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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

* Corresponding author: Sushila

Received  January 2020 Revised  April 2020 Published  October 2021 Early access  March 2021

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.

Citation: Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3589-3610. doi: 10.3934/dcdss.2021021
References:
[1]

A. K. AlomariM. 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.  Google Scholar

[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.  Google Scholar

[3]

G. AmadorK. ColonN. LunaG. MercadoE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. BabolianJ. 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.  Google Scholar

[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.  Google Scholar

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D. BaleanuJ. 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.   Google Scholar

[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.  Google Scholar

[10]

D. BaleanuJ. H. Asad and A. Jajarmi, The fractional model of spring pendulum: new features within different kernels, P. Romanian Acad. A, 19 (2018), 447-454.   Google Scholar

[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.  Google Scholar

[12]

J. BiazarR. AnsariK. 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.   Google Scholar

[13]

J. Cresser, Quantum Physics Notes, Department of Physics, Macquarie University, Australia, (2011). Google Scholar

[14] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 2018.  doi: 10.1017/9781316995433.  Google Scholar
[15]

A. GoswamiJ. SinghD. 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.  Google Scholar

[16]

A. GoswamiJ. SinghD. 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.  Google Scholar

[17]

A. GoswamiSu shilaJ. 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.  Google Scholar

[18]

A. GoswamiJ. 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.  Google Scholar

[19]

A. GoswamiJ. 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.  Google Scholar

[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.   Google Scholar

[21]

A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Soliton Fract., 39 (2009), 1486-1492.  doi: 10.1016/j.chaos.2007.06.034.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[24]

K. HosseiniF. SamadaniD. 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.  Google Scholar

[25]

K. HosseiniD. KumarM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. KaplanK. HosseiniF. 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.   Google Scholar

[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.  Google Scholar

[31]

A. NiknamA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. SinghD. Kumar and Su shila, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Appl. Math. Mech., 4 (2011), 165-175.   Google Scholar

[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.  Google Scholar

[36]

A. ShidfarA. MolabahramiA. 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.  Google Scholar

[37]

A. ShidfarA. MolabahramiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

L. ZhengT. WangX. 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.  Google Scholar

show all references

References:
[1]

A. K. AlomariM. 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.  Google Scholar

[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.  Google Scholar

[3]

G. AmadorK. ColonN. LunaG. MercadoE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. BabolianJ. 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.  Google Scholar

[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.  Google Scholar

[8]

D. BaleanuJ. 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.   Google Scholar

[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.  Google Scholar

[10]

D. BaleanuJ. H. Asad and A. Jajarmi, The fractional model of spring pendulum: new features within different kernels, P. Romanian Acad. A, 19 (2018), 447-454.   Google Scholar

[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.  Google Scholar

[12]

J. BiazarR. AnsariK. 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.   Google Scholar

[13]

J. Cresser, Quantum Physics Notes, Department of Physics, Macquarie University, Australia, (2011). Google Scholar

[14] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 2018.  doi: 10.1017/9781316995433.  Google Scholar
[15]

A. GoswamiJ. SinghD. 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.  Google Scholar

[16]

A. GoswamiJ. SinghD. 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.  Google Scholar

[17]

A. GoswamiSu shilaJ. 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.  Google Scholar

[18]

A. GoswamiJ. 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.  Google Scholar

[19]

A. GoswamiJ. 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.  Google Scholar

[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.   Google Scholar

[21]

A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Soliton Fract., 39 (2009), 1486-1492.  doi: 10.1016/j.chaos.2007.06.034.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[24]

K. HosseiniF. SamadaniD. 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.  Google Scholar

[25]

K. HosseiniD. KumarM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. KaplanK. HosseiniF. 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.   Google Scholar

[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.  Google Scholar

[31]

A. NiknamA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. SinghD. Kumar and Su shila, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Appl. Math. Mech., 4 (2011), 165-175.   Google Scholar

[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.  Google Scholar

[36]

A. ShidfarA. MolabahramiA. 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.  Google Scholar

[37]

A. ShidfarA. MolabahramiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

L. ZhengT. WangX. 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.  Google Scholar

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 $
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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