Abstract
We formulated the oscillators with positiondependent 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 nonlinear 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 nonlinear control vibration. For the purpose, one needs to consider the simple harmonic oscillator (SHO) with Hamiltonian [3,4]
In recent years, systems with positiondependent 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 massdependent oscillators having
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
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
A similar type of mass of the form [29,30]
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 BaFe_{2}(As_{1−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 positiondependent finite mass variation, i.e.
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
The Lagrangian is related to the Hamiltonian as
In our case, we have just one generalized coordinate
Solving equation (10), one can find the Lagrangian (
Using
Substituting equation (12) into the relation
2.1 Decreasing PDM
Let the mass be specified as
Figure 1 shows the variation in
Figure 1
Substituting equation (14) in equation (13), we have
The equation (15) can further be simplified to
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 selfcontrolled vibration. It is worth mentioning here that one can also be able to derive equation (16b and 16c) following the work of ElNabulsi and others [64,65,66,67,68,69,70]. When the value of
2.1.1 Analytical calculation on decreasing PDM
Let us rewrite the equation of motion (equation 16(b)) as
The equation (18) has the similarity with the Kryloff–Bogoliuboff autonomous system, which is expressed as [5]
It should be stated here that
In our analytical approach, we assume a similar type of solution as
To obtain the expression of the value of
Figure 2 showed the variation in frequency (
Figure 2
Figure 3
2.2 Increasing PDM
Here, we consider an increasing positiondependent mass, which is expressed as
Figure 4 shows the variation in
Figure 4
2.2.1 Analytical calculation on increasing PDM
Substituting equation (26) in equation (13) and simplifying, we have
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
The analytical results on the variation in
Figure 5
3 Quantum mechanical study
In this case, we solve the eigenvalue relation using matrix diagonalization method [22,63,74,75,76,77] as
Here,
Using the abovementioned procedure, one can get the recursion relation satisfied by
The energy eigenvalues of the Hamiltonian (equation (8)) with PDM (equation (14)) are obtained following the abovementioned procedure for different values of
Energy level (


Eigenvalue (


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 
Energy level (


Eigenvalue (


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
Figure 6
Figure 7
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,
In conclusion, the present model analysis can be extended to PDM that varies between any finite values of
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.
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