HIGHLY ACCURATE NUMERICAL ENERGY FOR THE GROUND STATE OF A HYDROGEN ATOM IN A PLASMA WITH THE PRESENCE OF A UNIFORM MAGNETIC FIELD
Main Article Content
Abstract
The FK operator method has been recently applied successfully to the problem of a hydrogen atom in a magnetic field with any intensity. In this work, this method is applied more generally, considering the screening effect. Screening effect plays an important role in practical astronomy research on hydrogen atom in a plasma environment. In this paper, the FK operator method is combined with the Kustaanheimo-Stiefel transformation, so the wavefunction is expanded in terms of the eigenfunctions of a four-dimensional harmonic oscillator. There is an improvement in algebraic calculations to obtain the matrix elements of the screening term in the Hamiltonian, which results in the explicit expressions of the matrix elements in the series expansion. These results help to build a FORTRAN program finding highly accurate numerical energies with the precision up to 32 decimal places. For an illustration purpose, energies for the ground state with the screening parameter ranging from 0 to 1 a.u. are shown in the paper.
Keywords
algebraic method, hydrogen atom, screened Coulomb potential, annihilation and creation operators, basic set
Article Details
References
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Bahar, M. K., & Soylu, A. (2015). Confinement effects of magnetic field on two-dimensional hydr--ogen atom in plasmas. Physics of Plasmas, 22(5), 052701. https://doi.org/10.1063/1.4919613
Bahar, M. K., & Soylu, A. (2016). Probe of hydrogen atom in plasmas with magnetic, electric, and Aharonov-Bohm flux fields. Physics of Plasmas, 23(9), 092712. https://doi.org/10.1063/1.4963772
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Cao, T. X. H., Ly, D. N., Hoang, N.T. D., & Le, V. H. (2019). High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength. Computer Physics Communications, 240, 138-151. https://doi.org/10.1016/j.cpc.20 19.02.013
Feranchuk, I., Ivanov, A., Le, V.-H., & Ulyanenkov, A. (2015). Non-perturbative Description of Quantum Systems (Vol. 894). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-13006-4
Hoang, N. T. D., Nguyen, D. A. P., Hoang, V. H., & Le, V. H. (2016). Highly accurate analytical energy of a two-dimensional exciton in a constant magnetic field. Physica B: Condensed Matter, 495, 16-20. https://doi.org/10.1016/j.physb.2016.04.038
Kar, S., & Ho, Y. K. (2005). Electron affinity of the hydrogen atom and a resonance state of the hydrogen negative ion embedded in Debye plasmas. New Journal of Physics, 7, 141-141. https://doi.org/10.1088/1367-2630/7/1/141
Le, D. N., Hoang, N.T. D., & Le, V.H. (2017). Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field. Journal of Mathematical Physics, 58(4), 042102. https://doi.org/10.1063/1.4979618
Le, D. N., & Hoang, D. N. T. (2017). Binding Energy of Exciton in Monolayer Semiconductor Ws 2 With Yukawa-Like Screening Potential. Ho Chi Minh City University of Education Journal of Science, 14(9), 43-50. Retrieved from
http://journal.hcmue.edu.vn/index.php/hcmuejos/issue/archive
Lin, C. Y., & Ho, Y. K. (2011). The photoionization of excited hydrogen atom in plasmas. Computer Physics Communications, 182(1), 125-129. https://doi.org/10.1016/j.cpc.20 10.06.013
Main, J., Schwacke, M., & Wunner, G. (1998). Hydrogen atom in combined electric and magnetic fields with arbitrary mutual orientations. Physical Review A, 57(2), 1149-1157. https://doi. org/10.1103/PhysRevA.57.1149
Netlib.org. LAPACK: Linear Algebra PACKage. (n.d.). Subroutine DSYGVX.f. Retrieved from http://www.netlib.org/lapack/explore-ht-ml/d2/d97/dsyevx-8f.html
Paul, S., & Ho, Y. K. (2009). Hydrogen atoms in Debye plasma environments. Physics of Plasmas, 16(6), 063302. https://doi.org/10.1063/1.3152602
Paul, S., & Ho, Y. K. (2010). Two-colour three-photon transitions in a hydrogen atom embedded in Debye plasmas. Journal of Physics B: Atomic, Molecular and Optical Physics, 43(6), 065701. https://doi.org/10.1088/0953-4075/43/6/065701
Paul, S., & Ho, Y. K. (2011). Solution of the generalized exponential cosine screened Coulomb potential. Computer Physics Communications, 182(1), 130-133. https://doi.org/10.1016/j. cpc.2010.06.014
Saha, B., Mukherjee, P. K., & Diercksen, G. H. F. (2002). Energy levels and structural properties of compressed hydrogen atom under Debye screening. Astronomy & Astrophysics, 396(1), 337–344. https://doi.org/10.1051/0004-6361:20021350
Saha, J. K., Mukherjee, T. K., Mukherjee, P. K., & Fricke, B. (2011). Hyperpolarizability of hydrogen atom under spherically confined Debye plasma. The European Physical Journal D, 62(2), 205-211. https://doi.org/10.1140/epjd/e2011-10668-4
Sasmal, G. P. (2014). On computation for a hydrogen atom in arbitrary magnetic fields using finite volume method. Journal of Atomic and Molecular Sciences, 5(3), 187-205. https://doi.org/10. 4208/jams.110813.021414a
Shukla, P. K., & Eliasson, B. (2008). Screening and wake potentials of a test charge in quantum plasmas. Physics Letters A, 372(16), 2897-2899. https://doi.org/10.1016/j.physleta.2007.12. 067
Soylu, A. (2012). Plasma screening effects on the energies of hydrogen atom. Physics of Plasmas, 19(7), 072701. https://doi.org/10.1063/1.4736947
Thirumalai, A., & Heyl, J. S. (2009). Hydrogen and helium atoms in strong magnetic fields. Physical Review A, 79(1), 012514. https://doi.org/10.1103/PhysRevA.79.012514
Bahar, M. K., & Soylu, A. (2015). Confinement effects of magnetic field on two-dimensional hydr--ogen atom in plasmas. Physics of Plasmas, 22(5), 052701. https://doi.org/10.1063/1.4919613
Bahar, M. K., & Soylu, A. (2016). Probe of hydrogen atom in plasmas with magnetic, electric, and Aharonov-Bohm flux fields. Physics of Plasmas, 23(9), 092712. https://doi.org/10.1063/1.4963772
Bartsch, T. (2003). The Kustaanheimo–Stiefel transformation in geometric algebra. Journal of Physics A: Mathematical and General, 36(25), 6963-6978. https://doi.org/10.1088/0305-4470/36/25/305
Cao, H. T. X., Ly, D. N., & Hoang, D. N. T. (2016). Nang luong trang thai co ban cua nguyen tu hydro trong tu truong deu co cuong do bat ki [Ground state energy of a hydrogen atom in a uniform magnetic field with arbitrary strength]. Ho Chi Minh City University of Education Journal of Science, 12, 39-51.
Cao, T. X. H., Ly, D. N., Hoang, N.T. D., & Le, V. H. (2019). High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength. Computer Physics Communications, 240, 138-151. https://doi.org/10.1016/j.cpc.20 19.02.013
Feranchuk, I., Ivanov, A., Le, V.-H., & Ulyanenkov, A. (2015). Non-perturbative Description of Quantum Systems (Vol. 894). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-13006-4
Hoang, N. T. D., Nguyen, D. A. P., Hoang, V. H., & Le, V. H. (2016). Highly accurate analytical energy of a two-dimensional exciton in a constant magnetic field. Physica B: Condensed Matter, 495, 16-20. https://doi.org/10.1016/j.physb.2016.04.038
Kar, S., & Ho, Y. K. (2005). Electron affinity of the hydrogen atom and a resonance state of the hydrogen negative ion embedded in Debye plasmas. New Journal of Physics, 7, 141-141. https://doi.org/10.1088/1367-2630/7/1/141
Le, D. N., Hoang, N.T. D., & Le, V.H. (2017). Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field. Journal of Mathematical Physics, 58(4), 042102. https://doi.org/10.1063/1.4979618
Le, D. N., & Hoang, D. N. T. (2017). Binding Energy of Exciton in Monolayer Semiconductor Ws 2 With Yukawa-Like Screening Potential. Ho Chi Minh City University of Education Journal of Science, 14(9), 43-50. Retrieved from
http://journal.hcmue.edu.vn/index.php/hcmuejos/issue/archive
Lin, C. Y., & Ho, Y. K. (2011). The photoionization of excited hydrogen atom in plasmas. Computer Physics Communications, 182(1), 125-129. https://doi.org/10.1016/j.cpc.20 10.06.013
Main, J., Schwacke, M., & Wunner, G. (1998). Hydrogen atom in combined electric and magnetic fields with arbitrary mutual orientations. Physical Review A, 57(2), 1149-1157. https://doi. org/10.1103/PhysRevA.57.1149
Netlib.org. LAPACK: Linear Algebra PACKage. (n.d.). Subroutine DSYGVX.f. Retrieved from http://www.netlib.org/lapack/explore-ht-ml/d2/d97/dsyevx-8f.html
Paul, S., & Ho, Y. K. (2009). Hydrogen atoms in Debye plasma environments. Physics of Plasmas, 16(6), 063302. https://doi.org/10.1063/1.3152602
Paul, S., & Ho, Y. K. (2010). Two-colour three-photon transitions in a hydrogen atom embedded in Debye plasmas. Journal of Physics B: Atomic, Molecular and Optical Physics, 43(6), 065701. https://doi.org/10.1088/0953-4075/43/6/065701
Paul, S., & Ho, Y. K. (2011). Solution of the generalized exponential cosine screened Coulomb potential. Computer Physics Communications, 182(1), 130-133. https://doi.org/10.1016/j. cpc.2010.06.014
Saha, B., Mukherjee, P. K., & Diercksen, G. H. F. (2002). Energy levels and structural properties of compressed hydrogen atom under Debye screening. Astronomy & Astrophysics, 396(1), 337–344. https://doi.org/10.1051/0004-6361:20021350
Saha, J. K., Mukherjee, T. K., Mukherjee, P. K., & Fricke, B. (2011). Hyperpolarizability of hydrogen atom under spherically confined Debye plasma. The European Physical Journal D, 62(2), 205-211. https://doi.org/10.1140/epjd/e2011-10668-4
Sasmal, G. P. (2014). On computation for a hydrogen atom in arbitrary magnetic fields using finite volume method. Journal of Atomic and Molecular Sciences, 5(3), 187-205. https://doi.org/10. 4208/jams.110813.021414a
Shukla, P. K., & Eliasson, B. (2008). Screening and wake potentials of a test charge in quantum plasmas. Physics Letters A, 372(16), 2897-2899. https://doi.org/10.1016/j.physleta.2007.12. 067
Soylu, A. (2012). Plasma screening effects on the energies of hydrogen atom. Physics of Plasmas, 19(7), 072701. https://doi.org/10.1063/1.4736947
Thirumalai, A., & Heyl, J. S. (2009). Hydrogen and helium atoms in strong magnetic fields. Physical Review A, 79(1), 012514. https://doi.org/10.1103/PhysRevA.79.012514