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Journal articles on the topic 'Manning-Rosen potential'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

B., I. Ita, Louis H., I. Amos P., O. Magu T., and A. Nzeata-Ibe N. "Bound State Solutions of the Klein-Gordon Equation with Manning-Rosen Plus Yukawa Potential Using Pekeris-Like Approximation of the Coulomb Term and Parametric Nikiforov-Uvarov." Physical Science International Journal 15, no. 3 (2017): 1–6. https://doi.org/10.9734/PSIJ/2017/34330.

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The solutions of the klein-gordon equation with Manning-Rosen plus Yukawa potential (MRYP) has been presented using the Pekeris-like approximation of the coulomb term and parametric Nikiforov-Uvarov (NU) method. The bound state energy eigenvalues and the corresponding un-normalized eigen functions were obtained in terms of Jacobi polynomials. So also, Yukawa, Manning-Rosen and coulomb potentials have been recovered from the mixed potentials and their eigen values obtained.

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2

AHMADOV, A. I., C. AYDIN, and O. UZUN. "ANALYTICAL SOLUTIONS OF THE KLEIN–FOCK–GORDON EQUATION WITH THE MANNING–ROSEN POTENTIAL PLUS A RING-SHAPED LIKE POTENTIAL." International Journal of Modern Physics A 29, no. 01 (2014): 1450002. http://dx.doi.org/10.1142/s0217751x1450002x.

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In this work, on the condition that scalar potential is equal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Manning–Rosen plus ring-shaped like potential are obtained by Nikiforov–Uvarov method. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary l states. The conclusion also contain central Manning–Rosen, central and noncentral Hulthén potential.

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3

Hassanabadi, H., M. Kamali, and B. H. Yazarloo. "Spin-one Duffin–Kemmer–Petiau equation in the presence of Manning–Rosen potential plus a ring-shaped-like potential." Canadian Journal of Physics 92, no. 6 (2014): 465–71. http://dx.doi.org/10.1139/cjp-2013-0112.

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We present the solution of the Duffin–Kemmer–Petiau equation for Manning–Rosen potential plus a ring-shaped-like potential in (1+3)-dimensional space–time for spin-one particles within the framework of an exponential approximation for the centrifugal term. We have used the Nikiforov–Uvarov method in our calculations. The radial wavefunction and the angular wavefunctions are expressed in terms of Jacobi polynomials. We have also represented some numerical results for the Manning–Rosen potential plus a ring-shaped-like potential.

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4

Ikot, A. N., L. E. Akpabio, and E. B. Umoren. "Exact Solution of Schrödinger Equation with Inverted Woods-Saxon and Manning-Rosen Potentials." Journal of Scientific Research 3, no. 1 (2010): 25. http://dx.doi.org/10.3329/jsr.v3i1.5310.

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We have analytically solved the radial Schrödinger equation with inverted Woods-Saxon and Manning-Rosen Potentials. With the ansatz for the wave function, we obtain the generalized wave function and the negative energy spectrum for the system.Keywords: Inverted Woods-Saxon Potentials; Manning-Rosen Potential; Schrödinger Equation.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i1.5310 J. Sci. Res. 3 (1), 25-33 (2011)

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5

Ikot, A. N., E. Maghsoodi, C. N. Isonguyo, S. Zarrinkamar, and H. Hassanabadi. "Relativistic symmetries of Schioberg and general Manning-Rosen potentials and the effects of tensor coupling." Journal of Research in Physics 37, no. 1 (2013): 1–17. http://dx.doi.org/10.2478/jrp-2013-0001.

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Abstract In this paper we solve the Dirac equation with Schioberg and general Manning- Rosen potentials including the Coulomb-like tensor interaction. The approximate analytical bound state solutions of the Dirac equation with the Schioberg and Manning-Rosen potential, energy equations and the corresponding unnormalized wave functions are obtained in a closed form using SUSYQM. We have also reported the numerical results to show the effect of the tensor interaction.

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6

Nasser, I., M. S. Abdelmonem, and Afaf Abdel-Hady. "The Manning–Rosen potential using J-matrix approach." Molecular Physics 111, no. 1 (2012): 1–8. http://dx.doi.org/10.1080/00268976.2012.698026.

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7

Falaye, B. J., K. J. Oyewumi, T. T. Ibrahim, M. A. Punyasena, and C. A. Onate. "Bound state solutions of the Manning–Rosen potential." Canadian Journal of Physics 91, no. 1 (2013): 98–104. http://dx.doi.org/10.1139/cjp-2012-0330.

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Using the asymptotic iteration method (AIM), we have obtained analytical approximations to the ℓ-wave solutions of the Schrödinger equation with the Manning–Rosen potential. The energy eigenvalues equation and the corresponding wavefunctions have been obtained explicitly. Three different Pekeris-type approximation schemes have been used to deal with the centrifugal term. To show the accuracy of our results, we have calculated the eigenvalues numerically for arbitrary quantum numbers n and ℓ for some diatomic molecules (HCl, CH, LiH, and CO). It is found that the results are in good agreement w

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8

Bayramova, G. A. "Analytical solution of the Shredinger equation for the linear combination of the Manning-Rosen and the class of Yukawa potential." Izvestiya vysshikh uchebnykh zavedenii. Fizika, no. 9 (2021): 157–69. http://dx.doi.org/10.17223/00213411/64/9/157.

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In the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hyper

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9

Ikhdair, Sameer M. "Approximate l-States of the Manning-Rosen Potential by Using Nikiforov-Uvarov Method." ISRN Mathematical Physics 2012 (April 5, 2012): 1–20. http://dx.doi.org/10.5402/2012/201525.

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The approximately analytical bound state solutions of the l-wave Schrödinger equation for the Manning-Rosen (MR) potential are carried out by a proper approximation to the centrifugal term. The energy spectrum formula and normalized wave functions expressed in terms of the Jacobi polynomials are both obtained for the application of the Nikiforov-Uvarov (NU) method to the Manning-Rosen potential. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary principal and orbital quantum numbers n and l with two different values of the potential screening parameter

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10

Roy, Amlan K. "Studies on bound-state spectra of Manning–Rosen potential." Modern Physics Letters A 29, no. 09 (2014): 1450042. http://dx.doi.org/10.1142/s0217732314500424.

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Accurate ro-vibrational energies, eigenfunctions, radial densities, expectation values are presented for the exponential-type Manning–Rosen (MR) potential. Bound states, accurate up to ten significant figure are obtained by employing a simple, reliable generalized pseudospectral (GPS) method. All 55 eigenstates with n ≤10 are treated for arbitrary values of potential parameters, covering a wide range of interaction, through a non-uniform, optimal spatial radial discretization. A detailed investigation has been made on energy changes with respect to screening and other potential parameters. A s

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11

Yanar, Hilmi, and Ali Havare. "Spin and Pseudospin Symmetry in Generalized Manning-Rosen Potential." Advances in High Energy Physics 2015 (2015): 1–17. http://dx.doi.org/10.1155/2015/915796.

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Spin and pseudospin symmetric Dirac spinors and energy relations are obtained by solving the Dirac equation with centrifugal term for a new suggested generalized Manning-Rosen potential which includes the potentials describing the nuclear and molecular structures. To solve the Dirac equation the Nikiforov-Uvarov method is used and also applied the Pekeris approximation to the centrifugal term. Energy eigenvalues for bound states are found numerically in the case of spin and pseudospin symmetry. Besides, the data attained in the present study are compared with the results obtained in the previo

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12

Khirali, B., A. K. Behera, J. Bhoi, and U. Laha. "Scattering with Manning–Rosen potential in all partial waves." Annals of Physics 412 (January 2020): 168044. http://dx.doi.org/10.1016/j.aop.2019.168044.

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13

AHMADOV, H. I., C. AYDIN, N. SH HUSEYNOVA, and O. UZUN. "ANALYTICAL SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH THE MANNING–ROSEN POTENTIAL PLUS A RING-SHAPED-LIKE POTENTIAL." International Journal of Modern Physics E 22, no. 10 (2013): 1350072. http://dx.doi.org/10.1142/s0218301313500729.

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The analytical solution of the Schrödinger equation for the Manning–Rosen potential plus a ring-shaped-like potential is obtained by applying the Nikiforov–Uvarov method by using the improved approximation scheme to the centrifugal potential for arbitrary l states. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary l states.

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14

Yanar, Hilmi. "More accurate ro-vibrational energies for SiF ⁺(X ¹Σ⁺) molecule". Physica Scripta 97, № 4 (2022): 045404. http://dx.doi.org/10.1088/1402-4896/ac576d.

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Abstract The most appropriate potential energy function for the X 1Σ+ state of SiF+ molecule has been specified by comparing the vibrational energies obtained via special cases of the general molecular potential (GMP) which are Morse, improved Rosen-Morse, modified Rosen-Morse, improved Manning-Rosen and Tietz potentials with the vibrational energies obtained in the presence of improved generalized Pöschl-Teller (IGPT) potential and experimental data. It has been shown that the improved Rosen-Morse potential is better than the other well-known potential energy functions in fitting experimental

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15

Sahoo, P., and U. Laha. "Nucleon–nucleus inelastic scattering by Manning–Rosen distorted nonlocal potential." Canadian Journal of Physics 100, no. 1 (2022): 68–74. http://dx.doi.org/10.1139/cjp-2021-0184.

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Within the framework of non-relativistic quantum scattering theory, we treat the charged hadron scattering by replacing the nuclear interaction by a separable nonlocal one and the electromagnetic part by the Manning–Rosen potential. The off-energy-shell scattering is rigorously studied by this additive interaction by including the effect of electromagnetic interaction. The exact analytical expressions for the off-shell solutions and half-shell T-matrix are obtained in maximal reduced form. The half-shell T-matrix for the proton–oxygen system is computed and the resultant phase shifts are found

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16

Wei, Gao-Feng, Chao-Yun Long, and Shi-Hai Dong. "The scattering of the Manning–Rosen potential with centrifugal term." Physics Letters A 372, no. 15 (2008): 2592–96. http://dx.doi.org/10.1016/j.physleta.2007.12.042.

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17

Pingak, Redi Kristian, Albert Zicko Johannes, Fidelis Nitti, and Meksianis Zadrak Ndii. "A Theoretical Study on Vibrational Energies of Molecular Hydrogen and Its Isotopes Using a Semi-classical Approximation." Indonesian Journal of Chemistry 21, no. 3 (2021): 725. http://dx.doi.org/10.22146/ijc.63294.

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This study aims to apply a semi-classical approach using some analytically solvable potential functions to accurately compute the first ten pure vibrational energies of molecular hydrogen (H2) and its isotopes in their ground electronic states. This study also aims at comparing the accuracy of the potential functions within the framework of the semi-classical approximation. The performance of the approximation was investigated as a function of the molecular mass. In this approximation, the nuclei were assumed to move in a classical potential. The Bohr-Sommerfeld quantization rule was then appl

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18

Khirali, B., A. K. Behera, J. Bhoi, and U. Laha. "On- and off-shell Jost functions for the Manning-Rosen potential." Physica Scripta 95, no. 7 (2020): 075308. http://dx.doi.org/10.1088/1402-4896/ab95ae.

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19

Khirali, B., A. K. Behera, J. Bhoi, and U. Laha. "Regular and Jost states for the S-wave Manning–Rosen potential." Journal of Physics G: Nuclear and Particle Physics 46, no. 11 (2019): 115104. http://dx.doi.org/10.1088/1361-6471/ab4118.

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20

Chabab, M., A. Lahbas, and M. Oulne. "Closed analytical solutions of Bohr Hamiltonian with Manning-Rosen potential model." International Journal of Modern Physics E 24, no. 11 (2015): 1550089. http://dx.doi.org/10.1142/s0218301315500895.

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In the present paper, we have obtained closed analytical expressions for eigenvalues and eigenfunctions of the Bohr Hamiltonian with the Manning–Rosen potential for [Formula: see text]unstable nuclei as well as exactly separable rotational ones with [Formula: see text]. Some heavy nuclei with known [Formula: see text] and [Formula: see text] bandheads have been fitted by using two parameters in the [Formula: see text]unstable case and three parameters in the axially symmetric prolate deformed one. A good agreement with experimental data has been achieved.

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21

Dong, Shi-Hai, and J. García-Ravelo. "Exact solutions of thes-wave Schrödinger equation with Manning–Rosen potential." Physica Scripta 75, no. 3 (2007): 307–9. http://dx.doi.org/10.1088/0031-8949/75/3/013.

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22

Chabab, M., A. El Batoul, A. Lahbas, and M. Oulne. "Electric quadrupole transitions of the Bohr Hamiltonian with Manning–Rosen potential." Nuclear Physics A 953 (September 2016): 158–75. http://dx.doi.org/10.1016/j.nuclphysa.2016.05.012.

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23

Min-Cang, Zhang, and An Bo. "Analytical Solutions of the Manning-Rosen Potential In the Tridiagonal Program." Chinese Physics Letters 27, no. 11 (2010): 110301. http://dx.doi.org/10.1088/0256-307x/27/11/110301.

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24

Wea, Kristiana Nathalia, A. Suparmi, C. Cari, and Wahyulianti. "The construction of partner potential from the general potential Rosen-Morse and Manning Rosen in 4 dimensional Schrodinger system." Journal of Physics: Conference Series 909 (November 2017): 012028. http://dx.doi.org/10.1088/1742-6596/909/1/012028.

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25

Tokmehdashi, Hadi, Ali Akbar Rajabi, and Majid Hamzavi. "Hulthén and Coulomb-Like Potentials as a Tensor Interaction within the Relativistic Symmetries of the Manning-Rosen Potential." Advances in High Energy Physics 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/870523.

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The bound-state solutions of the Dirac equation for the Manning-Rosen potential are presented approximately for arbitrary spin-orbit quantum numberκwith the Hulthén and Coulomb-like potentials as a tensor interaction. The generalized parametric Nikiforov-Uvarov (NU) method is used to obtain energy eigenvalues and corresponding two-component spinors of the two Dirac particles and these are obtained in the closed form by using the framework of the spin symmetry and p-spin symmetry concept. We have also shown that tensor interaction removes degeneracies between spin and p-spin doublets. Some nume

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26

Bharali, A. "Ring potential generated from the central hyperbolic Manning-Rosen potential using the transformation method." Progress of Theoretical and Experimental Physics 2013, no. 3 (2013): 33A01–0. http://dx.doi.org/10.1093/ptep/ptt001.

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27

Sahoo, P., and U. Laha. "Exact solutions of the Schrödinger equation for Manning-Rosen plus Yamaguchi potential." Physica Scripta 96, no. 9 (2021): 095302. http://dx.doi.org/10.1088/1402-4896/ac07b8.

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28

Onyeaju, M. C., J. O. A. Idiodi, A. N. Ikot, M. Solaimani, and H. Hassanabadi. "Linear and nonlinear optical properties in spherical quantum dots: Manning-Rosen potential." Journal of Optics 46, no. 3 (2016): 254–64. http://dx.doi.org/10.1007/s12596-016-0359-9.

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29

CHEN, ZHAO-YOU, MIN LI, and CHUN-SHENG JIA. "APPROXIMATE ANALYTICAL SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH THE MANNING–ROSEN POTENTIAL MODEL." Modern Physics Letters A 24, no. 23 (2009): 1863–74. http://dx.doi.org/10.1142/s0217732309030345.

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By approximating the centrifugal term in terms of a new approximation scheme, we solve the Schrödinger equation with the arbitrary angular momentum for the Manning–Rosen potential. The bound state energy eigenvalues and the unnormalized radial wave functions have been approximately obtained by using the supersymmetric shape invariance approach and the function analysis method. The numerical experiments show that our approximate analytical results are in better agreement with those obtained by using the numerical integration procedure than the analytical results obtained by using the convention

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30

Louis, Hitler, Ita B. Iserom, Magu T. Odey, et al. "Solutions to the Dirac Equation for Manning-Rosen Plus Shifted Deng-Fan Potential and Coulomb-Like Tensor Interaction Using Nikiforov-Uvarov Method." International Journal of Chemistry 10, no. 3 (2018): 99. http://dx.doi.org/10.5539/ijc.v10n3p99.

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We solve the Dirac equation for the Manning-Rosen plus shifted Deng-Fan potential including a Coulomb-like tensor potential with arbitrary spin–orbit coupling quantum number κ. In the framework of the spin and pseudospin (pspin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov–Uvarov method. Also Special cases of the potential as been considered and their energy eigen values as well as their corresponding eigen functions are obtained for both relativistic and non-relativistic scope.

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31

Onate, C. A., and M. C. Onyeaju. "Sustainability of Molecular Studies for Feinberg-Horodecki Equation under Eckart-Manning-Rosen Potential." IOP Conference Series: Materials Science and Engineering 1107, no. 1 (2021): 012136. http://dx.doi.org/10.1088/1757-899x/1107/1/012136.

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32

Benamira, F., L. Guechi, and A. Zouache. "Comment on ‘Exact solutions of thes-wave Schrödinger equation with Manning–Rosen potential’." Physica Scripta 80, no. 1 (2009): 017001. http://dx.doi.org/10.1088/0031-8949/80/01/017001.

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33

Qiang, Wen-Chao, and Shi-Hai Dong. "Analytical approximations to the solutions of the Manning–Rosen potential with centrifugal term." Physics Letters A 368, no. 1-2 (2007): 13–17. http://dx.doi.org/10.1016/j.physleta.2007.03.057.

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34

Arum Sari, Resita, A. Suparmi, and C. Cari. "Solution of Dirac equation for Eckart potential and trigonometric Manning Rosen potential using asymptotic iteration method." Chinese Physics B 25, no. 1 (2016): 010301. http://dx.doi.org/10.1088/1674-1056/25/1/010301.

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35

Aslanova, S. M. "Analytical solution of the Duffin - Kemmer - Petiau equation for the sum of the Manning - Rosen and the Yukawa class potential." Izvestiya vysshikh uchebnykh zavedenii. Fizika, no. 7 (2021): 140–50. http://dx.doi.org/10.17223/00213411/64/7/140.

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This paper presents an analytical bound-state solution to the Duffin - Kemmer - Petiau equation for the new putative combined Manning - Rosen and Yukawa class potentials. Using the developed scheme to approximate and overcome the difficulties arising in the centrifugal part of the potential, the bound-state solution of the modified Duffin - Kemmer - Petiau equation is found. Analytical expressions of energy eigenvalue and the corresponding radial wave functions are obtained for an arbitrary value of the orbital quantum number l . Also, eigenfunctions are expressed in terms of hypergeometric fu

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36

Onate, C. A., M. C. Onyeaju, A. N. Ikot, J. O. A. Idiodi, and J. O. Ojonubah. "Eigen solutions, Shannon entropy and fisher information under the Eckart Manning Rosen potential model." Journal of the Korean Physical Society 70, no. 4 (2017): 339–47. http://dx.doi.org/10.3938/jkps.70.339.

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37

Ikhdair, S. M., and R. Sever. "Approximate l‐state solutions of the D‐dimensional Schrödinger equation for Manning‐Rosen potential." Annalen der Physik 520, no. 11 (2008): 897–910. http://dx.doi.org/10.1002/andp.20085201107.

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38

Ikot, A. N., H. Hassanabadi, B. H. Yazarloo, A. D. Antia, and S. Zarrinkamar. "Generalized Tensor Interaction and Relativistic Spin and Pseudospin Symmetries with the Manning–Rosen Potential." Ядерная физика 77, no. 03 (2014): 306–13. http://dx.doi.org/10.7868/s0044002714020111.

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39

Qiang, Wen-Chao, and Shi-Hai Dong. "The Manning–Rosen potential studied by a new approximate scheme to the centrifugal term." Physica Scripta 79, no. 4 (2009): 045004. http://dx.doi.org/10.1088/0031-8949/79/04/045004.

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40

Suparmi, A., C. Cari, Beta Nur Pratiwi, and Dewanta Arya Nugraha. "The Shannon entropy information for mixed Manning Rosen potential in D-dimensional Schrodinger equation." Journal of Physics: Conference Series 795 (January 2017): 012001. http://dx.doi.org/10.1088/1742-6596/795/1/012001.

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41

Ikot, A. N., H. Hassanabadi, B. H. Yazarloo, A. D. Antia, and S. Zarrinkamar. "Generalized tensor interaction and relativistic spin and pseudospin symmetries with the Manning-Rosen potential." Physics of Atomic Nuclei 77, no. 3 (2014): 282–89. http://dx.doi.org/10.1134/s1063778814020100.

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42

Behera, A. K., J. Bhoi, U. Laha, and B. Khirali. "Study of nucleon–nucleon and alpha-nucleon elastic scattering by the Manning–Rosen potential." Communications in Theoretical Physics 72, no. 7 (2020): 075301. http://dx.doi.org/10.1088/1572-9494/ab8a1a.

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43

Jia, Chun-Sheng, Guang-Chuan Liang, Xiao-Long Peng, Hong-Ming Tang, and Lie-Hui Zhang. "Relationship of the Williams-Poulios and Manning-Rosen Potential Energy Models for Diatomic Molecules." Few-Body Systems 55, no. 11 (2014): 1159–65. http://dx.doi.org/10.1007/s00601-014-0903-6.

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44

Hassanabadi, H., L. L. Lu, S. Zarrinkamar, G. H. Liu, and H. Rahimov. "Approximate Solutions of Schrödinger Equation under Manning-Rosen Potential in Arbitrary Dimension via SUSYQM." Acta Physica Polonica A 122, no. 4 (2012): 650–54. http://dx.doi.org/10.12693/aphyspola.122.650.

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45

Zhang, Min-Cang, та Guo-Qing Huang-Fu. "Analytical arbitrary ℓ-wave solutions of the manning-rosen potential in the tridiagonalization program". International Journal of Quantum Chemistry 112, № 4 (2011): 1036–40. http://dx.doi.org/10.1002/qua.23096.

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46

Ortakaya, S., H. Hassanabadi, and E. Maghsoodi. "Scattering phase shifts of Dirac equation with Manning-Rosen potential and Yukawa tensor interaction." Indian Journal of Physics 89, no. 4 (2014): 307–16. http://dx.doi.org/10.1007/s12648-014-0592-5.

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47

Ikhdair, S. M., and R. Sever. "Approximate l-state solutions of the D-dimensional Schrödinger equation for Manning-Rosen potential." Annalen der Physik 17, no. 11 (2008): 897–910. http://dx.doi.org/10.1002/andp.200810322.

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48

Taş, Ahmet, Soner Alpdoğan, and Ali Havare. "The Scattering and Bound States of the Schrödinger Particle in Generalized Asymmetric Manning-Rosen Type Potential." Advances in High Energy Physics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/619241.

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We solve exactly one-dimensional Schrödinger equation for the generalized asymmetric Manning-Rosen (GAMAR) type potential containing the different types of physical potential that have many application fields in the nonrelativistic quantum mechanics and obtain the solutions in terms of the Gauss hypergeometric functions. Then we determine the solutions for scattering and bound states. By using these states we calculate the reflection and transmission coefficients for scattering states and achieve a correlation that gives the energy eigenvalues for the bound states. In addition to these, we sho

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49

Qiang, Wen-Chao, Kai Li, and Wen-Li Chen. "New bound and scattering state solutions of the Manning–Rosen potential with the centrifugal term." Journal of Physics A: Mathematical and Theoretical 42, no. 20 (2009): 205306. http://dx.doi.org/10.1088/1751-8113/42/20/205306.

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50

Ikot, A. N., H. Hassanabadi, H. P. Obong, Y. E. Chad Umoren, C. N. Isonguyo, and B. H. Yazarloo. "Approximate solutions of Klein—Gordon equation with improved Manning—Rosen potential in D -dimensions using SUSYQM." Chinese Physics B 23, no. 12 (2014): 120303. http://dx.doi.org/10.1088/1674-1056/23/12/120303.

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