EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 238-248 ISSN 1307-5543 – ejpam.com Published by New York Business Global Investigation of the Infinite Discrete Levels from Finite discrete Levels in Position Dependent Mass Quantum Systems Biswanath Rath1, Jihad Asad2,∗, Hussein Shanak2, Rabab Jarrar2, Mohammed K. A. Kaabar3 1 Department of Physics, Maharaja Sriram Chandra Bhanj Deo University, Takatpur, Baripada 757003, Odisha, India 2 Department of Physics,Faculty of Applied Sciences, Palestine Technical University-Kadoorie, Tulkarm, Palestine 3 Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia Abstract. We find quantum systems having finite distinct discrete energy levels can reflect infinite distinct energy levels under suitable form of position dependent mass systems. A model example has been investigated considering the fractional Harmonic Oscillator [6].In addition to this,we also notice the same behaviour in other quantum models. In all the cases, we find finite distinct discrete levels becoming infinite distinct discrete levels under position dependent mass quantum systems without the change in the potential energy. 2020 Mathematics Subject Classifications: 35J10, 34L40, 11C20, 15B57 Key Words and Phrases: Infinite levels, Position dependent mass,Matrix diagonalisation method, Fractional potentials, Exponential potentials, von Roos model 1. Introduction Bounded quantum structures are associated with actual strength levels [2]—. These systems can have infinite or finite distinct discrete levels. A model infinite quantum system is [6]: h = p2 + λx2 + x2 (1 + gx2) (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4190 Email addresses: biswanathrath10@gmail.com (B. Rath), j.asad@ptuk.edu.ps (J. Asad), h.shanak@ptuk.edu.ps (H. Shanak), r.jarrar@ptuk.edu.ps (R. Jarrar), mohammed.kaabar@wsu.edu (M. K. A. Kaabar) http://www.ejpam.com 238 © 2022 EJPAM All rights reserved. J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 239 This systems reflect infinite levels as long as λ ≫ 0. However a drastic change in energy levels is noticed in the limit λ = 0. Under this condition model Hamiltonian becomes H = p2 + x2 (1 + x2) (2) The model potential satisfies the condition V (x) = x2 (1+x2) ≪ 1 ,which is basically the reason for ”distinct discrete levels”(DDL). This mannequin has been studied detail by Rath and Kaabar [11]. However, we consider here a slightly modified DDL as: H1 = p2 + x2 (1 + 2x2) (3) which is recast as = [p2 +M(x)x2] (4) where M(x) = 1 (1+2x2) . Energy levels of this oscillator has been reflected in Fig. 1. Below, we focus our attention on new models without the change in potential energy as follows. 0 2 4 6 8 10 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 + M x 2 ; M=1/(1 + 2x 2 ) Im a x is Figure 1: Energy levels of H = [p2 +Mx2]; withM = 1 (1+2x2) J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 240 2. Position dependent mass system without potential change At the outset, we would like to state that all our equations are valid subject to validity of commutation relation [x, p] = i (5) HPDM = p2/M(x) +M(x)x2N = p2 M + V (x) (6) Hence, the corresponding M(x) can be written as M(x) = M = V (x) x2N N = 1, 2, 3... (7) and so on. In fact, for N=1 model Hamiltonian was also previously proposed by Mathews and Lakshnman for the study of nonlinear analysis in view of its potential applications in quantum field theory [4]. As such the Hamiltonian is not self-adjoint(HPDM ̸= H† PDM ) [1, 3, 7, 9, 13–15]. In its self adjoint form, it is expressed as [15]: HPDMS = H(2) = [− 1 M1/4 ∂x 1 M1/2 ∂x 1 M1/4 +M(x)x2] (8) Hence the resultant Hamiltonian becomes von Roos model Hamiltonian [13, 14]. Energy levels of this Hamiltonian are reflected in Fig. 2. In subsequent applications of von Roos model we use p2/M = T = [− 1 M1/4 ∂x 1 M1/2 ∂x 1 M1/4 ] (9) Hence in all the PDM calculations, the corresponding Hamiltonian is self-adjoint in nature(T = T †). Apart from this we present a few models as Case-I: Trivial exponential model H(3) = p2 + 10(1− exp(−x2))) (10) Hence, the corresponding PDM operator can be written as H (4) PDM = H(4) = p2 M + 10(1− exp(−x2)); withM = 10 [1− exp(−x2)] x2 (11) The respective energy levels are reflected in Fig. 3 and Fig. 4, respectively. Similarly, we select a model PDM Hamiltonian as H(5) = p2 M + 10(1− exp(−x4)); withM = 10[1− exp(−x4))] x4 (12) The respective energy levels are reflected in Fig. 5. One can also extend this approach to H(6) = p2 M + 10(1− exp(−x6)); withM = 10[1− exp(−x6))] x6 (13) J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 241 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M + M x 2 ; M=1/(1 + 2x 2 ) Im a x is Figure 2: Energy levels of H = [p2/M +Mx2]; withM = 1 (1+2x2) The respective energy levels are reflected in Fig. 6. Case-II:Non-trivial exponential model Here, we consider the model Hamiltonian as H(7) = p2 + (1− exp(−x2)) (1 + exp(−x2)) (14) whose energy levels has finite distinct discrete levels as seen in Fig. 7. The corresponding PDM operators as H(8) = p2 M + (1− exp(−x2)) (1 + exp(−x2)) ; withM (1− exp(−x2)) x2(1 + exp(−x2)) (15) whose energy levels are reflected in Fig. 8. Apart from this we also consider similar cases as J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 242 0 2 4 6 8 10 12 14 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 First thirty energy levels:H=[p 2 + M x 2 ] Im a x is Figure 3: Energy levels of H = p2 +Mx2; withM = 10 (1−exp(−x2) x2 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M +(1−exp(−x 2 ))/(1+exp(−x 2 );M=(1−exp(−x 2 ))/x 2 (1+exp(−x 2 ) Im a x is Figure 8: Energy levels ofH = p2/M +Mx2; M = (1−exp(−x2) x2(1+exp−x2) J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 243 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M+10(1−exp(−x 2 ));M=10(1−exp(−x 2 ))/x 2 Im a x is Figure 4: Energy levels of H = p2/M +Mx2; withM = 10 (1−exp(−x2) x2 H(9) = p2 M + (1− exp(−x2)) (1 + exp(−x2)) ; withM (1− exp(−x2)) x4(1 + exp(−x2)) (16) and H(10) = p2 M + (1− exp(−x2)) (1 + exp(−x2)) ; withM (1− exp(−x2)) x6(1 + exp(−x2)) (17) Here, the energy levels are reflected in Figs. 9,10, respectively. These models are non- trivial because of the potential nature. The finite levels are due to the validity of condition V (x) ≪ 1. J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 244 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M+10(1−exp(−x 2 ));M=10(1−exp(−x 2 ))/x 4 Im a x is Figure 5: Energy levels of H = p2/M +Mx4; withM = 10 (1−exp(−x2) x4 0 10 20 30 40 50 60 70 80 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M +(1−exp(−x 2 ))/(1+exp(−x 2 );M=(1−exp(−x 2 ))/x 4 (1+exp(−x 2 ) Im a x is Figure 9: Energy levels ofH = p2/M +Mx4; M = (1−exp(−x2) x4(1+exp−x2) J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 245 0 20 40 60 80 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M+10(1−exp(−x 2 )); M=10(1−exp(−x 2 )/x 6 ) Im a x is Figure 6: Energy levels of H = p2/M +Mx6; with M = 10 (1−exp(−x2) x6 0 5 10 15 20 25 30 35 40 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 /M +(1−exp(−x 2 ))/(1+exp(−x 2 );M=(1−exp(−x 2 ))/x 6 (1+exp(−x 2 ) Im a x is Figure 10: Energy levels ofH = p2/M +Mx6; M = (1−exp(−x2) x6(1+exp−x2) J. Asad et al. / Eur. J. Pure Appl. Math, 15 (1) (2022), 238-248 246 0 2 4 6 8 10 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real axis:Spectra of H=p 2 +(1−exp(−x 2 ))/(1+exp(−x 2 ) Im a x is Figure 7: Energy levels of H = p2 + (1−exp−x2) (1+exp−x2) 3. Method of calculation In order to calculate the energy levels of above Hamiltonians, we use matrix diago- nalisation method [10, 12] to mirror convergent energy levels, on solving the eigenvalue relation H|Ψ >= E|Ψ > (18) with |Ψ >= ∑ m Am|m > (19) where |m > satisfies the relation [H0 = p2 + x2]|m >= (2m+ 1)|m > (20) 4. Conclusion In this paper we have shown that DDL quantum systems can reflect infinite distinct discrete energy levels under suitable form of position dependent mass. The model mass REFERENCES 247 Table 1 : Fractional Harmonic Oscillator and energy levels comparison n g Present Previous [6] 0 0.1 1.380 531 8 1.380 53 1 4.079 883 0 4.079 8 2 6.667 919 1 6.667 3 9.166 567 4 0 1 1.232 350 7 1.232 35 1 3.507 388 3 3.507 38 2 5.589 778 9 5.589 77 7.648 201 2 was actually proposed by Cruz et al. [15] and used by others [1, 3, 7, 9]. It is worth mentioning that only this typical form of T can be derived [8]. No other form of von Roos model Hamiltonian [13, 14] has that advantage. In other words, all other forms can only be based on approximation [5]. This universality feature has been exploited in the above model Hamiltonians. All figures have plotted from the respective numerical calculation of energy levels. It should be borne in mind that all the figures on PDM actually we have plotted with T i.e p2 M = T . In order to convince the reader about our method [1, 3], we consider two simple models as H (1) Mitra = p2 + x2 + x2 (1 + 0.1x2) (21) and H (2) Mitra = p2 + x2 + x2 (1 + x2) (22) and compare the present result with that of earlier computation by Mitra [6] in Table 1. Summarizing the above, we conclude that FDL( Finite Discrete Levels) =⇒ IDLPDM(Infinite Discrete Levels in Position Dependent Mass ) in quantum operators. 5. Acknowledgements The authors Rabab Jarrar, Hussein Shanak, Jihad Asad would like to thank Palestine Technical university- Kadoorie for funding this work. 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