Abstract
A model physical problem is studied in which a system of two electrons is subject either to soft confinement by means of attractive oscillator potentials or by entrapment within an impenetrable spherical box of finite radius R. When hard confinement is present the oscillators can be switched off. Exact analytical solutions are found for special parameter sets, and highly accurate numerical solutions (18 decimal places) are obtained for general cases. Some interesting degeneracy questions are discussed at length.
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References
J.R. Sabin, E.J. Brändas, S.A. Cruz (Editors), Advances in quantum chemistry: theory of confined quantum systems - Part one, book 57 (Academic Press, New York, 2009).
K.D. Sen (Editor), Electronic structure of quantum confined atoms and molecules (Springer, UK, 2014).
Y. Sajeev, N. Moiseyev, Phys. Rev. B 78, 075316 (2008).
M. Genkin, E. Lindroth, Phys. Rev. B 81, 125315 (2010).
S. Chakraborty, Y.K. Ho, Phys. Rev. A 84, 032515 (2011).
M. Taut, Phys. Rev. A 48, 3561 (1993).
A.I. Pupyshev, A.V. Scherbinin, Chem. Phys. Lett. 295, 217 (1998).
A.I. Pupyshev, A.V. Scherbinin, Phys. Lett. A 299, 371 (2002).
D.R. Herrick, J. Math. Phys. 16, 281 (1975).
D.R. Herrick, F.H. Stillinger, Phys. Rev. A 11, 42 (1975).
K.D. Sen, H.E. Montgomery Jr., N.A. Aquino, Int. J. Quantum Chem. 107, 798 (2007).
K.D. Sen, V.I. Pupyshev, H.E. Montgomery Jr., Ad. Quantum Chem. 57, 25 (2009).
Muzaian A. Shaqqor, Sami M. AL-Jaber, Int. J. Theor. Phys. 48, 2462 (2009).
H.E. Montgomery Jr., G. Campoy, N. Aquino, Phys. Scr. 81, 045010 (2010).
Xiao-Yan Gu, Jian-Qiang Sun, J. Math. Phys. 51, 022106 (2010).
D. Agboola, Pramana 76, 875 (2011).
J.D. Louck, J. Mol. Spectrosc. 4, 298 (1960).
A. Chatterjee, Phys. Rep. 186, 249 (1990).
K. Atkinson, W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction (Springer, New York, 2012).
D.J. Doren, D.R. Herschbach, J. Chem. Phys. 85, 4557 (1986).
R.L. Hall, Phys. Rev. D 22, 2062 (1980).
R.L. Hall, J. Math. Phys. 24, 324 (1983).
R.L. Hall, J. Math. Phys. 25, 2708 (1984).
R.L. Hall, Phys. Rev. A 39, 550 (1989).
R.L. Hall, J. Math. Phys. 33, 1710 (1992).
R.L. Hall, J. Math. Phys. 34, 2779 (1993).
S.J. Gustafson, I.M. Sigal, Mathematical concepts of quantum mechanics (Springer, New York, 2006) (the operator inequality is proved for dimensions d ≥ 3 on page 32).
M. Reed, B. Simon, Methods of modern mathematical physics II: Fourier analysis and self-adjointness (Academic Press, New york, 1975) (the operator inequality is proved for d = 3 on p. 169).
A.K. Common, J. Phys. A 18, 2219 (1985).
W. Thirring, A Course in Mathematical Physics 3: Quantum Mechanics of Atoms and Molecules (Springer, New York/Wien, 1990) (the local energy theorem is discussed on p. 154).
H. Ciftci, R.L. Hall, N. Saad, J. Phys. A: Math. Gen. 36, 11807 (2003).
N. Saad, R.L. Hall, H. Ciftci, J. Phys. A: Math. Gen. 39, 13445 (2006).
H. Ciftci, R.L. Hall, N. Saad, E. Dogu, J. Phys. A: Math. Theor. 43, 415206 (2010).
L.D. Landau, E.M. Lifshitz, Quantum Mechanics: non-relativistic theory (Pergamon, London, 1981).
F.M. Arscott, Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions (Pergamon Press, 1964).
A. Hautot, Bull. Soc. R. Sci. Liége 38, 1969 (654).
A. Hautot, Bull. Soc. R. Sci. Liége 38, 1969 (660).
J. Rovder, Mat. Căs. 24, 15 (1974).
R.L. Hall, N. Saad, K. Sen, J. Math. Phys. 52, 092103 (2011).
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Hall, R.L., Saad, N. & Sen, K.D. Soft and hard confinement of a two-electron quantum system. Eur. Phys. J. Plus 129, 274 (2014). https://doi.org/10.1140/epjp/i2014-14274-0
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DOI: https://doi.org/10.1140/epjp/i2014-14274-0