-q$ increases (see Fig. 1). We have found that the deformation does not affect the classical limit but rather the way in which the probability density is distributed along the position space (top panel of Fig. 2). Due to the particular forms of the Tsallis and Kaniadakis mass functionals, the probability density concentrates mostly in the region where the corresponding mass functionals have big values (near to the $x=0$), as physically expected.

Finally, we consider that the results of the present work can be useful for extending the group entropy approach in other directions as diffusion phenomena, field theory, etc. and also for complementing others recently obtained [30].

Acknowledgments

ISG and EPB acknowledge support received from the National Institute of Science and Technology for Complex Systems (INCT-SC), and from the National Council for Scientific and Technological Development (CNPq) (at Universidade Federal da Bahia), Brazil.