Figure 1.
Surface shows the imaginary part of the 1-D wave function $ \psi \left(\xi ,\eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
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October
2021, 14(10): 3589-3610.
doi: 10.3934/dcdss.2021021
Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator
| 1. | Department of Physics, Jagan Nath University, Jaipur-303901, Rajasthan, India |
| 2. | Department of Physics, Vivekananda Global University, Jaipur-303012, Rajasthan, India |
| 3. | Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India |
| 4. | Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India |
In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motion of quantum oscillator, lattice vibration, propagation of electromagnetic waves, fluid flow, etc. The main objective of this study is to show the effectiveness of HPSTM, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results reveal that proposed scheme is a powerful tool for study large class of problems. This study shows that the results obtained by the HPSTM are accurate and effective for analysis the nonlinear behaviour of complex systems and efficient over other available analytical schemes.
Keywords:
Fractional Schrödinger equations,
quantum mechanics,
harmonic oscillator,
Sumudu transform,
HPSTM.
Mathematics Subject Classification:
Primary: 26A33, 35Q41, 35R11.
Citation:
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete and Continuous Dynamical Systems - S,
2021, 14
(10)
: 3589-3610.
doi: 10.3934/dcdss.2021021
![]()
References:
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A. K. Alomari, M. S. Noorani and R. Nazar,
Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Commun. Nonlin. Sci. Numer. Simul., 14 (2009), 1196-1207.
doi: 10.1016/j.cnsns.2008.01.008. |
| [2] |
Z. Alijani, D. Baleanu, B. Shiri and G. C. Wu, Spline collocation methods for systems of fuzzy fractional differential equations, Chaos Soliton Fract., 131 (2020), 109510, 12 pp.
doi: 10.1016/j. chaos.2019.109510. |
| [3] |
G. Amador, K. Colon, N. Luna, G. Mercado, E. Pereira and E. Suazo,
On solutions for linear and nonlinear Schrödinger equations with variable coefficients: A computational approach, Symmetry, 8 (2016), 38-54.
doi: 10.3390/sym8060038. |
| [4] |
J. Biazar and H. Ghazvini,
Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method, Phys. Lett. A, 366 (2007), 79-84.
doi: 10.1016/j.physleta.2007.01.060. |
| [5] |
A. Borhanifar and R. Abazari,
Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Optics Commun., 283 (2010), 2026-2031.
doi: 10.1016/j.optcom.2010.01.046. |
| [6] |
E. Babolian, J. Saeidian and M. Paripour,
Application of the homotopy analysis method for solving equal-width wave and modified equal-width wave equations, Z. Naturforsch, 64a (2009), 685-690.
doi: 10.1515/zna-2009-1103. |
| [7] |
F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Prob. Eng., (2003), 103-118.
doi: 10.1155/S1024123X03207018. |
| [8] |
D. Baleanu, J. H. Asad and A. Jajarmi,
New aspects of the motion of a particle in a circular cavity, Proceedings of the Romanian Academy, Series A, 19 (2018), 361-367.
|
| [9] |
D. Baleanu, S. S. Sajjadi, A. Jajarmi and J. H. Asad, New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator, Europ. Phys. J. Plus, 134 (2019), 181.
doi: 10.1140/epjp/i2019-12561-x. |
| [10] |
D. Baleanu, J. H. Asad and A. Jajarmi,
The fractional model of spring pendulum: new features within different kernels, P. Romanian Acad. A, 19 (2018), 447-454.
|
| [11] |
D. Baleanu and B. Shiri,
Collocation methods for fractional differential equations involving non-singular kernel, Chaos Soliton Fract., 116 (2018), 136-145.
doi: 10.1016/j.chaos.2018.09.020. |
| [12] |
J. Biazar, R. Ansari, K. Hosseini and P. Gholamin,
Solution of the linear and non-linear Schrödinger equations using homotopy perturbation and Adomian decomposition methods, Int. Math. Forum, 3 (2008), 1891-1897.
|
| [13] |
J. Cresser, Quantum Physics Notes, Department of Physics, Macquarie University, Australia, (2011). |
| [14] |
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 2018.
doi: 10.1017/9781316995433.
|
| [15] |
A. Goswami, J. Singh, D. Kumar and S. Gupta,
An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma, J. Ocean Eng. Sci., 4 (2019), 85-99.
doi: 10.1016/j.joes.2019.01.003. |
| [16] |
A. Goswami, J. Singh, D. Kumar and S. Rathore,
An analytical approach to the fractional Equal Width equations describing hydro-magnetic waves in cold plasma, Physica A, 524 (2019), 563-575.
doi: 10.1016/j.physa.2019.04.058. |
| [17] |
A. Goswami, Su shila, J. Singh and D. Kumar,
Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math., 5 (2020), 2346-2368.
doi: 10.3934/math.2020155. |
| [18] |
A. Goswami, J. Singh and D. Kumar,
A reliable algorithm for KdV equations arising in warm plasma, Nonlin. Eng., 5 (2016), 7-16.
doi: 10.1515/nleng-2015-0024. |
| [19] |
A. Goswami, J. Singh and D. Kumar,
Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves, Ain Shams Eng. J., 9 (2018), 2265-2273.
doi: 10.1016/j.asej.2017.03.004. |
| [20] |
A. Ghorbani and J. Saberi-Nadjafi,
He's homotopy perturbation method for calculating adomian polynomials, Int. J. Nonlin. Sci. Num. Simul., 8 (2007), 229-232.
|
| [21] |
A. Ghorbani,
Beyond Adomian polynomials: He polynomials, Chaos Soliton Fract., 39 (2009), 1486-1492.
doi: 10.1016/j.chaos.2007.06.034. |
| [22] |
J. H. He,
Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79.
doi: 10.1016/S0096-3003(01)00312-5. |
| [23] |
K. Hosseini, A. Zabihi, F. Samadani and R. Ansari, New explicit exact solutions of the unstable nonlinear Schrödinger equation using the expa and hyperbolic function methods, Opt. Quant. Electron., 50 (2018). |
| [24] |
K. Hosseini, F. Samadani, D. Kumar and M. Faridi,
New optical solitons of cubic-quartic nonlinear Schrödinger equation, Optik, 157 (2018), 1101-1105.
doi: 10.1016/j.ijleo.2017.11.124. |
| [25] |
K. Hosseini, D. Kumar, M. Kaplan and E. Y. Bejarbaneh,
New exact traveling wave solutions of the unstable nonlinear Schrödinger equations, Commun. Theor. Phys., 68 (2017), 761-767.
doi: 10.1088/0253-6102/68/6/761. |
| [26] |
E. K. Jaradat, O. Alomari, M. Abudayah and A. Al-Faqih, An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method, Adv. Math. Phys., 2018 (2018), 6765021, 11 pp.
doi: 10.1155/2018/6765021. |
| [27] |
A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Front. Phys., 26 (2019), 196.
doi: 10.3389/fphy.2019.00196. |
| [28] |
Y. Khan and Q. Wu,
Homotopy perturbation transform method for nonlinear equations using He's polynomials, Comput. Math Appl., 61 (2011), 1963-1967.
doi: 10.1016/j.camwa.2010.08.022. |
| [29] |
M. Kaplan, K. Hosseini, F. Samadani and N. Raza,
Optical soliton solutions of the cubic-quintic non-linear Schrödinger equation including an anti-cubic term, J. Moder. Opt., 65 (2018), 1431-1436.
|
| [30] |
R. I. Nuruddeen,
Elzaki decomposition method and its applications in solving linear and nonlinear Schrödinger equations, Sohag J. Math., 4 (2017), 31-35.
doi: 10.18576/sjm/040201. |
| [31] |
A. Niknam, A. A. Rajabi and M. Solaimani,
Solutions of D-dimensional Schrödinger equation for Woods-Saxon potential with spin-orbit, coulomb and centrifugal terms through a new hybrid numerical fitting Nikiforov-Uvarov method, J. Theor. Appl. Phys., 10 (2016), 53-59.
doi: 10.1007/s40094-015-0201-9. |
| [32] |
A. Sadighi and D. D. Ganji,
Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy perturbation and Adomian decomposition methods, Phys. Lett. A, 372 (2008), 465-469.
doi: 10.1016/j.physleta.2007.07.065. |
| [33] |
B. Shiri and D. Baleanu,
System of fractional differential algebraic equations with applications, Chaos Soliton Fractal, 120 (2019), 203-212.
doi: 10.1016/j.chaos.2019.01.028. |
| [34] |
J. Singh, D. Kumar and Su shila,
Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Appl. Math. Mech., 4 (2011), 165-175.
|
| [35] |
J. Singh, D. Kumar and A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using sumudu transform, Abst. Appl. Anal., 2013 (2013), 934060, 8 pp.
doi: 10.1155/2013/934060. |
| [36] |
A. Shidfar, A. Molabahrami, A. Babaei and A. Yazdanian,
A study on the d-dimensional Schrödinger equation with a power-law nonlinearity, Chaos Soliton Fract., 42 (2009), 2154-2158.
doi: 10.1016/j.chaos.2009.03.139. |
| [37] |
A. Shidfar, A. Molabahrami, A. Babaei and A. Yazdanian,
A series solution of the Cauchy problem for the generalized d-dimensional Schrödinger equation with a power-law nonlinearity, Comput. Math. Appl., 59 (2010), 1500-1508.
doi: 10.1016/j.camwa.2009.11.017. |
| [38] |
A. M. Wazwaz,
A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos Soliton Fract., 37 (2008), 1136-1142.
doi: 10.1016/j.chaos.2006.10.009. |
| [39] |
G. K. Watugala,
Sumudu transform- a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Edu. Sci. Tech., 24 (1993), 35-43.
doi: 10.1080/0020739930240105. |
| [40] |
L. Zheng, T. Wang, X. Zhang and L. Ma,
The nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances, Appl. Math. Lett., 26 (2013), 463-468.
doi: 10.1016/j.aml.2012.11.009. |
show all references
References:
| [1] |
A. K. Alomari, M. S. Noorani and R. Nazar,
Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Commun. Nonlin. Sci. Numer. Simul., 14 (2009), 1196-1207.
doi: 10.1016/j.cnsns.2008.01.008. |
| [2] |
Z. Alijani, D. Baleanu, B. Shiri and G. C. Wu, Spline collocation methods for systems of fuzzy fractional differential equations, Chaos Soliton Fract., 131 (2020), 109510, 12 pp.
doi: 10.1016/j. chaos.2019.109510. |
| [3] |
G. Amador, K. Colon, N. Luna, G. Mercado, E. Pereira and E. Suazo,
On solutions for linear and nonlinear Schrödinger equations with variable coefficients: A computational approach, Symmetry, 8 (2016), 38-54.
doi: 10.3390/sym8060038. |
| [4] |
J. Biazar and H. Ghazvini,
Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method, Phys. Lett. A, 366 (2007), 79-84.
doi: 10.1016/j.physleta.2007.01.060. |
| [5] |
A. Borhanifar and R. Abazari,
Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Optics Commun., 283 (2010), 2026-2031.
doi: 10.1016/j.optcom.2010.01.046. |
| [6] |
E. Babolian, J. Saeidian and M. Paripour,
Application of the homotopy analysis method for solving equal-width wave and modified equal-width wave equations, Z. Naturforsch, 64a (2009), 685-690.
doi: 10.1515/zna-2009-1103. |
| [7] |
F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Prob. Eng., (2003), 103-118.
doi: 10.1155/S1024123X03207018. |
| [8] |
D. Baleanu, J. H. Asad and A. Jajarmi,
New aspects of the motion of a particle in a circular cavity, Proceedings of the Romanian Academy, Series A, 19 (2018), 361-367.
|
| [9] |
D. Baleanu, S. S. Sajjadi, A. Jajarmi and J. H. Asad, New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator, Europ. Phys. J. Plus, 134 (2019), 181.
doi: 10.1140/epjp/i2019-12561-x. |
| [10] |
D. Baleanu, J. H. Asad and A. Jajarmi,
The fractional model of spring pendulum: new features within different kernels, P. Romanian Acad. A, 19 (2018), 447-454.
|
| [11] |
D. Baleanu and B. Shiri,
Collocation methods for fractional differential equations involving non-singular kernel, Chaos Soliton Fract., 116 (2018), 136-145.
doi: 10.1016/j.chaos.2018.09.020. |
| [12] |
J. Biazar, R. Ansari, K. Hosseini and P. Gholamin,
Solution of the linear and non-linear Schrödinger equations using homotopy perturbation and Adomian decomposition methods, Int. Math. Forum, 3 (2008), 1891-1897.
|
| [13] |
J. Cresser, Quantum Physics Notes, Department of Physics, Macquarie University, Australia, (2011). |
| [14] |
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 2018.
doi: 10.1017/9781316995433.
|
| [15] |
A. Goswami, J. Singh, D. Kumar and S. Gupta,
An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma, J. Ocean Eng. Sci., 4 (2019), 85-99.
doi: 10.1016/j.joes.2019.01.003. |
| [16] |
A. Goswami, J. Singh, D. Kumar and S. Rathore,
An analytical approach to the fractional Equal Width equations describing hydro-magnetic waves in cold plasma, Physica A, 524 (2019), 563-575.
doi: 10.1016/j.physa.2019.04.058. |
| [17] |
A. Goswami, Su shila, J. Singh and D. Kumar,
Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math., 5 (2020), 2346-2368.
doi: 10.3934/math.2020155. |
| [18] |
A. Goswami, J. Singh and D. Kumar,
A reliable algorithm for KdV equations arising in warm plasma, Nonlin. Eng., 5 (2016), 7-16.
doi: 10.1515/nleng-2015-0024. |
| [19] |
A. Goswami, J. Singh and D. Kumar,
Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves, Ain Shams Eng. J., 9 (2018), 2265-2273.
doi: 10.1016/j.asej.2017.03.004. |
| [20] |
A. Ghorbani and J. Saberi-Nadjafi,
He's homotopy perturbation method for calculating adomian polynomials, Int. J. Nonlin. Sci. Num. Simul., 8 (2007), 229-232.
|
| [21] |
A. Ghorbani,
Beyond Adomian polynomials: He polynomials, Chaos Soliton Fract., 39 (2009), 1486-1492.
doi: 10.1016/j.chaos.2007.06.034. |
| [22] |
J. H. He,
Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79.
doi: 10.1016/S0096-3003(01)00312-5. |
| [23] |
K. Hosseini, A. Zabihi, F. Samadani and R. Ansari, New explicit exact solutions of the unstable nonlinear Schrödinger equation using the expa and hyperbolic function methods, Opt. Quant. Electron., 50 (2018). |
| [24] |
K. Hosseini, F. Samadani, D. Kumar and M. Faridi,
New optical solitons of cubic-quartic nonlinear Schrödinger equation, Optik, 157 (2018), 1101-1105.
doi: 10.1016/j.ijleo.2017.11.124. |
| [25] |
K. Hosseini, D. Kumar, M. Kaplan and E. Y. Bejarbaneh,
New exact traveling wave solutions of the unstable nonlinear Schrödinger equations, Commun. Theor. Phys., 68 (2017), 761-767.
doi: 10.1088/0253-6102/68/6/761. |
| [26] |
E. K. Jaradat, O. Alomari, M. Abudayah and A. Al-Faqih, An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method, Adv. Math. Phys., 2018 (2018), 6765021, 11 pp.
doi: 10.1155/2018/6765021. |
| [27] |
A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Front. Phys., 26 (2019), 196.
doi: 10.3389/fphy.2019.00196. |
| [28] |
Y. Khan and Q. Wu,
Homotopy perturbation transform method for nonlinear equations using He's polynomials, Comput. Math Appl., 61 (2011), 1963-1967.
doi: 10.1016/j.camwa.2010.08.022. |
| [29] |
M. Kaplan, K. Hosseini, F. Samadani and N. Raza,
Optical soliton solutions of the cubic-quintic non-linear Schrödinger equation including an anti-cubic term, J. Moder. Opt., 65 (2018), 1431-1436.
|
| [30] |
R. I. Nuruddeen,
Elzaki decomposition method and its applications in solving linear and nonlinear Schrödinger equations, Sohag J. Math., 4 (2017), 31-35.
doi: 10.18576/sjm/040201. |
| [31] |
A. Niknam, A. A. Rajabi and M. Solaimani,
Solutions of D-dimensional Schrödinger equation for Woods-Saxon potential with spin-orbit, coulomb and centrifugal terms through a new hybrid numerical fitting Nikiforov-Uvarov method, J. Theor. Appl. Phys., 10 (2016), 53-59.
doi: 10.1007/s40094-015-0201-9. |
| [32] |
A. Sadighi and D. D. Ganji,
Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy perturbation and Adomian decomposition methods, Phys. Lett. A, 372 (2008), 465-469.
doi: 10.1016/j.physleta.2007.07.065. |
| [33] |
B. Shiri and D. Baleanu,
System of fractional differential algebraic equations with applications, Chaos Soliton Fractal, 120 (2019), 203-212.
doi: 10.1016/j.chaos.2019.01.028. |
| [34] |
J. Singh, D. Kumar and Su shila,
Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Appl. Math. Mech., 4 (2011), 165-175.
|
| [35] |
J. Singh, D. Kumar and A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using sumudu transform, Abst. Appl. Anal., 2013 (2013), 934060, 8 pp.
doi: 10.1155/2013/934060. |
| [36] |
A. Shidfar, A. Molabahrami, A. Babaei and A. Yazdanian,
A study on the d-dimensional Schrödinger equation with a power-law nonlinearity, Chaos Soliton Fract., 42 (2009), 2154-2158.
doi: 10.1016/j.chaos.2009.03.139. |
| [37] |
A. Shidfar, A. Molabahrami, A. Babaei and A. Yazdanian,
A series solution of the Cauchy problem for the generalized d-dimensional Schrödinger equation with a power-law nonlinearity, Comput. Math. Appl., 59 (2010), 1500-1508.
doi: 10.1016/j.camwa.2009.11.017. |
| [38] |
A. M. Wazwaz,
A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos Soliton Fract., 37 (2008), 1136-1142.
doi: 10.1016/j.chaos.2006.10.009. |
| [39] |
G. K. Watugala,
Sumudu transform- a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Edu. Sci. Tech., 24 (1993), 35-43.
doi: 10.1080/0020739930240105. |
| [40] |
L. Zheng, T. Wang, X. Zhang and L. Ma,
The nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances, Appl. Math. Lett., 26 (2013), 463-468.
doi: 10.1016/j.aml.2012.11.009. |
Figure 2.
Comparison of profile of imaginary part of $ \psi \left(\xi ,\eta \right) $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
Figure 3.
Surface shows the real part of the 1-D wave function $ \psi \left(\xi ,\eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
Figure 4.
Comparison of profile of real part of $ \psi \left(\xi ,\eta \right) $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
Figure 5.
Surface shows the imaginary part of the 2-D wave function $ \psi \left(\xi ,\zeta, \eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
Figure 6.
Comparison of profile of imaginary part of $ \psi \left(\xi ,\zeta,\eta \right) $ for $ \zeta = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
Figure 7.
Surface shows the real part of the 2-D wave function $ \psi \left(\xi ,\zeta, \eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
Figure 8.
Comparison of profile of real part of $ \psi \left(\xi ,\zeta,\eta \right) $ for $ \zeta = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
Figure 9.
Surface shows the imaginary part of the 3-D wave function $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
Figure 10.
Comparison of profile of imaginary part of $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ for $ \zeta = \chi = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
Figure 11.
Surface shows the real part of the 3-D wave function $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ at (a) $ \alpha = 1 $ , (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $
Figure 12.
Comparison of profile of real part of $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ for $ \zeta = \chi = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $
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