Semi-exact Solutions of Konwent Potential
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摘要
In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).
In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).
In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).
引文
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[2]L.D.Landau and E.M.Lifshitz,Quantum Mechanics(Non-Relativistic Theory),3rd ed.,Pergamon,New York(1977).
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[4]F.Cooper,A.Khare,and U.Sukhatme,Phys.Rep.251(1995)267.
[5]S.H.Dong,Factorization Method in Quantum Mechanics,Springer,Dordrecht,Netherlands(2007).
[6]Z.Q.Ma,A.Gonzalez-Cisneros,B.W.Xu,and S.H.Dong,Phys.Lett.A 371(2007)180.
[7]W.C.Qiang and S.H.Dong,EPL 89(2010)10003.
[8]N.Rosen and P.M.Morse,Phys.Rev.42(1932)210.
[9]C.S.Jia,Y.F.Diao,X.J.Liu,et al.,J.Chem.Phys.137(2012)014101.
[10]X.Q.Song,C.W.Wang,and C.S.Jia,Chem.Phys.Lett.673(2017)50.
[11]C.S.Jia,C.W.Wang,L.H.Zhang,et al.,Chem.Eng.Sci.183(2018)26.
[12]C.S.Jia,R.Zeng,X.L.Peng,et al.,Chem.Eng.Sci.190(2018)1.
[13]X.L.Peng,R.Jiang,C.S.Jia,et al.,Chem.Eng.Sci.190(2018)122.
[14]W.L.Chen,Y.W.Shi,and G.F.Wei,Commun.Theor.Phys.66(2016)196.
[15]M.F.Manning,J.Chem.Phys.3(1935)136.
[16]M.Baradaran and H.Panahi,Advances in High Energy Physics 2017(2017)Article ID 2181532.
[17]M.Razavy,Am.J.Phys.48(1980)285.
[18]H.Konwent,P.Machnikowski,and A.Radosz,J.Phys.A:Math.Gen.28(1995)3757.
[19]Q.T.Xie,J.Phys.A:Math.Theor.45(2012)175302.
[20]B.H.Chen,Y.Wu,and Q.T.Xie,J.Phys.A:Math.Theor.46(2013)035301.
[21]R.R.Hartmann,J.Math.Phys.55(2014)012105.
[22]A.E.Sitniksky,Chem.Phys.Lett.676(2017)169.
[23]Dai-Nam Le,Ngoc-Tram D.Hoang,and Van-Hoang Le,J.Math.Phys.59(2018)032101.
[24]H.Konwent,Phys.Lett.A 118(1986)467.
[25]S.H.Dong,Wave Equations in Higher Dimensions,Springer,Dordrecht,Heidelberg,London,New York(2011).
[26]S.Dong,Q.Fang,B.J.Falaye,et al.,Mod.Phys.Lett.A 31(2016)1650017.
[27]S.Dong,G.H.Sun,B.J.Falaye,and S.H.Dong,The European Physical Journal Plus 131(2016)176.
[28]G.H.Sun,S.H.Dong,K.D.Launey,et al.,Int.J.Quan.Chem.115(2015)891.
[29]A.Ronveaux,ed.,Heun’s Differential Equations,Oxford University Press,Oxford(1995).
[30]C.A.Downing,J.Math.Phys.54(2013)072101.
[31]P.P.Fiziev,J.Phys.A:Math.Theor.43(2010)035203.
[32]R.Hartmann and M.E.Portnoi,Phys.Rev.A 89(2014)012101.
[33]D.Agboola,J.Math.Phys.55(2014)052102.
[34]F.K.Wen,Z.Y.Yang,C.Liu,et al.,Commun.Theor.Phys.61(2014)153.