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.
The study of non-linear vibration has become important in designing the flexible structures associated with aircraft, bridge, satellite, etc. . This study can also be extended to acoustics, biology , and other branches of engineering such as electronics, robotics, and mechatronics . 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]
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 , molecular physics , neutrino mass oscillations , and quantum information . 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].
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
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 . The increased effective mass can be used on the basis of fluid to explain some phenomena in quantum field theory . 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 . Enhancement of mass in the quasiparticle led to the entanglement in Kondo problem . In a similar manner, the enhancements of energy of electrons in quasicrystal , in hydrogen atom , and in quantum LC circuit  have also been reported. In addition, the enhancement of mass of electron in periodic lattice using nonlocal approach , implication of PDM for semiconductor as well as molecular physics , generation of massive photon in a magnetic material , 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 . Considering the above literature, we propose a new model position-dependent 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 and one generalized momentum . Therefore, we have
Solving equation (10), one can find the Lagrangian ( ) as
Using , and , equation (11) becomes
Substituting equation (12) into the relation with , we obtained:
2.1 Decreasing PDM
Let the mass be specified as
Figure 1 shows the variation in as stated above with for and 0.1, 0.5, and 1. In this case, the is decreasing from 1 and progressively saturating to 0.5 as we move away from origin. In other words, the varies between two finite limits, i.e., 1 and 0.5. Further, the nature of variation in is symmetric with respect to origin of the co-ordinate system. It is worth mentioning here that the value of constant parameter ( ) plays the major role in the variation in . For example: one will see that the decreases very fast for 1, whereas the same decreases very slow for 0.1. Interested reader will see that the variation in will look like a delta function for 1. However, the variation in in all the cases is confined between two limits, i.e., between 1 and 0.5.
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 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 , the above equation (16b and 16c) will be reduced to
2.1.1 Analytical calculation on decreasing PDM
Let us rewrite the equation of motion (equation 16(b)) as
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 for the system using the He’s approach [63,71,72,73], one needs to replace and in the final expression of equation 24(a). Based on our initial boundary condition (i.e., and ), the two equations (16b and 16c) would effectively give rise to same expression of frequency as
Figure 2 showed the variation in frequency ( ) with respect to amplitude (A). Initially, the 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 , the also varies between two finite values. The analytical results on the variation in with respect to t, with respect to t, and trajectory of phase space ( vs x) for 0.1, 0.5, and 1 with are shown in Figure 3.
2.2 Increasing PDM
Here, we consider an increasing position-dependent mass, which is expressed as
Figure 4 shows the variation in as stated in (equation (26)) with for and 0.1, 0.5, and 1. In this case, the 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 , i.e., the variation in looks like a volcanic type feature around small values of .
2.2.1 Analytical calculation on increasing PDM
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 with respect to , with respect to , and trajectory of phase space ( vs ) for and 0.1, 0.5, and 1 with 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 values (Figure 5(c)) for increasing mass case.
3 Quantum mechanical study
Here, satisfies the exact eigenvalue relation as
Using the above-mentioned procedure, one can get the recursion relation satisfied by as
The energy eigenvalues of the Hamiltonian (equation (8)) with PDM (equation (14)) are obtained following the above-mentioned procedure for different values of and (Table 1). We also calculated the energy eigenvalues of the Hamiltonian (equation (8)) with PDM (equation (26)) with different values of and (Table 2). It is to be noted here that the Hamiltonian is used as
|Energy level ( )||Eigenvalue ( )|
|Energy level ( )||Eigenvalue ( )|
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 and (Figure 6) and (ii) with PDM (equation (26)) with different values of and (Figure 7) considering 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 and , respectively. This feature could be because of the inherent property associated with corresponding mass.
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, 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 . 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., ) for any value of , 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 . 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 . The close view reveals that the difference between any two adjacent energy levels is no longer a constant factor, i.e., . This could be useful to establish a common link between classical and quantum mechanics .
In conclusion, the present model analysis can be extended to PDM that varies between any finite values of to 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.
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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