Resumo

1. Introduction

Systems endowed with position-dependent mass have attracted scientists’ attention in recent decades due to a wide range of applications in fundamental areas of physics. These systems can be studied from both classical and quantum perspectives. Systems with position-dependent mass are present in the study of electronic and optical properties of semiconductors [¹[1] O. von Roos, Phys. Rev. B 27, 7547 (1983)., ²[2] R.A. El-Nabulsi, J. of Physics and Chemistry of Solids 140, 109384 (2020)., ³[3] G. Bastard, Wave mechanics applied to semiconductor heterostructures (Les Editions de Physique, Les Ulis, 1988).], of graded mixed semiconductors [⁴[4] T. Gora and F. Williams, Phys. Rev. 177, 1179 (1969).], and with abrupt heterostructures [⁵[5] G.T. Einevoll and P.C. Hemmer, J. of Phys. C: Solid State Phys. 21, L1193 (1988)., ⁶[6] R.A. Morrow and K.R. Brownstein, Phys. Rev. B 30, 678 (1984).], in quantum dots and wells [²[2] R.A. El-Nabulsi, J. of Physics and Chemistry of Solids 140, 109384 (2020)., ⁷[7] P. Harrison and A. Valavanis, Quantum wells, wires and dots: theoretical and computational physics of semiconductor nanostructures (John Wiley & Sons, Hoboken, 2016)., ⁸[8] R. Khordad, Physica B: Condensed Matter 406, 3911 (2011).], quantum liquids [⁹[9] F.A. Saavedra, J. Boronat, A. Polls and A. Fabrocini, Phys. Rev. B 50, 4248 (1994).] and within lattices and superlattices [¹⁰[10] G. Bastard, Phys. Rev. B 24, 5693 (1981)., ¹¹[11] R.A. El-Nabulsi, Physica E: Low-dimensional Systems and Nanostructures 127, 114525 (2021).]. They are also present in the research of the inversion potential for the $N{H}_3$ molecule through density functional theory [¹²[12] N. Aquino, G. Campoy and H. Yee-Madeira, Chemical Phys. Lett. 296, 111 (1998).], in many-body systems [¹³[13] K. Bencheikh, K. Berkane and S. Bouizane, J. of Phys. A: Mathematical and General 37, 10719 (2004).], energy spectrum and transition rate for atomic nuclei [¹⁴[14] M. Alimohammadi, H. Hassanabadi and S. Zare, Nuclear Physics A 960, 78 (2017).], magnetic monopoles [¹⁵[15] A.L. Jesus and A.G.M. Schmidt, J. of Mathematical Physics 60, 122102 (2019).], classical and quantum nonlinear oscillators [¹⁶[16] P.M. Mathews and M. Lakshmanan, Quarterly of Applied Mathematics 32, 215 (1974)., ¹⁷[17] J.F. Cariñena, A.M. Perelomov, M.F. Ranada and M. Santander, J. of Phys. A: Mathematical and Theoretical 41, 085301 (2008)., ¹⁸[18] Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni, International J. of Theoretical Physics 50, 2268 (2011)., ¹⁹[19] O. Mustafa, The Euro. Phys. Journal Plus 136, 249 (2021).], and curved spaces [²⁰[20] C. Quesne and V.M. Tkachuk, J. of Phys. A: Mathematical and General 37, 4267 (2004).]. Classically, these systems can be described in the analysis of the hydrodynamic impact of a rigid body against a liquid-free surface [²¹[21] C.P. Pesce, J. of Applied Mechanics 70, 751 (2003).] and in the dynamics of a falling chain wrapped around a table [²²[22] A. Cayley, Proceedings of the Royal Society of London 8, 506 (1856).], among others.

Traditionally, the solution methods for a system with position-dependent mass (PDM) are not those usually used for solving a constant mass system. In the case of classical systems with PDM, Newton’s Second Law must be extended to allow the analysis of the system’s dynamics [²³[23] W.C. Guttner and C.P. Pesce, J. of the Brazilian Society of Mechanical Sciences and Engineering 39, 1969 (2017).]. However, the equations of motion obtained from the Second Law are nonlinear, making their analytical solution difficult.

Therefore, the unusual methods of Hamiltonian factorization and canonical transformations can facilitate finding the system’s solution. Furthermore, the application of canonical transformations aims to relate the system endowed with PDM to a system with constant mass. Besides, Hamiltonian factorization allows the separation of a second-order differential equation into two first-order differential equations [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).].

Similarly, a quantum PDM system can be analyzed through uncommon methods since the usual kinetic operator $\widehat{T}$ for this system is non-Hermitian. Then, since kinetic energy is required to be a physical observable, the operator representing it must be Hermitian [¹[1] O. von Roos, Phys. Rev. B 27, 7547 (1983).]. Thus, to study the dynamics of these quantum systems, the generalized kinetic operator, proposed in Ref. [¹[1] O. von Roos, Phys. Rev. B 27, 7547 (1983).], can be used in conjunction with the Hamiltonian factorization method and canonical transformations. With these tools, the dynamics of position-dependent mass systems become workable in the quantum realm.

The harmonic oscillator model is of fundamental importance in physics because many systems with complicated analysis can be reduced or approximated by this model, such as the vibration of atoms in molecules [²⁵[25] S.H. Dong, Factorization Method in Quantum Mechanics (Springer, Berlin, 2007).]. An example of such model is the nonlinear Mathews-Lakshmanan oscillator [¹⁶[16] P.M. Mathews and M. Lakshmanan, Quarterly of Applied Mathematics 32, 215 (1974).], which can be treated as a PDM system with $m(x)=1/[1+\kappa x^2]$. Mathews-Lakshmanan-type oscillators were previously studied as PDM systems by different methods inRefs. [¹⁹[19] O. Mustafa, The Euro. Phys. Journal Plus 136, 249 (2021)., ²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007)., ²⁶[26] J.F. Cariñena, M.F. Rañada and M. Santander, Reports on Mathematical Physics 54, 285 (2004)., ²⁷[27] J.F. Cariñena, M.F. Rañada and M. Santander, Ann. of Phys. 322, 434 (2007)., ²⁸[28] S. Cruz y Cruz, Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9, 4 (2013)., ²⁹[29] B. Bagchi, S. Das, S. Ghosh and S. Poria, J. of Phys. A: Mathematical and Theoretical 46, 032001 (2012)., ³⁰[30] B.G. Costa, I.S. Gomez and M. Portesi, J. of Mathematical Physics 61, 082105 (2020)., ³¹[31] O. Mustafa, Physica Scripta 96, 065205 (2021).].

This paper aims to investigate classical and quantum PDM systems using Hamiltonian factorization and canonical transformations, specifically applied to a Mathews-Lakshmanan-type oscillator. Building upon the work presented in Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).], which employed these methods on an oscillator with the same position-dependent mass, our study seeks to provide a detailed exposition of the Hamiltonian factorization and canonical transformations methods.

In the classical analysis, the phase space for a Matthew-Lakshmanan-type oscillator was previously studied in Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).] for a given energy range. Our study intends to extend those results by presenting the phase space for higher energies and thoroughly interpreting the graphical outcomes.

In the quantum case, Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).] examined the probability density for the state ${\psi }_4$ compared to the classical presence. In contrast, this paper analyzes the probability density for lower states considering the potential curve for the system. Furthermore, this study undertook a comprehensive analysis of the tunneling probabilities for a few quantum states of the Mathews-Lakshmanan-type oscillator, introducing a novel perspective on this system’s tunneling behavior.

2. Theory of position-dependent mass systems

2.1. Classical case

Consider a classical oscillator-type system with position-dependent mass. The Hamiltonian function for this system is given by

where $m(x)$ is a position-dependent function.

The equation of motion for the system is obtained from the Hamiltonian function and is a second-order differential equation, which can be factored into two first-order differential equations. This method is called Hamiltonian factorization and aims to facilitate finding the solution to the system [²⁵[25] S.H. Dong, Factorization Method in Quantum Mechanics (Springer, Berlin, 2007).].

The factorization is applied using the following ladder functions [³²[32] S. Cruz y Cruz, J. Negro and L.M. Nieto, J. of Phys.: Conference Series 128, 012053 (2008).]

where $W(x)$ represents a position-dependent function to be determined.

As in the classical case with constant mass, the ladder functions $A^{+}$ and $A^{-}$ satisfy the commutative property. Then, the factorization method gives rise to a unique Hamiltonian function given by

Comparing equations (1) and (2), it is possible to conclude that the potential relates to the function $W(x)$ in a way such that $V(x)=W^2(x)$.

Furthermore, the ladder functions have fundamental relations via Poisson brackets, among them¹ 1 These fundamental relations are associated with a Heisenberg algebra. [³²[32] S. Cruz y Cruz, J. Negro and L.M. Nieto, J. of Phys.: Conference Series 128, 012053 (2008).]

where $W^{\prime }(x)$ refers to the derivative of $W(x)$ with respect to position $x$.

From equation (3), one has

The differential equation (4) admits solutions of the form

where $C$ refers to the integration constant.

The constant $C$ determines the origin of the potential $V(x)$. Usually, a spatial shift is performed such that $V(x=0)=0$ and, consequently, $C=0$. Therefore, in this case

Since $V(x)=W^2(x)$, then

and the Hamiltonian of the PDM oscillator can be rewritten in a more general way as

Performing strategic canonical transformations on equation (7) allows a connection between the Hamiltonian function of the system with PDM and that of the harmonic oscillator with constant mass, which has known solutions. Thus, the following point canonical transformations are defined as [³²[32] S. Cruz y Cruz, J. Negro and L.M. Nieto, J. of Phys.: Conference Series 128, 012053 (2008).]

Applying the point canonical transformations, equations (8) and (9), into equation (7), it is possible to obtain the general form of the Hamiltonian function for constant mass systems, i.e.,

where $E$ represents the energy of the system. Because $\omega =m_0=1$ was considered unitary, the trajectories in the phase space are concentric circles and not the usual ellipses.

From equation (10), it is possible to obtain the well-known trajectories $X(t)$ and $P(t)$ for the oscillator with constant mass as

where $\varphi$ represents the initial phase of the system.

Because the canonical transformations are invertible, it is possible to relate the solutions of the harmonic oscillator with constant mass to the original system with PDM. Then, the trajectories $x(t)$ and $p(t)$ of the system with position-dependent mass are [³²[32] S. Cruz y Cruz, J. Negro and L.M. Nieto, J. of Phys.: Conference Series 128, 012053 (2008).]

with $\sqrt{2E}\cos (t+\varphi )$ representing the argument of the inverse function $X^{-1}$.

Therefore, point canonical transformations applied on a system with position-dependent mass highlight the proximity between the Hamiltonian with PDM and the usual one for a constant mass system. Moreover, this method allows a more straightforward way to find $x(t)$ and $p(t)$ for a PDM system because obtaining them from already-known trajectories is possible.

2.1.1. A classical Mathews-Lakshmanan-typeoscillator

Consider a Mathews-Lakshmanan-type oscillator with a mass function $m(x)$ of the form

where $m_0$ represents a positive constant value with mass dimension and $\lambda$ a parameter also positive and constant with the inverse of length dimension.

The mass function $m(x)$ chosen was not completely arbitrary. As already seen, through canonical transformations, equations (8) and (9), it is possible to relate the oscillator with constant mass to the one with PDM. According to Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).], for certain position-dependent mass functions, the canonical transformation referring to $X(x)$ may not map its domain $𝒟(m(x))$ onto the entire real line. In this case, the mass function is not physically acceptable for relating the oscillators since the position of the particle $X(x)$ does not take real values.

Notice that in the limit where $\lambda \to 0$, one has $m(x)\to m_0$, and there is a tendency to return to a system with constant mass. Moreover, when $x=0$, the maximum value $m_0$ is obtained for $m(x)$. On the other hand, when $|x|\to \mathrm{\infty }$, then $m(x)\to 0$. Therefore, the function defined by equation (13) is bounded and has a real domain, i.e., it is physically acceptable.

The features above can be observed in Fig. 1(a), in which the mass function $m(x)$ was plotted considering $m_0=1$ unitary. From this plot, varying the parameter $\lambda$, it is possible to observe that the function $m(x)$ has no singularities. Moreover, when $\lambda \to 0$, the shape of $m(x)$ is smoothed, and $m(x)\to m_0=1$, as expected. In addition, for $x=0$, the curve approaches its maximum value $m(x)=m_0=1$, regardless of the $\lambda$ value. These conclusions obtained by the graphical results corroborate the mathematical interpretations of equation (13).

Figure 1:
(a) Mass function $m(x)$, with $m_0=1$ unitary for different values of $\lambda$. (b) Potential $V(x)$ with $m_0=1$ unitary for different values of $\lambda$. (c) Zoom of the plot (b) for the region around $x=0$. The parameter was taken as $\lambda =0.03,0.27,0.51,0.76$, and $1.00$ for the three plots.

By inserting the mass function, equation (13), into equation (6) for the potential of the oscillator with position-dependent mass, one obtains

Thus, equation (14) represents the generalized potential energy for the Mathews-Lakshmanan-type oscillator defined in this section. The potential $V(x)$ for the PDM oscillator given by equation (14) was plotted in Fig.1(b), where $V(x=0)=0$, respecting the condition initially imposed that $C=0$ in equation (6).

It can also be noticed from Fig. 1(b) that when $\lambda \to 0$, the curve representing the potential approaches a parabola, similar to the potential curve for the harmonic oscillator with constant mass. In fact, taking the limit of $V(x)$ when $\lambda \to 0$, one has

and there is a tendency for the potential $V(x)$ to return to that one associated with the classical harmonic oscillator system, i.e., described by a parabola.

The same result evidenced in equation (15) also occurs for $x\to 0$, i.e., values of $x$ around the origin² 2 Another way to notice that is by using Taylor expansion around x = 0, that is, considering small oscillations around the origin of the system. . Further, Fig. 1(c) indicates that for values of $x$ close to zero, there is an overlap region for the plots of $V(x)$, regardless of $\lambda$ values. These curves around $x=0$ are similar to parabolas. As the horizontal displacement from $x=0$ occurs and $\lambda$ increases, the deformations become more evident, and the potential curve departs from the parabolic shape, as observed in Fig. 1(b).

Furthermore, it is important to observe in Fig. 1(b) that when $\lambda =1$, the system exhibits a larger classically allowed region for movement in comparison to the oscillator with $\lambda =0.03$. This disparity arises because the potential becomes less confining for higher values of$\lambda$.

It is also possible to obtain the trajectories $x(t)$ and $p(t)$ for the oscillator with $m(x)$ through the point canonical transformations represented in equations (8) and (9), with the mass function given by equation (13), in such a way that

Then, the trajectory $x(t)$ is obtained using equations (12a) and (12b), such that

Analogous, the trajectory $p(t)$ is obtained by substituting the mass function (13) and equation (17) into equation (12b). Then,

Considering $m_0=1$ unitary and phase shift $\varphi =0$, the trajectories $x(t)$ and $p(t)$ are given by

Furthermore, the Hamiltonian function represented in equation (7) becomes

The trajectory $x(t)$ represented in equation (19) for the Mathews-Lakshmanan-type oscillator agrees with the trajectory equation obtained in Ref. [¹⁹[19] O. Mustafa, The Euro. Phys. Journal Plus 136, 249 (2021).], where the author used the same mass function $m(x)$ as the one in this paper. However, the author obtained the expression for the trajectory $x(t)$ by a different method from the one used in this paper, solving directly the nonlinear differential equation of motion for the system.

The trajectories $x(t)$ and $p(t)$ of the system with position-dependent mass were compared to those for the oscillator with constant mass in Figs. 2(a) and 2(b). In both plots, the parameter $\lambda$ was considered equal to one for the differences in the behavior of the trajectories of both oscillators be better evidenced.

Figure 2:
(a) Trajectories $x(t)$, equation (17), in yellow for the oscillator with the mass function $m(x)$ and $X(t)$, equation (11a), in red for the oscillator with constant mass.(b) Trajectories $p(t)$, equation (18), in blue for the oscillator with the mass function $m(x)$ and $P(t)$, equation (11b), in black for the oscillator with constant mass. (c) Zoom of the plot (b) for $t$ values between $-$2 and 2, showing the inflection points in purple at $t=1.03$, $t=0$, and $t=-1.03$.

It can be noticed in Fig. 2(a) that the trajectory $x(t)$ for the system with PDM has a higher amplitude, and the position of the minima and maxima do not vary when compared to the trajectory $X(t)$ for the oscillator with constant mass. In addition, Fig. 2(a) coincides with the trajectories obtained in Fig. 1(b) of Ref. [¹⁹[19] O. Mustafa, The Euro. Phys. Journal Plus 136, 249 (2021).], where $x(t)$ was obtained directly by solving the nonlinear equation of motion of the PDM system with the same $m(x)$ used in this paper.

Moreover, the trajectory $p(t)$ for the PDM system is represented in Fig. 2(b). Observe that $p(t)$ has the same amplitude and, also, the same minima and maxima values compared to the trajectory $P(t)$ for the oscillator with constant mass.

Furthermore, the functions sine and cosine in terms of time only have inflection points localized in $t=0$. In contrast, the hyperbolic cosine has no inflection points in its domain. However, when these trigonometric functions are combined in equation (18), there are more inflection points in the trajectory $p(t)$ besides at $t=0$ (Fig. 2(c)).

The trajectories in phase space for the PDM oscillator, considering the same energy range as Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).], were obtained using equation (21) and are represented in Fig. 3. From Fig. 3(a), notice that when $\lambda =0.03$, i.e., $\lambda \to 0$, the trajectories tend to those of the oscillator with constant mass, i.e., concentric circles.

Figure 3:
(a) Phase space of the oscillator with function mass $m(x)$ with $m_0=1$ and $\lambda =0.03$ for different energy values. (b) Phase space of the oscillator with PDM with function mass $m(x)$ with $m_0=1$ and $\lambda =1.00$ for different energy values. In both plots, energy values were $E=0.05,0.30,0.55,0.80$, and $1.00$.

Additionally, Fig. 3(b) shows the trajectories in phase space for $\lambda =1$ and for the same energy values as in Fig. 3(a). Observe that when $\lambda =1$, the trajectories’ deformations become evident, with the deformity increasing as the energy value $E$ increases. Besides, from Fig. 3(b), it is noticeable that the particle’s velocity only cancels out at the return points, regardless of the $\lambda$ parameter.

Furthermore, from Fig. 3(b), observe that as energy values increase, the curves near the x-axis are more abruptly curved, flattening the trajectory in this direction. As for critical low-energies in Fig. 3(b), for example, $E=0.05$, even though $\lambda =1$, the trajectory approaches the one for a constant mass oscillator. In addition, the deformed trajectories in Fig. 3(b) resemble a diamond plot with rounded vertices, being more evidenced for $E\ge 0.30$. Besides, the trajectories in phase space have similar behavior to those obtained in Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).] for the same mass function used in this study.

Moreover, Fig. 4 shows the trajectories in phase space considering higher energies than those in Fig. 3. The phase space with $\lambda =0.03$ is represented in Fig. 4(a). Even though the energy values are higher, the trajectories in this phase space in Fig. 4(a) remain like those for the constant mass oscillator because $\lambda \to 0$. Therefore, the $\lambda$ parameter has a greater impact on the phase space deformation than the energy values.

Figure 4:
(a) Phase space of the oscillator with function mass $m(x)$ with $m_0=1$ and $\lambda =0.03$ for different energy values. (b) Phase space of the oscillator with PDM with function mass $m(x)$ with $m_0=1$ and $\lambda =1.00$ for different energy values. In both plots, energy values were $E=1.0,4.0,7.0$, and $10.0$.

In contrast, Fig. 4(b) shows the trajectories in phase space for $\lambda =1$ for higher energy values than those in Fig. 3. In this case, it is noticeable that the trajectories become highly deformed as $E$ increases. Additionally, one can observe that the $x$-axis range significantly and rapidly expands as the energy values increase. However, comparing Figs. 4(a) and 4(b), one can note that the maximum values in the $p$-axis do not vary when $\lambda$ value change. This result also occurs for lower energies in Fig. 3.

Besides, observe that the scales in Figs. 3(b) and 4(b) are significantly different, then it is not possible to properly interpret the shape of the curves in Fig. 4(b). The trajectories in phase space in Fig. 4(b) have similar behavior to those obtained by a different method than ours in Fig. 1 of Ref. [³³[33] M. Lakshmanan and V.K. Chandrasekar, Eur. Phys. J. Spec. Top. 222, 665 (2013).]. However, the authors did not specify the energy range used to plot the trajectories, which was done in this paper.

Hence, the differences between the behavior of the Mathews-Lakshmanan-type oscillator and the constant mass oscillator become more evident as energy and $\lambda$ values increase.

2.2. Quantum case

The Hamiltonian operator for a quantum system endowed with position-dependent mass in the usual form is³ 3 As the mass is dependent on the position operator x^, the mass in this case also becomes an operator m^⁢(x^).

Since the mass is a function of the position operator, then $\widehat{m}(\widehat{x})$ does not commute with $\widehat{p}$, and these quantities cannot be simultaneously measured. Therefore, the kinetic energy of this system is not a physical observable.

Thus, the kinetic energy operator $\widehat{T}$ represented in equation (22) is non-Hermitian. However, it is desired that the kinetic energy be a measurable quantity. Consequently, the operator $\widehat{T}$ representing this quantity must be Hermitian.

In 1983, Roos [¹[1] O. von Roos, Phys. Rev. B 27, 7547 (1983).] proposed a generalization of the kinetic energy operator to make the Hamiltonian operator $\widehat{H}$ Hermitian. In Roos generalization, the kinetic energy is written as⁴ 4 In this section, the mass operator m^⁢(x^) will be represented simply as m or m⁢(x) to simplify the notation. However, the mass remains an operator with spatial dependence on x^. [¹[1] O. von Roos, Phys. Rev. B 27, 7547 (1983).]

Then, the generalized Hamiltonian operator is given by

with $\widehat{p}=-d/dx$ and the parameters $\alpha ,\beta$ and $\gamma$ satisfying the constraint

The time-independent Schrödinger equation for quantum systems is

with $\widehat{H}=\widehat{T}+\widehat{V}$.

Then, substituting equation (23) into equation (25), the time-independent Schrödinger equation for the PDM system is generalized as

Therefore, the kinetic energy operator becomes Hermitian when the parameters obey the constraint in equation (24). However, the choice of the parameters $\alpha ,\beta$, and $\gamma$ is not unique [³⁴[34] O. Mustafa and S.H. Mazharimousavi, J. of Phys. A: Mathematical and General 39, 10537 (2006)., ³⁵[35] J.M. Lévy-Leblond, Phys. Rev. A 52, 1845 (1995).], i.e., different values of parameters generate distinct and non-equivalent Hamiltonian operators to give rise to an ambiguous ordering problem [³⁶[36] T. Bibi, S. Javed and S. Iqbal, Communications in Theoretical Phys. 75, 015102 (2023).].

Literature contains several studies advocating the choice of a specific set of parameters [⁴[4] T. Gora and F. Williams, Phys. Rev. 177, 1179 (1969)., ³⁷[37] D.J. BenDaniel and C.B. Duke, Phys. Rev. 152, 683 (1966)., ³⁸[38] T.L. Li and K.J. Kuhn, Phys. Rev. B 47, 12760 (1993)., ³⁹[39] Q.G. Zhu and H. Kroemer, Phys. Rev. B 27, 3519 (1983)., ⁴⁰[40] O. Mustafa and S.H. Mazharimousavi, International Journal of Theoretical Physics 46, 1786 (2007).]. Even before the generalization of the Hamiltonian operator $\widehat{H}$ performed by Roos, papers were published proposing a new form for the kinetic energy operator $\widehat{T}$ to study systems with position-dependent mass [⁴[4] T. Gora and F. Williams, Phys. Rev. 177, 1179 (1969)., ³⁷[37] D.J. BenDaniel and C.B. Duke, Phys. Rev. 152, 683 (1966).].

After the Roos generalization, researchers found that the previously proposed Hamiltonian operators represent a specific case of the generalized operator with a given set of parameters $\alpha ,\beta$, and $\gamma$. In addition, other sets of parameters were studied after the generalization that had not been previously analyzed [³⁸[38] T.L. Li and K.J. Kuhn, Phys. Rev. B 47, 12760 (1993)., ⁴⁰[40] O. Mustafa and S.H. Mazharimousavi, International Journal of Theoretical Physics 46, 1786 (2007).].

In 1966, BenDaniel and Duke [³⁷[37] D.J. BenDaniel and C.B. Duke, Phys. Rev. 152, 683 (1966).] proposed a modified Hamiltonian to study systems with position-dependent mass. The Hermitian operator suggested by these authors is recovered in the Roos generalization by considering $\alpha =\gamma =0$ and $\beta =-1$. In 1969, Gora and Williams [⁴[4] T. Gora and F. Williams, Phys. Rev. 177, 1179 (1969).] proposed a Hamiltonian operator such that, in the Roos generalization, $\alpha =-1$ and $\beta =\gamma =0$ to study semiconductors with smoothly graded composition, i.e., a system in which the effective mass is position-dependent.

In 1983, Zhu and Kroemer [³⁹[39] Q.G. Zhu and H. Kroemer, Phys. Rev. B 27, 3519 (1983).] proposed a Hermitian Hamiltonian where $\alpha =\gamma =-1/2$ and $\beta =0$ to study the wave functions with position-dependent effective mass in an abrupt heterojunction between two semiconductors. Furthermore, in 1993, Li and Kuhn [³⁸[38] T.L. Li and K.J. Kuhn, Phys. Rev. B 47, 12760 (1993).] analyzed the case where $\alpha =0$ and $\beta =\gamma =-1/2$ to study a heterojunction in which the mass function varies smoothly with the position. Finally, in 2007, Mustafa and Mazharimousavi [⁴⁰[40] O. Mustafa and S.H. Mazharimousavi, International Journal of Theoretical Physics 46, 1786 (2007).] proposed a Hermitian Hamiltonian operator with the parameters $\alpha =\gamma =-1/4$ and $\beta =-1/2$ in the generalization performed by Roos.

There is an interesting discussion in the article written by Morrow and Brownstein [⁶[6] R.A. Morrow and K.R. Brownstein, Phys. Rev. B 30, 678 (1984).] in 1984, in which the authors showed that $\alpha =\gamma$. This equality is because of the condition of continuity of ${\psi }^{\prime }(x,t)/m(x)$ in abrupt heterojunctions between two crystals, i.e., when there is a non-finite discontinuity in the effective mass of the system [⁴¹[41] A.R. Plastino, A. Puente, M. Casas, F. Garcias and A. Plastino, Revista Mexicana de Física 46, 78 (2000).]. Therefore, in this case, the Hamiltonian operator becomes [⁶[6] R.A. Morrow and K.R. Brownstein, Phys. Rev. B 30, 678 (1984).]

with the constraint

In this study, although the analyzed system is not a semiconductor with abrupt heterojunction, $\alpha =\gamma$ will be used due to the physical viability of the condition.

To apply the Hamiltonian factorization method, the operator represented in equation (26) will be used, considering $\alpha =a$, $\beta =2b$, and $\widehat{p}=-d/dx$ to simplify the notation. Then

with the constraint

In addition, the following ladder operators are defined [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]

where $W_a(\widehat{x})$ is called “superpotential”.

Moreover, as the quantum harmonic oscillator with constant mass, the operators ${\widehat{A}}_a^{+}$ and ${\widehat{A}}_a^{-}$ obey the following commutation relation [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]

Equation (31) shows that the ladder operators ${\widehat{A}}_a^{+}$ and ${\widehat{A}}_a^{-}$ do not commute. This gives rise to two Hamiltonian operators, ${\widehat{H}}_a^{-}$ and ${\widehat{H}}_a^{+}$, such that [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]

and

where the kinetic energy operators ${\widehat{T}}_a^{+}$ and ${\widehat{T}}_a^{-}$ are given by

and the potentials are

The potential ${\widehat{V}}_a^{-}$ represented by equation (35) differs from that obtained in Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).], where their potential has an extra term⁵ 5 The potential used by the authors results in an approximation for solving the ambiguous ordering problem. . However, the potential ${\widehat{V}}_a^{-}$ equation (35) agrees with the one obtained in Ref. [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]. The derivation of the potential ${\widehat{V}}_a^{-}$ in equation (35) is presented in the Appendix of the Supplementary Material.

Expanding the commutation relation represented in equation (31), one has

with the Hamiltonian operator $\widehat{H}$ characteristic of a harmonic oscillator

According to equation (37), the supersymmetric partners ${\widehat{H}}_a^{-}$ and ${\widehat{H}}_a^{+}$ have the same spectrum, however, ${\widehat{H}}_a^{-}$ has one less bound state than its partner ${\widehat{H}}_a^{+}$ [⁴³[43] A.R. Plastino, A. Rigo, M. Casas, F. Garcias and A. Plastino, Phys. Rev. A 60, 4318 (1999).].

Substituting the equations (32) and (33) into the commutator relation, equation (36), one has that

In addition, equation (37) suggests that $\widehat{H}$ must be unique. Therefore, the following condition [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]

is imposed in equation (38).

The solution of the differential equation represented in equation (39) is given by

Since the potential terms in equation (38) were canceled out, then ${\widehat{T}}_a^{+}={\widehat{T}}_a^{-}$, i.e.,

and for equation (41) to be satisfied, it is necessary that $a=b$.

In addition, using the constraint equation (28), it is possible to infer that $a=b=-1/4$. In this case, since $\alpha =\gamma =a$ and $\beta =2b$, one has

and the ambiguous ordering problem is solved for the harmonic oscillator, as done in Ref. [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).]

Thus, with $a=-1/4$ and equation (34), the potential ${\widehat{V}}_{-1/4}^{+}$ become

using equation (39) for both cases to infer that

As evidenced by equations (43) and (44), the potentials ${\widehat{V}}_{-1/4}^{+}$ and ${\widehat{V}}_{-1/4}^{-}$ are supersymmetric partners, as proved in Ref. [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).].

Furthermore, it is possible to notice that the superpotential represented in equation (40) has the same form as the function $W(x)$ in equation (5), obtained by factoring the Hamiltonian of the classical oscillator with position-dependent mass.

To analyze the existing relationship between the superpotential $W_{-1/4}(\widehat{x})$ and the potential $\widehat{V}(\widehat{x})$, equation (37) and the previously obtained parameter $a=-1/4$ are used. Thus

with $\widehat{H}$ given by equation (27) and ${\widehat{H}}_{-1/4}^{+}$ given by equation (32).

Therefore, the potential $V(x)$ for the quantum oscillator with PDM is given by

which has the same form of the potential given by equation (6) obtained for the classical case of a PDM oscillator. Hence, with the parameter set of Mustafa and Mazharimousavi [⁴⁰[40] O. Mustafa and S.H. Mazharimousavi, International Journal of Theoretical Physics 46, 1786 (2007).] (equation (42)), a correspondence between the classical and quantum position-dependent mass oscillators could be established [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).].

Furthermore, the Hamiltonian operator with the parameter set of Mustafa and Mazharimousavi is given by

Then, using equation (46) and equation (45), one has

The kinetic energy operator for the quantum harmonic oscillator with position-dependent mass represented in the first term of equation (47) is now unique since the Roos parameters were established for this system [⁴²[42] O. Mustafa, Phys. Lett. A 384, 126265 (2020).].

Moreover, with $a=b=-1/4$, the ladder operators, equations (29) and (30), become

and when $m(x)\to m_0$, the ladder operators return to those usually used in solving the quantum harmonic oscillator with constant mass.

Therefore, the Hamiltonian factorization method for systems with position-dependent mass allows the finding of the appropriate ladder operators to analyze the dynamics of systems with PDM. Thus, this method becomes a powerful mathematical tool to simplify solving these systems.

Due to the properties of the ladder operators ${\widehat{A}}^{+}$ and ${\widehat{A}}^{-}$, the same relations for the case of the quantum harmonic oscillator with constant mass are obtained, that is

with the wave function given by

In addition, it is possible to relate the Hamiltonian operator with position-dependent mass represented in equation (47) to the operator ${\widehat{H}}_0$⁶ 6 The Hamiltonian operator for a system with constant mass is H^0=p^22⁢m0+V^⁢(x^), where m0 represents the constant mass and V^⁢(x^) the potential operator. that describes the oscillator with constant mass.

Initially, consider the eigenvalue equation for the Hamiltonian operator ${\widehat{H}}_{-1/4}$, given by

with $E_n=(n+1/2)$, in which was taken $\mathrm{\hslash }=\omega =1$. Notice that because of the form of $E_n$, it is possible to infer that the PDM oscillator is in the same class as the harmonic oscillator with constant mass because they have the same energy spectrum, although having different potentials [⁴⁴[44] J. Guo-Xing, C. Chang-Ying and R. Zhong-Zhou, Communications in Theoretical Physics 51, 797 (2009).].

Now, using the following canonical point transformations [⁴⁵[45] J.P.G. Nascimento and I. Guedes, Revista Brasileira de Ensino de Física 38, e2315 (2016).]

in the eigenvalue equation (48), and considering

equation (48) becomes

Equation (51) is similar to the eigenvalue equation for the quantum harmonic oscillator with constant mass. In fact, the wave functions ${\phi }_n(x)$ for equation (51) is the same as that of the harmonic oscillator with constant mass [⁴⁶[46] D.J. Griffiths and D.F. Schroeter, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2018), 3 ed.], i.e.

where ${\mathscr{H}}_n(x)$ represents the Hermite polynomials.

Returning to the solution for the position-dependent harmonic oscillator using the inverse canonical transformation of equations (49) and (50), one has that

Therefore, equation (52) represents the eigenfunctions of the quantum harmonic oscillator with position-dependent mass. Notice that the Hermite polynomial now has a position dependence as well.

2.2.1. A quantum Mathews-Lakshmanan-typeoscillator

As an application to the theory developed in the previous section, consider a Mathews-Lakshmanan-type oscillator with a mass operator defined as

with $m_0=1$ unitary.

Firstly, it is noticeable from equation (45) that the potential energy for this system will have the same shape as the potential of the classical Mathews-Lakshmanan-type oscillator, represented in equation (14), showing a similarity between both systems.

Besides, from the canonical transformation in equation (49), one has

and, thus, it is possible to obtain the eigenfunctions from equation (52), given by

In addition, the Hamiltonian operator for the oscillator with function mass defined in equation (53) is given by

From equation (54), the first three eigenfunctions for our Mathews-Lakshmanan-type oscillator are

With equations (55)–(58), it is possible to plot the wave functions and probability distributions for the Mathews-Lakshmanan-type oscillator, represented in Figs. 5 and 6, respectively.

Figure 5:
Wave functions (a) ${\psi }_0(x)$ and ${\phi }_0(x)$, (b) ${\psi }_1(x)$ and ${\phi }_1(x)$, (c) ${\psi }_2(x)$ and ${\phi }_2(x)$, (d) ${\psi }_3(x)$ and ${\phi }_3(x)$ with $m_0=\lambda =1$. The wave functions ${\psi }_n(x)$ describe the harmonic oscillator with $m(x)$ (in purple), and ${\phi }_n(x)$ describe the harmonic oscillator with constant mass (in black).

Figure 6:
Probability densities (a) ${|{\psi }_0(x)|}^2$ and ${|{\phi }_0(x)|}^2$, (b)${|{\psi }_1(x)|}^2$ and ${|{\phi }_1(x)|}^2$, (c) ${|{\psi }_2(x)|}^2$ and ${|{\phi }_2(x)|}^2$, (d)${|{\psi }_3(x)|}^2$ and ${|{\phi }_3(x)|}^2$ with $m_0=\lambda =1$. The probability densities ${|{\psi }_n(x)|}^2$ are related to the harmonic oscillator with $m(x)$ (in purple) and ${|{\phi }_n(x)|}^2$ with the harmonic oscillator with constant mass (in black).

The plots for the wave functions for the PDM system with $m(x)$ represented in Fig. 5 agree with the ones obtained for the same quantum system in Ref. [²⁴[24] S. Cruz y Cruz, J. Negro and L. Nieto, Phys. Lett. A 369, 400 (2007).]. However, in our study, the wave functions for the PDM system were plotted with those for the constant mass oscillator to enable a further investigation of the similarities and differences between both oscillators.

From Fig. 5, it is noticeable that the wave functions for the oscillator with position-dependent mass preserve symmetry about the $y$-axis. Analogous to the case of the quantum harmonic oscillator with constant mass, this symmetry is due to the Hermite polynomials ${\mathscr{H}}_n(x)$ having definite parity, with ${\mathscr{H}}_{2n}$ even and ${\mathscr{H}}_{2n+1}$ odd. Therefore, the wave functions ${\psi }_{2n}$ remain to have even parity, and ${\psi }_{2n+1}$ preserves its odd parity.

From equation (54), one can observe that when $\lambda \to 0$, the wave functions tend to those for the oscillator with constant mass. This result also occurs for $x$ values close to zero. Notice in Fig. 5 that around $x=0$ values, there is an overlap region of the curves representing the oscillator with constant mass and the one with PDM. The superposition regions indicate that both oscillators have similar behavior around the origin, even though $\lambda =1$.

The probability densities for the constant mass and the PDM oscillators were plotted in Fig. 6. It is noticeable that for the Mathews-Lakshmanan-type oscillator, the distribution seems to be more spread out in contrast to the probability density for the oscillator with constant mass. Then, from Fig. 6, one could infer that the PDM system is more spatially dispersed, i.e., has a less defined position. However, it is crucial to consider that the PDM system has a different potential shape than that of the constant mass oscillator.

Figure 7:
Potential (equation (45)) and probability densities of ${|{\psi }_0(x)|}^2$ (lower level), ${|{\psi }_1(x)|}^2$, ${|{\psi }_2(x)|}^2$, and ${|{\psi }_3(x)|}^2$ (higher level) for (a) $\lambda =0.03$ and (b) $\lambda =1$. The probability densities ${|{\psi }_n(x)|}^2$ were dislocated regarding the $y$-axis in a factor of $1/2$, the incremental of each energy level.

The probability densities were plotted with the potential curve in Fig. 7. The potential for the system with $\lambda =1$ represented in Fig. 7(a) is less confining compared to the oscillator with $\lambda =0.03$ in Fig. 1(b), which resemble a parabola. Then, this indicates that the oscillator with $\lambda =1$ has a larger classically allowed region of movement, as seen in Fig. 7(b) (notice the difference between the $x$-axis ranges in Figs. 7(a) and 7(b)).

Additionally, Fig. 7(a) shows that when $\lambda =0.03$, i.e., $\lambda \to 0$, the potential and probability densities approach those for the oscillator with constant mass. In this case, the particle has higher probability densities at the extremities of the classically allowed region delimited by the potential, as expected.

Contrary to what Fig. 6 might indicate, Fig. 7(b) suggests that the particle is more likely to be detected around the system’s origin and not at the extremities of the classically allowed region. This conclusion is not trivial to be inferred only by analyzing Fig. 6 because the probability densities are not shown with the potential curve in Fig. 1(b).

Further, as well-known, there is a close relation between the probability density of the system and the tunneling probability through the potential barrier. For the oscillator with $\lambda \to 0$, Fig. 7(a) indicates that the tunneling probability decreases slowly for higher excited states, as for the oscillator with constant mass.

In contrast, for the system with $\lambda =1$, Fig. 7(b) shows that for the state corresponding to $n=0$, the system has an equally high probability of tunneling compared to the same state in Fig. 7(a). This closeness of tunneling probabilities comes from the similarity between the probability density curves in Fig. 6(a). However, one can observe in Fig. 7(b) that for the quantum state $n=1$, the tunneling probability decreases significantly in contrast with the same quantum state in Fig. 7(a).

Yet, the tunneling probability decays more noticeably and even faster for higher excited states ($n=2$ and $3$). For instance, notice the difference in the quantum state corresponding to $n=3$ between the Figs. 7(a) and 7(b). Therefore, the PDM system with a lower $\lambda$ value is more likely to tunnel through the potential barrier in this quantum state.

3. Conclusions

This study enhanced the comprehension of classical and quantum systems with position-dependent mass. The analytical challenges of nonlinear equations of motion and non-Hermitian operators were addressed through Hamiltonian factorization and canonical transformations methods for both systems. The theory was applied on a Mathews-Lakshmanan-type oscillator with $m(x)=1/[1+{(\lambda x)}^2]$.

Classically, the trajectories in phase space for a Mathews-Lakshmanan-type oscillator were investigated. As the energy and $\lambda$ values increased, more evident were the trajectories’ deformities, and higher was the tendency of those to flatter in the $x$-direction. This analysis provided an understanding of the system’s dynamics and highlighted the significant impact of the $\lambda$ value in this PDM oscillator.

In the quantum realm, the generalized kinetic energy operator proposed by Von Roos was employed to address the kinetic operator being non-Hermitian, and the solution to the ambiguous ordering problem for the PDM oscillator was presented. Moreover, the wave functions and probability densities were analyzed for a few states of the Mathews-Lakshmanan-type oscillator. Further, this study also investigated the tunneling probabilities for this PDM oscillator. As the $\lambda$ value increased, the tunneling probability decreased, being more evident for higher excited states. This result offers novel insights into the behavior of the Mathews-Lakshmanan-type oscillator.

Thus, this study contributed to the broader understanding of PDM systems and their dynamics, with implications for various fields of physics where position-dependent mass plays a crucial role.

Acknowledgments

The authors would like to thank the Writing Center (CERTA – Centro de Escrita, Revisão e Tradução Acadêmica – http://www3.unicentro.br/centrodeescritaacademica – of the Midwestern State University of Paraná (UNICENTRO) for assistance with English language developmental editing.

Supplementary material

The following online material is available for this article: Appendix– The potential $V_a^{-}$