Open Access Published by De Gruyter Open Access May 18, 2021

Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study

Biswanath Rath, Pravanjan Mallick, Prachiprava Mohapatra, Jihad Asad, Hussein Shanak and Rabab Jarrar
From the journal Open Physics

Abstract

We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

1 Introduction

The study of non-linear vibration has become important in designing the flexible structures associated with aircraft, bridge, satellite, etc. [1]. This study can also be extended to acoustics, biology [2], and other branches of engineering such as electronics, robotics, and mechatronics [1]. It is therefore important to design the non-linear control vibration. For the purpose, one needs to consider the simple harmonic oscillator (SHO) with Hamiltonian [3,4]

(1) H = p 2 2 m + 1 2 m ω 2 x 2
as a model to design sustained non-linear vibration [ 2] as long as mass of the system is constant. Further, the closed contour of the phase space of the above system signifies the existence of the discrete energy in the system. At this point, we would like to state that the phase portrait of the operator [ 5]
(2) H = p 2 2 x 2 2
is not formed closed orbit and hence it does not possess discrete energy states.

In recent years, systems with position-dependent mass (PDM) have attracted the attention of many researchers and scientists because of their importance in many branches of physics. These systems were first introduced in the theory of semiconductor physics [6,7,8,9,10], especially in the study of the electronic properties and band structure. Subsequently, the applicability of PDM systems can be found in many fields such as quantum mechanics [11,12], classical mechanics [13,14,15], nuclear physics [16], molecular physics [17], neutrino mass oscillations [18], and quantum information [19]. It is worth mentioning here that the PDM study can mostly be related to semiconductors as well as other solid state physics problem. Because of the wide range of applications of PDM, many efforts have been carried out in studying such systems [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49].

It is commonly seen that mass-dependent oscillators having m ( x ) x 0 . As an example of this, the m ( x ) can be expressed as [24,25,26,27,28]

(3) m ( x ) = m ( 1 + x 2 ) .

In this context, we would like to say that at large distance mass becomes zero and the particle possesses infinite kinetic energy and zero potential energy at this point. Therefore, the particle behaves like a free particle and becomes unbound. An unbound particle is most probably unsuitable for any spectral observation. Similarly, considering another form of mass variation as

(4) m ( x ) = m ( 1 + x 2 )
for which m ( x ) x . It is meaning less to discuss the quantum mechanical behaviour of such a particle at this point. It is therefore desired to think of finite mass variation with distance. In this case, there are two choices: (i) decreasing or (ii) increasing. An example of controlled decreasing mass has been discussed in ref. [ 50]. The model example from the above literature is
(5a) m ( x ) = m 1 + λ + x 2 1 + x 2 2 .

However, the authors [50] have considered a second model, where mass originates from zero and becomes constant at large distance. The proposed mass model [50] is

(5b) m ( x ) = m tanh 2 ( λ x ) .

A similar type of mass of the form [29,30]

(6a) m ( x ) = m sec h 2 ( a x )
has recently been reported. In a very recent work, another form of decreasing mass
(6b) m ( x ) = e x L α 1 1
has also been reported [ 37].

The applicability of increased PDM has also been seen in some experiments. The deuteron–deuteron scattering has successfully explained by considering enhancement of electron mass [51]. The increased effective mass can be used on the basis of fluid to explain some phenomena in quantum field theory [52]. The enhancement of mass of quasiparticle in BaFe2(As1−x P x )2 has been reported at quantum critical point which in turn affect the critical temperature of the superconducting state [53,54]. Effective mass of exciton in semiconductor coupled quantum well is predicted to be enhanced under the influence of electromagnetic field [55]. Enhancement of mass in the quasiparticle led to the entanglement in Kondo problem [56]. In a similar manner, the enhancements of energy of electrons in quasicrystal [57], in hydrogen atom [58], and in quantum LC circuit [59] have also been reported. In addition, the enhancement of mass of electron in periodic lattice using nonlocal approach [60], implication of PDM for semiconductor as well as molecular physics [61], generation of massive photon in a magnetic material [62], etc. have also been predicted recently.

In semiconductors and other problems related to solid state physics, we mainly deal with atoms whose masses cannot be infinite or zero. If it varies with distance, then it must be within the finite values. It is seen that the PDM in all the theoretical cases does not vary within the finite limit. However, in a very recent work, we demonstrated about the finite variation of mass under asymmetric condition [63]. Considering the above literature, we propose a new model position-dependent finite mass variation, i.e.

(7) m ( x ) x 0 c 1 ; m ( x ) x c 2
for symmetric cases comprising of both increasing and decreasing PDM. In view of the importance of vibration and PDM, we focused our attention to study the vibration of a newly designed finite symmetric increasing and decreasing mass harmonic like oscillators using classical and quantum mechanical approaches. The purpose of this work is to choose the PDM that varies between two values avoiding the ambiguity situation as discussed above, i.e., avoiding the values of m ( x ) x 0 and m ( x ) x .

The rest of the present work is organized as follows: In Section 2, a classical description of the harmonic like oscillator comprising of both symmetric decreasing and increasing mass is discussed by performing analytical calculations. In Section 3, quantum mechanical study is presented, and finally the discussions along with the important findings of the work are presented in Section 4.

2 Classical description of the system

Consider a particle whose Hamiltonian is given by

(8) H = p x 2 2 m ( x ) + m ( x ) 2 x 2 .

The Lagrangian is related to the Hamiltonian as

(9) H = i p i q ̇ i L .

In our case, we have just one generalized coordinate ( x ) and one generalized momentum ( p x ) . Therefore, we have

(10) H = p x x ̇ L p x 2 2 m ( x ) + m ( x ) 2 x 2 = p x x ̇ L .

Solving equation (10), one can find the Lagrangian ( L ) as

(11) L = p x x ̇ p x 2 2 m ( x ) m ( x ) 2 x 2 .

Using d x d t = H p x x ̇ = p x m ( x ) , and p x = x ̇ m ( x ) , equation (11) becomes

(12) L = m ( x ) 2 ( x ̇ 2 x 2 ) .

Substituting equation (12) into the relation L q i d d t L q ̇ i = 0 with q i = x , we obtained:

(13) x m ( x ) + ( x ̇ 2 x 2 ) 2 d m ( x ) d x x ̈ m ( x ) x ̇ d m ( x ) d t = 0 .

2.1 Decreasing PDM

Let the mass be specified as

(14) m ( x ) = m 2 ( 1 + e γ x ) .

Figure 1 shows the variation in m ( x ) as stated above with x for m = 1 and γ = 0.1, 0.5, and 1. In this case, the m ( x ) is decreasing from 1 and progressively saturating to 0.5 as we move away from origin. In other words, the m ( x ) varies between two finite limits, i.e., 1 and 0.5. Further, the nature of variation in m ( x ) is symmetric with respect to origin of the co-ordinate system. It is worth mentioning here that the value of constant parameter ( γ ) plays the major role in the variation in m ( x ) . For example: one will see that the m ( x ) decreases very fast for γ = 1, whereas the same decreases very slow for γ = 0.1. Interested reader will see that the variation in m ( x ) will look like a delta function for γ > 1. However, the variation in m ( x ) in all the cases is confined between two limits, i.e., between 1 and 0.5.

Figure 1 
                  Variation in 
                        
                           
                           
                              m
                              (
                              x
                              )
                           
                           m(x)
                        
                      as stated in equation (14) with x for 
                        
                           
                           
                              m
                              =
                              1
                           
                           m=1
                        
                      and 
                        
                           
                           
                              γ
                              =
                           
                           \gamma =
                        
                      0.1, 0.5, and 1.

Figure 1

Variation in m ( x ) as stated in equation (14) with x for m = 1 and γ = 0.1, 0.5, and 1.

Substituting equation (14) in equation (13), we have

(15) m 2 ( 1 + e γ x ) x ̈ + ( x ̇ 2 x 2 ) 2 m 2 ( γ e γ x ) d x d x + m 2 ( x + x e γ x ) x ̇ m 2 ( γ e γ x ) d x d t = 0 .

The equation (15) can further be simplified to

(16a) x ̈ + x + ( x ̇ 2 x 2 ) γ e γ x 2 ( 1 + e γ x ) d x d x + x ̇ 2 γ e γ x 2 ( 1 + e γ x ) = 0 ,
which leads to
(16b) x ̈ + x γ e γ x 2 ( 1 + e γ x ) ( x ̇ 2 + x 2 ) = 0 ; x 0
and
(16c) x ̈ + x + 3 x ̇ 2 γ e γ x 2 ( 1 + e γ x ) x 2 γ e γ x 2 ( 1 + e γ x ) = 0 ; x 0 .

This is essentially a modified harmonic oscillator that reflects the amplitude dependence frequency. In other words, vibration of particle is controlled by amplitude or vice versa. Further, it can be regarded as a self-controlled vibration. It is worth mentioning here that one can also be able to derive equation (16b and 16c) following the work of El-Nabulsi and others [64,65,66,67,68,69,70]. When the value of γ = 0 , the above equation (16b and 16c) will be reduced to

(17) x ̈ + x = 0 ,
which is the equation of motion for a free harmonic oscillator.

2.1.1 Analytical calculation on decreasing PDM

Let us rewrite the equation of motion (equation 16(b)) as

(18) x ̈ + x + F x , d x d t = 0 ,
where
(19a) F x , d x d t = Q ( x 2 + x ̇ 2 )
with
(19b) Q = γ 2 ( 1 + e γ x ) .

The equation (18) has the similarity with the Kryloff–Bogoliuboff autonomous system, which is expressed as [5]

(20) x ̈ + x + ε f x , d x d t = 0
with the solution is of the form
(21) x = a ( t ) sin ( ω t + ϕ ) .

It should be stated here that

(22) F ε f .

In our analytical approach, we assume a similar type of solution as

(23) x = A cos ω t
in which the frequency of oscillation ( ω ) is a function of A. Here, we also consider the same boundary condition as considered in numerical study, i.e., x ( t = 0 ) = 1 and x ̇ ( t = 0 ) = 0 to study classical dynamics using analytical approach. We use widely used ancient Chinese method based on He’s frequency formalism [ 63, 71, 72, 73] to derive the ω for the equation ( 18) using the following condition:
(24a) ω 2 = ω 1 2 R 2 ( 0 ) ω 2 2 R 1 ( 0 ) R 2 ( 0 ) R 1 ( 0 ) ,
where
(24b) R 1 ( 0 ) = A 2 2 γ e γ A 1 + e γ A
and
(24c) R 2 ( 0 ) = ( 1 ω 2 2 ) A A 2 2 γ e γ A 1 + e γ A .

To obtain the expression of the value of ω for the system using the He’s approach [63,71,72,73], one needs to replace ω 1 = 1 and ω 2 = ω in the final expression of equation 24(a). Based on our initial boundary condition (i.e., x ( t = 0 ) = 1 and x ̇ ( t = 0 ) = 0 ), the two equations (16b and 16c) would effectively give rise to same expression of frequency as

(25) ω = 1 A 2 γ e γ A 1 + e γ A .

Figure 2 showed the variation in frequency ( ω ) with respect to amplitude (A). Initially, the ω decreased from 1 to a certain value (∼0.92) and then the same showed increasing trend, which finally saturated to 1 with increasing A. Like the m ( x ) , the ω ( A ) also varies between two finite values. The analytical results on the variation in x ( t ) with respect to t, p ( t ) with respect to t, and trajectory of phase space ( p x vs x) for γ = 0.1, 0.5, and 1 with A = 1 are shown in Figure 3.

Figure 2 
                     Variation in frequency (
                           
                              
                              
                                 ω
                              
                              \omega 
                           
                        ) with amplitude (
                           
                              
                              
                                 A
                              
                              A
                           
                        ) for 
                           
                              
                              
                                 γ
                              
                              \gamma 
                           
                         = 0.1, 0.5, and 1.

Figure 2

Variation in frequency ( ω ) with amplitude ( A ) for γ = 0.1, 0.5, and 1.

Figure 3 
                     Variation in (a) x with respect to time t, (b) 
                           
                              
                              
                                 
                                    
                                       p
                                    
                                    
                                       x
                                    
                                 
                              
                              {p}_{x}
                           
                         with respect to t, and (c) 
                           
                              
                              
                                 
                                    
                                       p
                                    
                                    
                                       x
                                    
                                 
                              
                              {p}_{x}
                           
                         with respect to 
                           
                              
                              
                                 x
                              
                              x
                           
                         obtained from analytical calculation for 
                           
                              
                              
                                 γ
                              
                              \gamma 
                           
                         = 0.1, 0.5, and 1 with 
                           
                              
                              
                                 A
                                 =
                                 1
                              
                              A=1
                           
                        .

Figure 3

Variation in (a) x with respect to time t, (b) p x with respect to t, and (c) p x with respect to x obtained from analytical calculation for γ = 0.1, 0.5, and 1 with A = 1 .

2.2 Increasing PDM

Here, we consider an increasing position-dependent mass, which is expressed as

(26) m ( x ) = m 1 + e λ x .

Figure 4 shows the variation in m ( x ) as stated in (equation (26)) with x for m = 1 and λ = 0.1, 0.5, and 1. In this case, the m ( x ) is increasing from 0.5 and progressively saturating to 1 as we move away from origin. In this case, the mass also exhibits symmetric nature like the decreasing mass case. Here, we find an interesting feature on larger value of λ , i.e., the variation in m ( x ) looks like a volcanic type feature around small values of x .

Figure 4 
                  Variation in 
                        
                           
                           
                              m
                              (
                              x
                              )
                           
                           m(x)
                        
                      as stated in (equation (28)) with 
                        
                           
                           
                              x
                           
                           x
                        
                      for 
                        
                           
                           
                              m
                              =
                              1
                           
                           m=1
                        
                      and 
                        
                           
                           
                              λ
                              =
                           
                           \lambda =
                        
                      0.1, 0.5, and 1.

Figure 4

Variation in m ( x ) as stated in (equation (28)) with x for m = 1 and λ = 0.1, 0.5, and 1.

2.2.1 Analytical calculation on increasing PDM

Substituting equation (26) in equation (13) and simplifying, we have

(27a) x ̈ + x λ e λ x 2 ( 1 + e λ x ) ( x ̇ 2 + x 2 ) = 0 ; x 0 ,
(27b) x ̈ + x + 3 x ̇ 2 λ e λ x 2 ( 1 + e λ x ) x 2 λ e λ x 2 ( 1 + e λ x ) = 0 ; x 0 .

In this case, we also consider the formalism discussed above (in Section 2.1.1) to derive the frequency of oscillation for the solution to equations (27a and 27b) using the He’s formalism [63,71,72,73] as

(28) ω = 1 A 2 λ e λ A 1 + e λ A .

The analytical results on the variation in x ( t ) with respect to t , p ( t ) with respect to t , and trajectory of phase space ( p x vs x ) for and λ = 0.1, 0.5, and 1 with A = 1 are shown in Figure 5. The phase portrait exhibits egg structure like closed loop. The shrinking of loop of phase portrait in one side while expanding the other side is evident with increasing λ values (Figure 5(c)) for increasing mass case.

Figure 5 
                     Variation in (a) 
                           
                              
                              
                                 x
                              
                              x
                           
                         with respect to time 
                           
                              
                              
                                 t
                              
                              t
                           
                        , (b) 
                           
                              
                              
                                 
                                    
                                       p
                                    
                                    
                                       x
                                    
                                 
                              
                              {p}_{x}
                           
                         with respect to 
                           
                              
                              
                                 t
                              
                              t
                           
                        , and (c) 
                           
                              
                              
                                 
                                    
                                       p
                                    
                                    
                                       x
                                    
                                 
                              
                              {p}_{x}
                           
                         with respect to 
                           
                              
                              
                                 x
                              
                              x
                           
                         obtained from analytical calculation for 
                           
                              
                              
                                 λ
                              
                              \lambda 
                           
                         = 0.1, 0.5, and 1 with 
                           
                              
                              
                                 A
                                 =
                                 1
                              
                              A=1
                           
                        .

Figure 5

Variation in (a) x with respect to time t , (b) p x with respect to t , and (c) p x with respect to x obtained from analytical calculation for λ = 0.1, 0.5, and 1 with A = 1 .

3 Quantum mechanical study

In this case, we solve the eigenvalue relation using matrix diagonalization method [22,63,74,75,76,77] as

(29) H ψ = E ψ ,
where
(30) ψ = n A n n .

Here, n satisfies the exact eigenvalue relation as

(31) H 0 n = ( p 2 + x 2 ) n = ( 2 n + 1 ) n .

Using the above-mentioned procedure, one can get the recursion relation satisfied by A n as

(32a) k = 2 , 4 , 6 , P n k A n k + Q n A n + R n k A n + k = 0 ,
where
(32b) P n k = n H n k ,
(32c) R n k = n H n + k ,
(32d) Q n = n H n E .

The energy eigenvalues of the Hamiltonian (equation (8)) with PDM (equation (14)) are obtained following the above-mentioned procedure for different values of γ and m = 1 (Table 1). We also calculated the energy eigenvalues of the Hamiltonian (equation (8)) with PDM (equation (26)) with different values of λ and m = 1 (Table 2). It is to be noted here that the Hamiltonian is used as

(33) H = p 1 2 m ( x ) p + 1 2 m ( x ) ω 2 x 2
for our quantum mechanical calculation following the work of Wilkes and Muljarov [ 55].

Table 1

First five eigenvalues of Hamiltonian (equation (8)) with PDM (equation (14)) with different values of γ and m = 1

Energy level ( n ) γ Eigenvalue ( E n )
0 0.1 0.5
1 1.4725
2 2.4458
3 3.4060
4 4.3667
0 0.5 0.5
1 1.3846
2 2.2946
3 3.1659
4 4.0635
0 1 0.5
1 1.3281
2 2.2570
3 3.1322
4 4.0761
Table 2

First five eigenvalues of Hamiltonian (equation (8)) with PDM (equation (26)) with different values of λ and m = 1

Energy level ( n ) λ Eigenvalue ( E n )
0 0.1 0.5
1 1.536
2 2.5699
3 3.6203
4 4.6675
0 0.5 0.5
1 1.6353
2 2.7172
3 3.8568
4 4.9540
0 1 0.5
1 1.6922
2 2.7526
3 3.9041
4 4.9618

To have more information about the system, we plot quantum mechanical phase trajectories of the system (i) with PDM (equation (14)) with different values of γ and m = 1 (Figure 6) and (ii) with PDM (equation (26)) with different values of λ and m = 1 (Figure 7) considering E n = E = H of respective states. The formation of closed orbit (Figures 6 and 7) clearly indicated the stable behaviour of the system. Comparing the quantum phase portrait of both the case, one can see that the appearance of cusp in decreasing mass case and kink in increasing mass case near origin which is significant for higher values of γ and λ , respectively. This feature could be because of the inherent property associated with corresponding mass.

Figure 6 
               Phase trajectories of the Hamiltonian (equation (8)) with PDM (equation (14), i.e., decreasing mass case) considering E = H with 
                     
                        
                        
                           m
                           =
                           1
                        
                        m=1
                     
                   for (a) 
                     
                        
                        
                           γ
                           =
                           0.1
                        
                        \gamma =0.1
                     
                  , (b) 
                     
                        
                        
                           γ
                           =
                           0.5
                        
                        \gamma =0.5
                     
                  , and (c) 
                     
                        
                        
                           γ
                           =
                           1
                        
                        \gamma =1
                     
                  .

Figure 6

Phase trajectories of the Hamiltonian (equation (8)) with PDM (equation (14), i.e., decreasing mass case) considering E = H with m = 1 for (a) γ = 0.1 , (b) γ = 0.5 , and (c) γ = 1 .

Figure 7 
               Phase trajectories of the Hamiltonian (equation (8)) with PDM (equation (26), i.e., increasing mass case) considering E = H with 
                     
                        
                        
                           m
                           =
                           1
                        
                        m=1
                     
                   for (a) 
                     
                        
                        
                           λ
                           =
                           0.1
                        
                        \lambda =0.1
                     
                  , (b) 
                     
                        
                        
                           λ
                           =
                           0.5
                        
                        \lambda =0.5
                     
                  , and (c) 
                     
                        
                        
                           λ
                           =
                           1
                        
                        \lambda =1
                     
                  .

Figure 7

Phase trajectories of the Hamiltonian (equation (8)) with PDM (equation (26), i.e., increasing mass case) considering E = H with m = 1 for (a) λ = 0.1 , (b) λ = 0.5 , and (c) λ = 1 .

4 Discussion and conclusion

The Harmonic like oscillator under the influence of decreasing and increasing PDM has been studied. In the case of increasing mass, m ( x ) increased from 0.5 to 1 and in case of decreasing mass, the same decreased from 1 to 0.5. The purpose of selecting finite mass is that a mechanical or an oscillating particle can possess neither zero mass nor infinite mass at any point of motion. Particle with zero mass will act like a photon and is non-interacting in nature. Similarly the particle with infinite mass can hardly be physically viable. The equations of the investigated systems are compared well with the Kryloff–Bogoliuboff autonomous system [5]. The classical solution of the system for each case has been obtained analytically using He’s method where the frequency of oscillation is basically depends on amplitude. It is worth mentioning here that the frequency of oscillation (equations (25) and (28)) for both the cases are less than equal to 1 (i.e., ω 1 ) for any value of A , justifying the low frequency oscillation. We believe that the analytical approach as used in the present study is of appropriate. The existence of closed curve in classical phase portrait signifies that the system will admit discrete eigenvalues microscopically (quantum mechanically). It should be born in the mind that if the classical phase portrait is not closed, the quantum state of system will not possess discrete eigenvalue [5]. Further, in quantum mechanical study, the nature of closed phase portraits in two different cases clearly reflect the signature of mass involved in the system. It is worth mentioning here that the quantum phase portrait is the fingerprint of the PDM system reflecting the association of the nature of mass variation in the system even though the mass varies between 1 and 0.5 in both the cases. The discrete eigenvalues of the present system do not possess any similarity with that of the SHO having m = 1 . The close view reveals that the difference between any two adjacent energy levels is no longer a constant factor, i.e., E i + 1 E i E i E i 1 . This could be useful to establish a common link between classical and quantum mechanics [78].

In conclusion, the present model analysis can be extended to PDM that varies between any finite values of m ( x ) m 1 to m 2 and hence to study the dynamics of the system classically as well as quantum mechanically. We believe that the present analysis is new to the literature and the suggested method can also be used for any non-singular PDM system.

Acknowledgments

The authors sincerely thank the reviewer 1 for giving them valuable comments along with some relevant references on mass variation which helped them for the overall improvement of the manuscript. The authors Jihad Asad, Hussein Shanak, and Rabab Jarrar would like to thank Palestine Technical University – Kadoorie.

    Conflict of interest: Authors state no conflict of interest.

References

[1] David W, Simon N. Nonlinear Vibration with Control for Flexible and Adaptive Structures. Netherlands: Springer; 2010. Search in Google Scholar

[2] Rath B, Agarwalla S. Nonlinear oscillator: controlled and uncontrolled vibrations. Proc Natl Acad Sci, India, Sect A Phys Sci. 2014;84(1):83–66. Search in Google Scholar

[3] Resnick R, Halliday D. Physics, Part-I. New York: John Wiley & Sons. Inc; 1966. Search in Google Scholar

[4] Biswas SN. Classical mechanics. Kolkata: Books and Allied (P) Ltd.; 1998. Search in Google Scholar

[5] Gupta KC. Classical mechanics of particles and rigid body. New Delhi: New Age International (P) Ltd; 1997. Search in Google Scholar

[6] Oldwig von R. Position-dependent effective masses in semiconductor theory. Phys Rev. 1983;B27:7547. Search in Google Scholar

[7] Ganguly A, Kuru Ş, Negro J, Nieto L. A study of the bound states for square potential wells with position-dependent mass. Phys Lett A. 2006;360:228–33. Search in Google Scholar

[8] Silveririnha MG, Engheta N. Transformation electronics: Tailoring the effective mass of electrons. Phys Rev B. 2012;86:161104(R). Search in Google Scholar

[9] Chen Y, Yan Z, Mihalache D, Malomed BA. Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses. Sci Rep. 2017;7:1257. Search in Google Scholar

[10] Morrow RA. Establishment of an effective-mass Hamiltonian for abrupt heterojunctions. Phys Rev B. 1987;35:8074–79; Morrow RA. Effective-mass Hamiltonians for abrupt heterojunctions in three dimensions. Phys Rev B. 1987;36:4836. Search in Google Scholar

[11] de Saavedra, FA, Boronat J, Polls A, Fabrocini A. Effective mass of one He4 atom in liquid He3. Phys Rev B. 1994;50:4248–51. Search in Google Scholar

[12] Serra L, Lipparini E. Spin response of unpolarized quantum dots. Europhys Lett. 1997;40:667–72. Search in Google Scholar

[13] Lakshmanan M. On a non-linear harmonic oscillator. J Sound Vib. 1979;64:458–61. Search in Google Scholar

[14] Carinena JF, Ranada MF, Santander M. One-dimensional model of a quantum nonlinear harmonic oscillator. Rep Math Phys. 2004;54:285–93. Search in Google Scholar

[15] Flores J, Solovey G, Gil S. Variable mass oscillator. Am J Phys. 2003;71:721–55. Search in Google Scholar

[16] Bonatsos D, Georgoudis PE, Minkov N, Petrellis D, Quesne C. Bohr Hamiltonian with a deformation-dependent mass term for the Kratzer potential. Phys Rev C. 2013;88:034316. Search in Google Scholar

[17] Ovando G, Peña JJ, Morales J. Position-dependent mass Schrödinger equation for the Morse potential. J Phys: Conf Ser. 2017;792:012037. Search in Google Scholar

[18] Bethe HA. Possible explanation of the solar-neutrino puzzle. Phys Rev Lett. 1986;56:1305–8. Search in Google Scholar

[19] Falaye BJ, Serrano FA, Shi-Hai D. Fisher information for the position-dependent mass Schrödinger system. Phys Lett A. 2016;380:267–71. Search in Google Scholar

[20] Baleanu D, Jajarmi A, Sajjadi SS, Asad JH. The fractional features of a harmonic oscillator with position-dependent mass. Commun Theor Phys. 2020;72:055002. Search in Google Scholar

[21] Mustafa O. Position-dependent mass harmonic oscillator: classical-quantum mechanical correspondence and ordering-ambiguity. arXiv:1208.2109v3 [quant-ph]. Search in Google Scholar

[22] Rath B, Mallick P, Akande J, Mohapatra PP, Adjaї DKK, Koudahoun LH, et al. Asymmetric variation of a finite mass harmonic like oscillator. Proc Indian Natl Sci Acad. 2017;83:935–40. Search in Google Scholar

[23] Ruby VC, Senthilvelan M. On the construction of coherent states of position dependent mass Schrödinger equation endowed with effective potential. J Math Phys. 2010;51:052106-1–14. Search in Google Scholar

[24] Chargui Y, Dhahbi A, Trabelsi A. A novel approach for constructing kinetic energy operators with position dependent mass. Results Phys. 2019;13:102329. Search in Google Scholar

[25] Tiwari AK, Pandey SN, Santhilvelan M, Lakshmanan M. Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f(x)ẋ2 + g(x) = 0. J Math Phys. 2013;54:053506. Search in Google Scholar

[26] Lakshmanan M, Chandrasekar VK. Generating finite dimensional integrable nonlinear dynamical systems. Eur Phys J ST. 2013;222:665–88. Search in Google Scholar

[27] Musielak ZE. Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J Phys A: Math Theor. 2008;41:055205. Search in Google Scholar

[28] Mathews PM, Lakshmanan M. On a unique nonlinear oscillator. Quart Appl Math. 1974;32:215–88. Search in Google Scholar

[29] Yañez-Navarro G, Guo-Hua S, Dytrych T, Launey KD, Shi-Hai D, Draayer JP. Quantum information entropies for position-dependent mass Schrödinger problem. Ann Phys. 2014;348:153–60. Search in Google Scholar

[30] Guo-Hua S, Popov D, Camacho-Nieto O, Shi-Hai D. Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well. Chin Phys B. 2015;24:100303. Search in Google Scholar

[31] Mustafa O. PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian. Phys Lett A. 2020;384:126265. Search in Google Scholar

[32] Biswas K, Saha JP, Patra P. On the position-dependent effective mass Hamiltonian. Eur Phys J Plus. 2020;135:457. Search in Google Scholar

[33] Bravo R, Plyushchay MS. Position-dependent mass, finite-gap systems, and supersymmetry. Phys Rev D. 2016;93:105023. Search in Google Scholar

[34] Mustafa O, Algadhi Z. Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality. Eur Phys J Plus. 2019;134:228. Search in Google Scholar

[35] Quesne C. Deformed shape invariance symmetry and potentials in curved space with two known eigenstates. J Math Phys. 2018;59:042104. Search in Google Scholar

[36] Zhao FQ, Liang XX, Ban SL. Influence of the spatially dependent effective mass on bound polarons in finite parabolic quantum wells. Eur Phys J B. 2003;33:3–8. Search in Google Scholar

[37] El-Nabulsi RA. Inverse-power potentials with positive-bound energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments. Eur Phys J Plus. 2020;135:693. Search in Google Scholar

[38] El-Nabulsi RA. Scalar particle in new type of the extended uncertainty principle. Few Body Syst. 2020;61:1–10. Search in Google Scholar

[39] El-Nabulsi RA. A new approach to the Schrodinger equation with position-dependent mass and its implications in quantum dots and semiconductors. J Phys Chem Solid. 2020;140:109384. Search in Google Scholar

[40] El-Nabulsi RA. A generalized self-consistent approach to study position-dependent mass in semiconductors organic heterostructures and crystalline impure materials. Phys E: Low Dim Syst Nanostruct. 2020;124:114295. Search in Google Scholar

[41] El-Nabulsi RA. Dynamics of position-dependent mass particle in crystal lattices microstructures. Phys E: Low Dim Syst Nanostruct. 2020;127:114525. Search in Google Scholar

[42] El-Nabulsi RA. Path integral method for quantum dissipative systems with dynamical friction: applications to quantum dots/zero-dimensional nanocrystals. Superlattices Microstruct. 2020;144:106581. Search in Google Scholar

[43] El-Nabulsi RA. On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc R Soc A. 2020;476:20190729. Search in Google Scholar

[44] El-Nabulsi RA. Some implications of three generalized uncertainty principles in statistical mechanics of an ideal gas. Eur Phys J Plus. 2020;135:34. Search in Google Scholar

[45] El-Nabulsi RA. Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics. Eur Phys J Plus. 2019;134:192. Search in Google Scholar

[46] Von Roos O. Position-dependent effective masses in semiconductor theory. Phys Rev. 1983;B27:7547–52. Search in Google Scholar

[47] Yu J, Dong SH, Sun GH. Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential. Phys Lett A. 2004;322:290–77. Search in Google Scholar

[48] Dong SH, Pena JJ, Pacheco-Garcia C, Garcia-Ravelo J. Vasodilatory mechanism of levobunolol on vascular smooth muscle cells. Mod Phys Lett A. 2007;22:1039–45. Search in Google Scholar

[49] Mustafa O, Algadhi Z. Position-dependent mass charged particles in magnetic and Aharonov–Bohm flux fields: separability, exact and conditionally exact solvability. Eur Phys J Plus. 2020;135:559. Search in Google Scholar

[50] Negro J, Nieto LM. On position-dependent mass harmonic oscillators. J Phys: Conf Ser. 2008;128:012053. Search in Google Scholar

[51] Davidson M. Variable mass theories in relativistic quantum mechanics as an explanation for anomalous low energy nuclear phenomena. J Phys: Conf Ser. 2015;615:012016. Search in Google Scholar

[52] Pinto MB. Introducing the notion of bare and effective mass via Newton’s second law of motion. Eur J Phys. 2007;28:171. Search in Google Scholar

[53] Walmsley P, Putzke C, Malone L, Guillamón I, Vignolles D, Proust C, et al. Quasiparticle mass enhancement close to the quantum critical point in BaFe2(As(1−x)P(x))2. Phys Rev Lett. 2013;110:257002. Search in Google Scholar

[54] Grinenko V, Iida K, Kurth F, Efremov DV, Drechsler SL, Cherniavskii I, et al. Selective mass enhancement close to the quantum critical point in BaFe2(As1−x Px)2. Sci Rep. 2017;7:4589. Search in Google Scholar

[55] Wilkes J, Muljarov EA. Exciton effective mass enhancement in coupled quantum wells in electric and magnetic fields. N J Phys. 2016;18:023032. Search in Google Scholar

[56] Pari NAÁ, García DJ, Cornaglia PS. Quasiparticle mass enhancement as a measure of entanglement in the Kondo problem. Phys Rev Lett. 2020;125:217601. Search in Google Scholar

[57] El-Nabulsi RA. Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic carrier concentrations. J Phys Chem Solids. 2019;127:224–30. Search in Google Scholar

[58] El-Nabulsi RA. Nonlocal uncertainty and its implications in quantum mechanics at ultramicroscopic scales. Quant Stud: Math Found. 2019;6:123–33. Search in Google Scholar

[59] El-Nabulsi RA. Modeling of electrical and mesoscopic circuits at quantum nanoscale from heat momentum operator. Phys E: Low Dim Syst Nanostruct. 2018;98:90–104. Search in Google Scholar

[60] El-Nabulsi RA. Nonlocal approach to energy bands in periodic lattices and emergence of electron mass enhancement. J Phys Chem Solids. 2018;122:167–73. Search in Google Scholar

[61] El-Nabulsi RA. Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors. Eur Phys J Plus. 2018;133:394. Search in Google Scholar

[62] El-Nabulsi RA. Massive photons in magnetic materials from nonlocal quantization. J Magn Magn Mater. 2018;458:213–66. Search in Google Scholar

[63] Asad J, Mallick P, Samei ME, Rath B, Mohapatra P, Shanak H, et al. Asymmetric variation of a finite mass harmonic like oscillator. Results Phys. 2020;19:103335. Search in Google Scholar

[64] El-Nabulsi RA. A generalized nonlinear oscillator from non-standard degenerate Lagrangians and its consequent Hamiltonian formalism. Proc Natl Acad Sci, India, Sect A Phys Sci. 2014;84:563–99. Search in Google Scholar

[65] Cariñena JF, Rañada MF, Santander M, Senthilvelan M. A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators. Nonlinearity. 2004;17:1941–63. Search in Google Scholar

[66] Chandrasekar VK, Senthilvelan M, Lakshmanan M. Unusual Liénard-type nonlinear oscillator. Phys Rev E. 2005;72:066203–11. Search in Google Scholar

[67] El-Nabulsi RA. Non-standard Lagrangians with higher-order derivatives and the Hamiltonian formalism. Proc Natl Acad Sci, India, Sect A Phys Sci. 2015;85:247–52. Search in Google Scholar

[68] El-Nabulsi RA. Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Comp Appl Math. 2014;33:163–79. Search in Google Scholar

[69] El-Nabulsi RA. Non-standard Lagrangians in quantum mechanics and their relationship with attosecond laser pulse formalism. Lasers Eng (Old City Publ). 2018;40(4–6):347–74. Search in Google Scholar

[70] El-Nabulsi RA. Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J Stat Phys. 2018;172(6):1617–40. Search in Google Scholar

[71] He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B. 2006;20:1141–99. Search in Google Scholar

[72] He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int J Mod Phys B. 2008;22(21):3487–78. Search in Google Scholar

[73] Rath B. Some studies on: ancient Chinese formalism, He's frequency formulation for nonlinear oscillators and “Optimal Zero Work” method. Orissa J Phys. 2011;18(1):109. Search in Google Scholar

[74] Rath B, Mallick P, Samal PK. Real spectra of isospectral non-hermitian hamiltonians. Afr Rev Phys. 2014;9(0027):201–5. Search in Google Scholar

[75] Rath B. Iso-spectral instability of harmonic oscillator: breakdown of unbroken pseudo-hermiticity and PT symmetry condition. Afr Rev Phys. 2015;10(0051):427–34. Search in Google Scholar

[76] Chaudhuri RN, Mondal M. Hill determinant method with a variational parameter. Phys Rev A. 1989;40:6080–3. Search in Google Scholar

[77] Banerjee K, Bhatnagar SP, Choudhry V, Kanwal SS. The anharmonic oscillator. Proc R Soc Lond A. 1978;360:575–86. Search in Google Scholar

[78] Lanczewski T. Motion of a classical object with oscillating mass. arxiv:1103.3402v1 [physics.gen-ph]. Search in Google Scholar

Received: 2020-12-07
Revised: 2021-02-28
Accepted: 2021-03-14
Published Online: 2021-05-18

© 2021 Biswanath Rath et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.