#### 3.1. Free Particle and the q-Plane Wave Solution

Recently, a nonlinear Schroedinger equation (NLSE) for a free particle of mass

m was proposed [

67],

where

q is a real number, as defined in

Section 1, in such a way to recover the linear one of Equation (

8), for

$V\left(\overrightarrow{x}\right)=0$, in the particular case

$q=1$. One sees that the right-hand side of this equation is nonlinear for any

$q\ne 1$, so that, in contrast to Equation (

8), the amplitude

${\mathsf{\Psi}}_{0}$ does not cancel and should be considered in these NL cases, guaranteeing the correct physical dimensionalities for all terms.

One should notice that the NLSE of Equation (

23) presents the same structure of the NL Fokker–Planck equation proposed in [

72,

73] in the absence of an external potential, which appears in nonextensive statistical mechanics [

6,

9,

10]. Essentially, it represents the analogue of the porous-medium equation [

81], very common in the framework of anomalous-diffusion phenomena, but with an imaginary time.

Consistently, the energy and momentum operators are generalized as:

which, when acting on the

q-plane wave of Equation (

6), yield the relations of Equation (

11), namely those for the energy,

$E=\hslash \omega $, and momentum,

$\overrightarrow{p}=\hslash \overrightarrow{k}$. Now, considering

$\overrightarrow{k}\to \overrightarrow{p}/\hslash $ and

$\omega \to E/\hslash $, one verifies that this new form is a solution of the equation above, with

$E={p}^{2}/2m$, for all values of

q.

Due to the peculiar properties of this type of solution, the

q-generalized quantum mechanics is expected to be useful for addressing complex phenomena, such as dark matter, nonlinear quantum optics and others. In particular, some of the properties of the

q-plane wave make it potentially relevant from the physical point of view, like: (i) it presents an oscillatory behavior; (ii) it is localized for certain values of

q. Indeed, for

$q\ne 1$, the

q-exponential

${exp}_{q}\left(iu\right)$ is characterized by an amplitude

${r}_{q}\left(u\right)\ne 1$ [

82],

so that

${r}_{q}\left(u\right)$ decreases for increasing arguments, if

$q>1$. From Equations (

25)–(

28), one notices that

${cos}_{q}\left(u\right)$ and

${sin}_{q}\left(u\right)$ cannot be zero simultaneously (even though their moduli do tend to zero simultaneously as

$u\to \pm \infty $) yielding

${exp}_{q}(\pm iu)\ne 0$. In

Figure 1, we represent

${cos}_{q}\left(u\right)$ and

${sin}_{q}\left(u\right)$, respectively, for two different values of

$q>1$, showing that the amplitude

${r}_{q}\left(u\right)$ produces a modulation of these functions, which get strongly weakened as

$\left|u\right|\to \infty $. Such a property makes these types of solutions appropriate for many types of physical phenomena, which occur in limited intervals of space and time. Let us stress that, in the particular situation where

$\overrightarrow{k}\xb7\overrightarrow{x}=\omega t$, one has

$\mathsf{\Psi}(\overrightarrow{x},t)={\mathsf{\Psi}}_{0}\phantom{\rule{4pt}{0ex}}(\forall t)$, and consequently, the

q-plane wave behaves like a soliton. Indeed, in the one-dimensional case, one has a soliton propagating with a velocity

$c=\omega /k$. This enables the approach of nonlinear excitations that do not deform in time and should be relevant, e.g., in nonlinear optics and plasma physics.

Additionally,

${exp}_{q}\left(iu\right)$ presents further peculiar properties,

for any

$\alpha \phantom{\rule{4pt}{0ex}}\mathrm{real}$. By integrating Equation (

30) from

$-\infty $ to

$+\infty $, one obtains [

84],

leading to the physically-important property of square integrability for

$1<q<3$; as some typical examples, one has

${\mathcal{I}}_{3/2}={\mathcal{I}}_{2}=\pi $. One should notice that this integral diverges in both limits

$q\to 1$ and

$q\to 3$. Hence, the

q-plane wave of Equation (

6) presents a modulation, characteristic of a localized wave, for

$1<q<3$.

Applying the complex conjugate in Equation (

23) and using the property of Equation (

32), one obtains the equation for

${\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$,

Next, we will discuss the subject of a probability density and its corresponding continuity equation, related to the present NLSE. As will be shown, this analysis turns out to be rather nontrivial, as examined in detail in [

85,

86].

#### 3.2. Continuity Equation and Classical Field Theory

Let us now address the matter of the continuity equation; we start by following the standard procedure, i.e., considering the probability density of Equation (

12), together with the pair of Equations (

23) and (

34). One has:

so that, using this pair of equations on the right-hand side, one readily sees that the continuity equation is not fulfilled. Instead, one gets the following balance equation:

where:

These results show that the continuity equation of Equations (

15) and (

16) is recovered for

$q=1$; in general, for

$q\ne 1$, one has

$R(\overrightarrow{x},t)\ne 0$. Indeed, considering the

q-plane wave solution of Equation (

6), one obtains:

showing that for

$q\ne 1$, only in the particular case of the soliton, where

$\overrightarrow{k}\xb7\overrightarrow{x}=\omega t$ (leading to

$\mathsf{\Psi}(\overrightarrow{x},t)={\mathsf{\Psi}}_{0}\phantom{\rule{4pt}{0ex}}(\forall t)$), one has

$R=0$, so that probability is preserved in time.

Moreover, one may see that by considering a Lagrangian density in the form of Equation (

18), characterized by

$\mathsf{\Psi}(\overrightarrow{x},t)$ and

${\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$, and following a procedure similar to the one carried put in the linear case [

78,

79,

80]), one does not obtain Equations (

23) and (

34). A way to overcome this difficulty was proposed in [

85], where an additional field

$\mathsf{\Phi}(\overrightarrow{x},t)$ was introduced. Hence, we will now develop an exact classical field theory, through the definition of a Lagrangian density

$\mathcal{L}$, which will depend on the fields

$\mathsf{\Psi}(\overrightarrow{x},t)$ and

$\mathsf{\Phi}(\overrightarrow{x},t)$, their complex conjugates, as well as on their spatial and time derivatives,

Let us then consider the following Lagrangian density,

where

$A\equiv 1/\left(2q\mathsf{\Omega}\right)$ is a multiplicative constant and, as before,

${\mathsf{\Psi}}_{0}$ and

${\mathsf{\Phi}}_{0}$ represent the amplitudes of the fields

$\mathsf{\Psi}(\overrightarrow{x},t)$ and

$\mathsf{\Phi}(\overrightarrow{x},t)$, respectively. One should notice that the Lagrangian density above recovers the one of Equation (

19) in the particular case

$q=1$.

From the above Lagrangian density, one may construct a classical action, which may be extremized to yield the Euler–Lagrange equations for each field [

78,

79,

80]. The Euler–Lagrange equation for the field Φ,

leads to:

which corresponds to the NLSE of Equation (

23). Carrying out the same procedure for the field

$\mathsf{\Psi}(\overrightarrow{x},t)$, one obtains,

with similar equations holding for the complex-conjugate fields,

${\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$ and

${\mathsf{\Phi}}^{*}(\overrightarrow{x},t)$.

It is important to notice that Equation (

44) becomes the complex conjugate of Equation (

23) (i.e., Equation (

34)) only for

$q=1$, in which case

$\mathsf{\Phi}(\overrightarrow{x},t)={\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$. For all

$q\ne 1$, one has that

$\mathsf{\Phi}(\overrightarrow{x},t)$ is distinct from

${\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$, with the fields

$\mathsf{\Phi}(\overrightarrow{x},t)$ and

$\mathsf{\Psi}(\overrightarrow{x},t)$ being related by Equation (

44).

Now, if one substitutes the

q-plane wave solution of Equation (

6) in Equation (

44), one finds,

Consistently with the above, we generalize Equation (

12) by defining the probability density for finding a particle at time

t, in a given position

$\overrightarrow{x}$, as:

for any value of

q. Hence, considering the

q-plane wave solution for a free particle, one has

$\rho (\overrightarrow{x},t)=1/\mathsf{\Omega}$, leading trivially to

$[\partial \rho (\overrightarrow{x},t)/\partial t]=0$, but yielding the same non-integrability difficulty of Equation (

13), typical of the standard plane-wave solution in full space.

In fact, the continuity equation is fulfilled for this particular solution, although in general, one has:

and using the equations for the fields

$\mathsf{\Psi}(\overrightarrow{x},t)$ and

$\mathsf{\Phi}(\overrightarrow{x},t)$ (Equations (

23) and (

44)), as well as their corresponding complex conjugates, one obtains a balance equation in the form of Equation (

36), where

and

One should notice that Equations (

48) and (

49) coincide with Equations (

37) and (

38), respectively, through the identifications

$\mathsf{\Phi}(\overrightarrow{x},t)\leftrightarrow {\mathsf{\Psi}}^{*}(\overrightarrow{x},t)$ and

${\mathsf{\Phi}}^{*}(\overrightarrow{x},t)\leftrightarrow \mathsf{\Psi}(\overrightarrow{x},t)$. Therefore, within this later frame, solutions must satisfy:

for the preservation of probability. Since herein we are dealing with nonlinear equations, which usually present more than one solution, some possible solutions may not satisfy the above requirement for

$q\ne 1$. Now, in contrast to Equation (

38), considering the pair of solutions in Equations (

6) and (

45), one shows that:

so that the continuity equation is fulfilled for the

q-plane wave and its auxiliary solution of Equation (

45). Next, we present other solutions for the pair of Equations (

23) and (

44); as will be shown, in many cases, the condition of Equation (

50) is not fulfilled trivially.

#### 3.4. Nonlinear Schroedinger Equations for a Particle in a Potential

The authors of [

96] investigated how the

q-plane wave solution of Equation (

6) gets transformed under two basic types of changes in the reference frame, namely a Galilean transformation connecting two inertial reference frames, as well as the case of a uniformly-accelerated reference frame. In these transformations, we will restrict ourselves to a one-dimensional space, for simplicity, although the extension to

d dimensions is straightforward. Hence, let us consider a Galilean transformation relating the original inertial frame

$({x}^{\prime},{t}^{\prime})$ with a second inertial frame

$(x,t)$ that moves with respect to the former one with a uniform velocity

v, so that:

In this case, it was shown that the

q-plane wave solution, in the second inertial frame

$(x,t)$, keeps the form of Equation (

6) by redefining the wave vector and frequency [

96],

Furthermore, for a free particle viewed from a uniformly-accelerated reference frame (with acceleration

a), the space-time coordinates are:

where, as before,

$({x}^{\prime},{t}^{\prime})$ represent the variables associated with the inertial frame, and

$F=ma$ stands for the force acting on the particle. Such an analysis suggested that in the corresponding NLSE, the potential

$V\left(x\right)$ should couple to

${\left[\mathsf{\Psi}(x,t)\right]}^{q}$, instead of coupling to

$\mathsf{\Psi}(x,t)$, as happens in the standard LSE.

Taking into account the results of [

96], the Lagrangian density for the free particle was modified by introducing a

d-dimensional potential

$V\left(\overrightarrow{x}\right)$ [

78],

where

${\mathcal{L}}_{\mathrm{free}}$ stands for the free-particle Lagrangian density of Equation (

41), with the same constant factor

A. Following the procedure described before [

78,

79,

80], the Euler–Lagrange equation for the field

$\mathsf{\Phi}(\overrightarrow{x},t)$ yields the NLSE:

whereas the Euler–Lagrange equation for the field

$\mathsf{\Psi}(\overrightarrow{x},t)$ leads to the auxiliary equation,

An interesting feature concerning Equations (

78) and (

79) is that in the first of them, the potential

$V\left(\overrightarrow{x}\right)$ appears multiplying

${\left[\mathsf{\Psi}(\overrightarrow{x},t)\right]}^{q}$, whereas in the second, it couples to

${\left[\mathsf{\Psi}(\overrightarrow{x},t)\right]}^{(q-1)}\mathsf{\Phi}(\overrightarrow{x},t)$. Considering these terms in the simple case of a constant potential,

$V\left(\overrightarrow{x}\right)={V}_{0}$, the

q-plane wave of Equation (

6) together with the solution for the second field

$\mathsf{\Phi}(\overrightarrow{x},t)$ of Equation (

45) satisfy both field equations, leading to

$\hslash \omega ={\hslash}^{2}{k}^{2}/2m+{V}_{0}$, where

${k}^{2}={\sum}_{n=1}^{d}{k}_{n}^{2}$ in

d dimensions.

To our knowledge, the first solutions of Equation (

78) were presented in [

91,

97]: (i) the Gaussian wave-packet solution of Equation (

62) was studied in [

91] by considering a one-dimensional harmonic potential; (ii) solutions for several potentials were analyzed in [

97], like the

d-dimensional quadratic, the shifted-attractive delta and the two-dimensional Moshinsky ones. This latter potential is defined as:

and working with the center-of-mass coordinates,

the authors of [

97] have shown that Equation (

78) admits the quasi-stationary solution:

where the parameters

${\lambda}_{1}$ and

${\lambda}_{2}$ are related to

ω,

κ and

B by means of two coupled nonlinear equations. The wave function of Equation (

82) represents the first case reported in the literature of a solution of Equation (

78) for a system of interacting particles. In the limit

$q\to 1$, the solution above reduces to the ground-state function associated with the Moshinsky model’s potential.

An extension of the work of [

97] was carried for the case where the particles are subjected to a Moshinsky-like potential with time-dependent coefficients [

93]. In this latter work, the authors have shown that the nonlinearity creates entanglement between the particles, which is not present in the usual (

$q=1$) scenario and, so, being potentially relevant for describing physical reality [

95].

A common feature in the studies of [

91,

93,

97] is the fact that by studying only Equation (

78), the norm is not preserved, in the sense that the continuity given by Equation (

15) is not fulfilled (for a discussion of the non-preservation of the norm, see, e.g., [

91]). A joint study of Equations (

78) and (

79) was carried in [

87], for a particle in an infinite one-dimensional rectangular potential well,

Hence, considering the same separation of variables of

Subsection 3.3.1, i.e.,

$\mathsf{\Psi}(x,t)={\psi}_{1}\left(x\right){\psi}_{2}\left(t\right)$ and

$\mathsf{\Phi}(x,t)={\varphi}_{1}\left(x\right){\varphi}_{2}\left(t\right)$, stationary-state solutions were found,

and by choosing

$\mu =(2-q)\u03f5$,

The stationary-state solutions for this potential were expressed in terms of

$y={\mathrm{Sin}}_{q}\left(x\right)$, defined by means of [

87]:

whose period is

$4{\tau}_{q}$, where:

The equations above express a generalization of the standard trigonometric function, recovered in the limit

$q=1$, i.e.,

${\mathrm{Sin}}_{1}\left(x\right)\equiv sin\left(x\right)$. One should notice that this generalization differs from the

${sin}_{q}\left(x\right)$ function that is currently used in nonextensive statistics [

82]; particularly, one important distinction concerns the fact that

$|{\mathrm{Sin}}_{q}\left(x\right)|\le 1$ for

$1\le q<2$ [

87]. In this way, the wave vectors in Equation (

84) are given by:

and considering the boundary condition

${\psi}_{1}\left(0\right)={\psi}_{1}\left(a\right)=0$, one finds

$\delta =0$ and

${k}_{q}a=2{\tau}_{q}n$, where

$n=1,2,3,..$, so that:

Then, the following expression:

generalizes the energy spectrum of the standard quantum well,

${\u03f5}_{n}\left(1\right)=({\hslash}^{2}/2m){(n\pi /a)}^{2}$.

Finally, we can write the probability density of Equation (

46) as:

where

$\mathrm{Re}\left\{s\right\}$ stands for the real part of

s, and we have used the normalization condition for finding the amplitude

${\tilde{A}}_{n,q}$. It is important to stress that

$\mathrm{Re}\{{\left[{\mathrm{Sin}}_{q}(2n{\tau}_{q}x/a)\right]}^{(3-q)/(2-q)}\}>0$ for

$1<q<4/3$ and that this quantity may be also positive for other values of

q outside this interval, e.g., whenever the parameter

q satisfies the inequalities

$(3/2)+2k<(3-q)/(2-q)<(5/2)+2k$, with

k integer and

$k\ge 1$. However, there are values of

q in the range

$4/3<q<2$ for which one obtains

$\rho \left(x\right)<0$, representing situations still not well understood. Such cases may be compared with what happens to the Wigner function, which may present negative values for some values of its arguments, and so, it cannot be considered as a simple probability distribution, being often called a quasidistribution (see, e.g., [

98]).

In

Figure 2, we present the dimensionless probability density

$a\rho \left(x\right)$, for a particle in an infinite potential well (cf. Equation (

83)) in the cases

$n=1$ (a) and

$n=2$ (b) and typical values of

q, namely

$q=1,1.25$ and

$1.8$. For

$n=1$, one has an argument

$0\le (2{\tau}_{q}x/a)\le 2{\tau}_{q}$, so that

${\mathrm{Sin}}_{q}(2{\tau}_{q}x/a)\ge 0$ [

87]. From

Figure 2a, one notices that

q plays an important role for a particle with an energy

${\u03f5}_{1}\left(q\right)$, as concerns its confinement around the central region of the well: by increasing

q in the range

$1<q<2$, the particle becomes more confined around

$(x/a)=1/2$. In this context, the present solution with an index

$q>1$ may be relevant for systems where one finds a low-energy particle localized in the central region of a confining potential. In

Figure 2b, we show

$a\rho \left(x\right)$ in the case

$n=2$ and the same values of

q considered in

Figure 2a. Now, one has an argument

$0\le (4{\tau}_{q}x/a)\le 4{\tau}_{q}$, so that

${\mathrm{Sin}}_{q}(4{\tau}_{q}x/a)$ may yield negative values for

$(x/a)>1/2$ [

87]. As mentioned above, in these cases, one has always real positive probabilities for

$1<q<4/3$, as well as other values of

q outside this interval (e.g.,

$q=1.8$). In these cases, the corresponding probability densities present a symmetry with respect to

$(x/a)=1/2$, with maxima at

$(x/a)=1/4$ and

$(x/a)=3/4$. Once again, the present solution with an index

$q>1$ may be relevant for systems where one finds a low-energy particle with the same probability for being found in two different regions, symmetrically localized around the central region of the well.