Asymmetric variation of a finite mass harmonic like oscillator
Jihad Asad, P. Mallick, B. Rath, M. E. Samei, Prachiparava Mohapatra, Hussein Shanak, Rabab Jarrar
11 Harmonic oscillator with finite mass variation under asymmetry condition
Jihad Asad , P. Mallick
2, ** , M.E. Samei , B. Rath
2, , Hussein Shanak , Rabab Jarrar Department of Physics, College of Applied Sciences, Palestine Technical University- PTUK, Tulkarm, Palestine ([email protected]; [email protected]; [email protected]) Department of Physics, North Orissa University, Takatpur, Baripada-757003, Odisha, India ([email protected]; [email protected]) Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran ([email protected]; [email protected])
Abstract:
Classical as well as quantum mechanical analysis have been carried out on harmonic oscillator with asymmetric position dependent mass. Classical and quantum phase space analysis are reflected for a plausible understanding of the subject.
PACS no : 03.65.Ge
Keywords: position dependent mass, phase space, classical, quantum, asymmetric dependence.
Corresponding author: * [email protected]; ** [email protected]; [email protected] Note : The first two authors contributed equally.
I. Introduction
Position dependent mass (PDM) has drawn considerable attention due to their applications in semiconductor physics [1-4], quantum dots [5], quantum liquids [6], nonlinear oscillators [7-9] etc. However, one needs to take proper care in defining the kinetic energy operator in PDM due to noncommutivity of momentum and position and the same can be avoided if one chooses suitable combination of position and momentum at classical level [3]. The authors [3] suggested the PDM is of the form )( bxax mexm . (1) The PDM stated above is of asymmetric in nature. In fact, it has been seen that mass is the asymmetric function of position ( x ) in semiconductor physics. If a and b are negative then the )( xm (Eq. (1)) will be 0 at large value of x . In other words a massive particle undergoes a drastic change and becomes a photon, which can hardly interact with any matter. Secondly, if the mass behaves as )1()( xmxm . (2) Then at large distance mass becomes infinite. If the mass becomes large while moving with distance then it becomes meaningless to discuss physics behind an infinite mass. In order to avoid this ambiguity, we choose the PDM is of the form xx emxm (3) where the parameter is of weak in nature. The purpose of considering this mass is that asymmetry in variation of mass and is confined within two finite limits such as (i) m and (ii) m (Fig. 1). Hence considering the above facts, we confined our focus on finite mass. Now we feel to introduce this simple asymmetry mass in Harmonic oscillator and study the system both classically and quantum mechanically. Fig. 1:
The variation of the mass (Eq. (1)) with position (x) with m . II. Classical Analysis
The Hamiltonian of the oscillator considered here is xxmxmpH x . (4) The Lagrangian is related to the Hamiltonian as: i ii LqpH . (5) In our case we have just one generalized coordinate ( ) x , and one generalized momenta ( ) x p . Therefore we have LxpH x ; Lxpxxmxmp xx . (6) On solving Eq. (6), one can find the expression of Lagrangian ( L ) as xxmxmpxpL xx . (7) Using )( xmpxpHdtdx xx , and )( xmxp x . Thus Eq. (7) becomes: )(2 )( xxxmL . (8) Our next step is deriving the equation of motion (i.e, known as: Euler Lagrange Equation). Substituting Eq. (8) into the following relation ii qLdtdqL with i q x , we got: dtxdmxxmxdxxdmxxxxm . (9) Substituting the expression of mass (Eq. (1)) into Eq. (9), we get the following equation of motion, which in a simplified form can be written as
01 212 xxxx ee xxxxx . (10) II.1 Analytical Study:
In order to solve the above equation analytically (Eq. (10)), we follow the He’s frequency formulation on ancient Chinese method [10-12] to get the frequency of oscillation using the following condition: (0) (0)(0) (0)
R RR R . (11) In the above formalism, the boundary condition for the solution )( tx will be tx and tx . We therefore considered tAtx cos)( (12) and following the He’s formalism [10-12] as stated above, we get the frequency of oscillation as AAAA ee AA . (13) The variation of ( ) x t vs time ( t ), ( ) p t vs t and trajectory of phase space ( p vs x ) are shown in Figure 2, 3 and 4 respectively. In classical analysis of phase space, we notice the clear asymmetry behaviour. Fig. 2:
Variation of ( ) x t with time ( t ) for different amplitude. Fig. 3:
Variation of ( ) p t with time ( t )for different amplitude. Fig. 4:
Trajectory of classical phase space obtained analytically.
II.2 Numerical Study:
In order to get better information on the classical dynamics of the system (Eq. (10)), the numerical study on the said system is under taken. We can rewrite the Eq. (10) as xQxxQx (14) where
12 211 2121 xxxxxx e xee xQ (15) Here, we also use the initial condition tx and tx for the values of = 0.1. Using the algorithm ( Appendix-A ), we find the numerical solution of Eq. (14). The variation of x and p with respect to time t and phase space trajectory ( p with respect to x ) obtained numerically for = 0.1 are shown in Figure 5 and 6 respectively. Figure 5: Variation of x and p obtained numerically for = 0.1 with respect to time t . Figure 6: Variation of p obtained numerically for = 0.1 with respect to x . III. Quantum Mechanical Analysis
Now we analyze the above problem quantum mechanically. In quantum analysis, we solve the eigenvalue relation [13-15] as n . (16) Where n nn A (17) and n satisfies the eigenvalue relation for position independent harmonic oscillator as nnn nxpH )12( . (18) Using the above formalism [13-15], we get the recurrence relation as ,.....6,4,2 knknnk nknkn ARASAP . (19) Where knHnP kn . (20) knHnR kn . (21) EnHnS n . (22) The eigenvalues of the Harmonic oscillator Hamiltonian (Eq. (4)) with position dependent mass (Eq. (1)) are obtained following above mentioned procedure for and m (Table 1). In this case, we also plot quantum mechanical phase space (Fig. 7) considering HE of respective states. Comparing the trajectory of quantum phase space (Fig. 7) with that of classical phase space obtained both analytically (Fig. 4)and numerically (Fig. 6), we noticed a great similarity between classical and quantum. Fig. 7:
Trajectory of quantum phase space.
Table 1:
Eigenvalues of Asymmetry mass Harmonic oscillator. Quantum number (n) Computed Eigenvalues 0 0.5 1 1.2789 2 2.2610 3 3.3735 4 4.5412
IV. Conclusion
We discuss about the finite mass variation under asymmetric condition. Interested reader will see that mass of the system varied asymmetrically between two finite values. Further, classical phase space nature remains practically the same as 0 quantum phase space. We believe that apart from theoretical analysis, the present model could be suitable for semiconductor physics where asymmetry is an important factor.
Acknowledgment:
The authors (
Jihad Asad, Hussein Shanak, and Rabab Jarrar ) would like to thank Palestine Technical University.
The author (
M.E. Samei ) was supported by Bu-Ali Sina University.
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Appendix-A : The proposed method for to find the solution for Eq. (14) using MATLAB 1 function [MatrixEq]= SolveMainEq(lambda, x0, xprime0, m0, tmin, tmax) 2 % gamma : [gamma]; 3 % Eq1(r, c): results of main equation ; 4 [xlambda, ylambda]=size(lambda); 5 column=2; 6 for j=1:ylambda 7 f1=@(t, x)[x(2); (1+2 _ lambda(j) _ x(1)) _ (x(2))^2/(2 _ (1+exp(x(1) + ... lambda(j) _ (x(1))^2))) + ... (1+2 _ lambda(j) _ x(1)) _ (x(1))^2/(2 _ (1+exp(x(1) + ... lambda(j) _ (x(1))^2))) - x(1)]; 8 [tr, xr]=ode23(f1, [tmin tmax], [x0, xprime0]); 9 maxrow=max(size(tr)); 10 for k=1:maxrow 11 MatrixEq(k, 1)=k; 12 MatrixEq(k, column)= tr(k); 13 MatrixEq(k, column+1)= round(xr(k, 1),6); 14 MatrixEq(k, column+2)= round(xr(k, 2),6); 15 MatrixEq(k, column+3)= round( (1+2 _ lambda(j) _ xr(k, 1))/(2 _ (1+ ... exp(xr(k, 1)+ lambda(j) _ (xr(k, 1))^2))),6); 16 mx = round(m0/(1+ exp((-1) _ xr(k, 1) - lambda(j) _ (xr(k, 1))^2)),6); 17 MatrixEq(k, column+4)=mx; 18 px = round(mx _ xr(k, 2),6); 19 MatrixEq(k, column+5)=px; 20 MatrixEq(k, column+6)=round( px^2/(2 _ mx) + mx _ xr(k, 1)/2,6); 21 end; 22 column=column+7; 23 end; 24 end Table A1 : Numerical results of Eq. 14 for and t . Sl. No. t x x )( xQ )( xm x p
1 0 1 0 0.14984 0.75026 0 2 9.41004E-5 1 -8E-5 0.14984 0.75026 -6E-5 3 5.64602E-4 1 -4.8E-4 0.14984 0.75026 -3.6E-4 4 0.00292 1 -0.00248 0.14984 0.75026 -0.00186 5 0.01468 0.99991 -0.01248 0.14985 0.75024 -0.00936 6 0.07349 0.9977 -0.06242 0.1501 0.74974 -0.0468 7 0.19277 0.98425 -0.16281 0.15158 0.74671 -0.12157 8 0.35773 0.94619 -0.29727 0.15578 0.73802 -0.21939 9 0.56268 0.86908 -0.45197 0.16432 0.72003 -0.32543 10 0.8076 0.73825 -0.61014 0.17879 0.68842 -0.42003 11 1.10091 0.53734 -0.7489 0.20051 0.63789 -0.47772 12 1.38849 0.30989 -0.82162 0.22344 0.5792 -0.47588 13 1.64128 0.10004 -0.83008 0.24213 0.52524 -0.43599 14 1.78967 -0.02192 -0.81097 0.25163 0.49453 -0.40105 15 1.93806 -0.13977 -0.77465 0.25973 0.4656 -0.36068 16 2.10593 -0.26498 -0.714 0.26712 0.43587 -0.31121 17 2.32048 -0.40736 -0.60902 0.27394 0.40353 -0.24576 18 2.58348 -0.54654 -0.44499 0.27895 0.37363 -0.16626 19 2.86371 -0.64272 -0.23898 0.28147 0.35402 -0.08461 20 3.10574 -0.67728 -0.04656 0.28219 0.34719 -0.01616 21 3.25751 -0.67501 0.07604 0.28215 0.34764 0.02644 22 3.40929 -0.65423 0.197 0.28172 0.35173 0.06929 23 3.59514 -0.60423 0.33906 0.28055 0.36176 0.12266 24 3.82111 -0.50932 0.4966 0.27777 0.38145 0.18943 25 4.08596 -0.35637 0.65049 0.27169 0.41492 0.2699 26 4.35082 -0.16817 0.76142 0.26152 0.45876 0.34931 27 4.52112 -0.03435 0.80632 0.25253 0.49144 0.39626 28 4.69141 0.10522 0.82871 0.2417 0.52656 0.43636 29 4.83679 0.22596 0.82932 0.23124 0.55751 0.46235 30 5.0241 0.37944 0.80462 0.21668 0.59721 0.48052 31 5.24693 0.55211 0.7385 0.19895 0.64166 0.47387 32 5.50401 0.7272 0.61608 0.18001 0.6857 0.42244 33 5.7995 0.88195 0.42394 0.16289 0.72307 0.30654 34 6.07287 0.96924 0.21135 0.15323 0.7433 0.1571 35 6.2513 0.99372 0.06257 0.15054 0.74885 0.04685 36 6.3851 0.99451 -0.05066 0.15045 0.74902 -0.03794 37 6.51889 0.98019 -0.16292 0.15202 0.74579 -0.1215 38 6.68406 0.94209 -0.297 0.15623 0.73708 -0.21891 39 6.88928 0.86499 -0.45123 0.16477 0.71906 -0.32446 40 7.1345 0.73425 -0.60881 0.17923 0.68744 -0.41852 41 7.42818 0.53357 -0.74691 0.20091 0.63692 -0.47572 42 7.71554 0.30698 -0.81893 0.22372 0.57845 -0.47371 3 43 7.96804 0.0981 -0.82707 0.24229 0.52475 -0.434 44 8.11563 -0.02277 -0.80802 0.25169 0.49432 -0.39942 45 8.26323 -0.13957 -0.77202 0.25972 0.46565 -0.35949 46 8.43119 -0.26442 -0.71157 0.26709 0.436 -0.31024 47 8.64579 -0.40635 -0.60695 0.27389 0.40376 -0.24506 48 8.90881 -0.54508 -0.44347 0.27891 0.37394 -0.16583 49 9.189 -0.64091 -0.23819 0.28143 0.35438 -0.08441 50 9.43098 -0.67535 -0.04643 0.28215 0.34757 -0.01614 51 9.58271 -0.6731 0.07572 0.28211 0.34801 0.02635 52 9.73444 -0.65241 0.19624 0.28168 0.35209 0.0691 53 9.92024 -0.60261 0.3378 0.28051 0.36209 0.12231 54 10 -0.57335 0.39547 0.27973 0.36808 0.145571 0 1 0 0.14984 0.75026 0 2 9.41004E-5 1 -8E-5 0.14984 0.75026 -6E-5 3 5.64602E-4 1 -4.8E-4 0.14984 0.75026 -3.6E-4 4 0.00292 1 -0.00248 0.14984 0.75026 -0.00186 5 0.01468 0.99991 -0.01248 0.14985 0.75024 -0.00936 6 0.07349 0.9977 -0.06242 0.1501 0.74974 -0.0468 7 0.19277 0.98425 -0.16281 0.15158 0.74671 -0.12157 8 0.35773 0.94619 -0.29727 0.15578 0.73802 -0.21939 9 0.56268 0.86908 -0.45197 0.16432 0.72003 -0.32543 10 0.8076 0.73825 -0.61014 0.17879 0.68842 -0.42003 11 1.10091 0.53734 -0.7489 0.20051 0.63789 -0.47772 12 1.38849 0.30989 -0.82162 0.22344 0.5792 -0.47588 13 1.64128 0.10004 -0.83008 0.24213 0.52524 -0.43599 14 1.78967 -0.02192 -0.81097 0.25163 0.49453 -0.40105 15 1.93806 -0.13977 -0.77465 0.25973 0.4656 -0.36068 16 2.10593 -0.26498 -0.714 0.26712 0.43587 -0.31121 17 2.32048 -0.40736 -0.60902 0.27394 0.40353 -0.24576 18 2.58348 -0.54654 -0.44499 0.27895 0.37363 -0.16626 19 2.86371 -0.64272 -0.23898 0.28147 0.35402 -0.08461 20 3.10574 -0.67728 -0.04656 0.28219 0.34719 -0.01616 21 3.25751 -0.67501 0.07604 0.28215 0.34764 0.02644 22 3.40929 -0.65423 0.197 0.28172 0.35173 0.06929 23 3.59514 -0.60423 0.33906 0.28055 0.36176 0.12266 24 3.82111 -0.50932 0.4966 0.27777 0.38145 0.18943 25 4.08596 -0.35637 0.65049 0.27169 0.41492 0.2699 26 4.35082 -0.16817 0.76142 0.26152 0.45876 0.34931 27 4.52112 -0.03435 0.80632 0.25253 0.49144 0.39626 28 4.69141 0.10522 0.82871 0.2417 0.52656 0.43636 29 4.83679 0.22596 0.82932 0.23124 0.55751 0.46235 30 5.0241 0.37944 0.80462 0.21668 0.59721 0.48052 31 5.24693 0.55211 0.7385 0.19895 0.64166 0.47387 32 5.50401 0.7272 0.61608 0.18001 0.6857 0.42244 33 5.7995 0.88195 0.42394 0.16289 0.72307 0.30654 34 6.07287 0.96924 0.21135 0.15323 0.7433 0.1571 35 6.2513 0.99372 0.06257 0.15054 0.74885 0.04685 36 6.3851 0.99451 -0.05066 0.15045 0.74902 -0.03794 37 6.51889 0.98019 -0.16292 0.15202 0.74579 -0.1215 38 6.68406 0.94209 -0.297 0.15623 0.73708 -0.21891 39 6.88928 0.86499 -0.45123 0.16477 0.71906 -0.32446 40 7.1345 0.73425 -0.60881 0.17923 0.68744 -0.41852 41 7.42818 0.53357 -0.74691 0.20091 0.63692 -0.47572 42 7.71554 0.30698 -0.81893 0.22372 0.57845 -0.47371 3 43 7.96804 0.0981 -0.82707 0.24229 0.52475 -0.434 44 8.11563 -0.02277 -0.80802 0.25169 0.49432 -0.39942 45 8.26323 -0.13957 -0.77202 0.25972 0.46565 -0.35949 46 8.43119 -0.26442 -0.71157 0.26709 0.436 -0.31024 47 8.64579 -0.40635 -0.60695 0.27389 0.40376 -0.24506 48 8.90881 -0.54508 -0.44347 0.27891 0.37394 -0.16583 49 9.189 -0.64091 -0.23819 0.28143 0.35438 -0.08441 50 9.43098 -0.67535 -0.04643 0.28215 0.34757 -0.01614 51 9.58271 -0.6731 0.07572 0.28211 0.34801 0.02635 52 9.73444 -0.65241 0.19624 0.28168 0.35209 0.0691 53 9.92024 -0.60261 0.3378 0.28051 0.36209 0.12231 54 10 -0.57335 0.39547 0.27973 0.36808 0.14557