su(1, 1) so(2, 1) Lie Algebraic Extensions of the Mie-type Interactions with Positive Constant Curvature

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1 arxiv: v1 [math-ph] 2 Jan 2013 su1, 1) so2, 1) Lie Algebraic Extensions of the Mie-type Interactions with Positive Constant Curvature Özlem Yeşiltaş 1 Department of Physics, Faculty of Science, Gazi University, Ankara, Turkey Abstract The Schrödinger equation in three dimensional space with constant positive curvature is studied for the Mie potential. Using analytic polynomial solutions, we have obtained whole spectrum of the corresponding system. With the aid of factorization method, ladder operators are obtained within the variable and function transformations. Using ladder operators, we have given the generators of so2, 1) algebra and the Casimir operator which are related to the Mie Oscillator on the positive curvature. keyword:positive curvature, mie potential PACS: w, Fd, Ge. 1 Introduction Recently, quantum mechanics in curved spherical spaces as a fundamental problem has become a subject of intense research efforts [1, 2, 3, 4, 5, 6, 7]. The notion of the constant curvature and the accidental degeneracy first began with Schrödinger [8], Infeld [9], Stevenson [10]. Essential advances of these systems with accidental degeneracy have been made by Nishino [11], Higgs [12], Leemon [13]. It has been found that the complete degeneracy of the energy of the Coulomb problem and harmonic oscillator on the three dimensional sphere in the orbital and azimuthal quantum number is caused by an 1 yesiltas@gazi.edu.tr 1

2 additional integral of motion. At the same time, some papers on curved spherical spaces are concerned with some applications of physics such as linear and non-linear optics [14], quantum dots [15, 16]. Furthermore, in [17], the authors studied liquid crystals using spherical geometries. Thus, molecular potentials such as Mie type interactions may be an interesting candidate for the topological applications of some molecules. Symmetry groups have come to play an important role in quantum physics. On the other hand, symmetry algebras enable one to understand the degenerate energy eigen-states of a system, exact solvability of the spectrum of a quantum system usually indicates the presence of symmetry. In [18], symmetry algebras are studied within exact solvability and so2, 2) algebras. To our knowledge, there has not been studied Mie-type interactions, which are used to determine molecular structures [19, 20, 21, 22, 23, 24], in constant positive curvature. Hence, this study is concerned with the extension of Mie potential to the spherical coordinates with spaces of constant curvature and to the symmetry algebras determining so2, 1) algebra for the Hamiltonian which is factorized and defined as Mie interactions on constant positive curvature. This paper is organized as follows. The Mie potential in spherical coordinates with spaces of constant curvature is given in Section 2. Section 3 presents the solutions of the eigenvalue equation which is derived from the Schrödinger equation with Laplace-Beltrami operator. Section 4 is assigned to discuss symmetry algebras which are more general than the potential algebras for the corresponding system. 2 Mie Potential on the Constant Curvature We will attend to the case of the three dimensional space of constant positive curvature which is geometrically given on the three dimensional sphere of radius R, S 3 embedded into the four dimensional Euclidean space when the equation of S 3 has the form S 3 = {ζ 0,ζ i ) R 4 : ζ 2 0 +ζ i ζ i = R 2 } 1) where i = 1,2,3 in the tangent space x i are the coordinates and ζ i is ζ i = ζ 0 = x i 1+ r2 R 2 2) R. 3) 1+ r2 R 2 2

3 The spherical coordinates are given by ζ 1 = Rsinψsinθcosφ 4) ζ 2 = Rsinψsinθsinφ 5) ζ 3 = Rsinψcosθ 6) ζ 4 = Rcosψ 7) where 0 ψ π, 0 θ π, 0 φ < 2π. Differentiating with respect to the arbitrary angles ψ,θ,φ gives a four dimensional vector and the squared length of this vector is ds 2 = R 2 dψ 2 +sin 2 ψdθ 2 +sin 2 θdφ 2 ) ) 8) which is called as Robertson-Walker metric for the positive curvature κ = 1. Define r 2 = x x x 2 3 and the potential Vr) which is known as Mie potential [21, 22, 23] given by, k a ) l l ) a k Vr) = ε ; l = 2k; k = 1, 9) l k r l k r) where ε is the interaction energy between the atoms in a molecule, a is the coordinate of the interaction, l > k. A special case that is k = 1 performed as 1 a ) ) 2 a Vr) = V 0, V 0 = 2εk. 10) 2 r r Inserting the above dependence of r on ζ gives or we may give Vψ) as Vζ) = V 0 a2 2 1 ζ2 R 2 ζ 2 a 1 ζ2 R 2, 11) ζ ) ) 2 1 a a Vψ) = V 0. 12) 2 Rtanψ Rtanψ In fact, this potential 12) is known as trigonometric Rosen- Morse I potential [25]. 3

4 3 Eigenvalue Equation and Solutions Here we give the Schrödinger equation for 11) on the constant curvature, ) 2 2µ +V Ψ = EΨ, 13) where is the Laplace-Beltrami operator which is a restriction of the Laplace operator on the sphere, then we have the following formula for and define the metric which is = 1 g 3 i,k=1 gg ik ), 14) x i x k ds 2 = g ik dx i dx k 15) where g = det g ik and by the chain rule g ik = g ik ) 1. Thus, using 4), 5), 6) and 7), 14), Schrödinger equation takes the form 1 sin 2 ψ ψ sin2 ψ )Ψ+ 2µR2 ψ 2 ) )) 2 16) E 2 mm+1) 2µR 2 sin 2 ψ V 1 a a 0 Ψ = 0. 2 Rtanψ Rtanψ Using a transformation of the wave-function in 16) Ψψ) = φψ) sinψ 17) and 16) turns into C 1 = 2µR2 2 C 2 = 2µR2 2 C 3 = 2µR2 av 0 2 R ) E + a2 V 0 2R 2 2 mm+1) + V ) 0a 2 2µR 2 2R 2 18) 19) 20) φ +C 1 +2+C 3 cotψ C 2 +2)csc 2 ψ)φ = 0. 21) 4

5 Another transformation of the variables which are lead to φψ) = e αψ/2 Fψ), z = cotψ 22) 1+z 2 ) 2 F z)+21+z 2 )α+z)f z)+ ) C 3 z C 2 +2)1+z 2 )+C α2 +1 Fz) = ) Finally, we shall use an ansatze in above equation as Fz) = 1+z 2 ) 1 β 2 fz), 24) then we can obtain f z)+ α+2βz 1 z)+ 1+z 2 f 1+z 2 ) 2C 1 C 2 +β + α C3 α+αβ)z +β 2 β C 2 2) ) fz) = 0. 25) Let us arrange the coefficient of fz) in 25) as 1 1+z 2 C 1 + α β β 2 +zc 3 α+αβ) 4 and the coefficients of z and z 2 can be terminated in 26) if ) 2 C 2 β +β 2 26) C 3 α1 β) = 0 27) C 1 +2β β 2 + α2 +5 = ) Then, we may continue to search to obtain a hypergeometric equation, hence we use in 25) and this yields z = it, α iα 29) 1 t 2 )f t) α+2βt)f t)+β1 β)+2+c 2 )ft) = 0. 30) Jacobi differential equation is given as [26] 1 x 2 )y x)+b a a+b+2)x)y x)+nn+a+b+1)yx) = 0. 31) 5

6 We now compare 30) and 31) in order to express the solutions ft) in terms of Jacobi polynomials, and then we have a = 2β α 2, b = 2 2β +α 2. 32) 2 Thus, our solutions ft) can be given as ft) = P n a,b) t). Moreover, let us substitute in 25), one can see that C 2 +2 = jj +1) 33) nn+2β 1) = β1 β)+jj +1). 34) Shifting n to n n 1 and using 27); we find the followings which are n-dependent constants: α n = C 3 35) n+j Finally, 28) leads to find our energy eigenvalues as n+j) 2 E n = 2 2µR 2 β n = 1 n+j). 36) C 2 3 4n+j) 2 µa2 V We may also give j in terms of parameters of the potential as, j = 1 2 ± mm+1)+ µv 0a 2 ). 37) ) And, we can write the un-normalized eigenfunction solutions of 13) which are complex as Ψψ) = N sinψ e iαψ/2 1+cot 2 ψ) 1 β 2 P a,b) n icotψ). 39) where P n a,b) R, are corresponding Jacobi polynomials. On the other hand, in the limit of R, E n µ2 a 2 V ) 4 n+j) 2 which means we have energy eigenvalues 40) in flat space and this agrees with the results in [22]. When ψ 0, ψ π, we obtain Ψ n 0. 6

7 4 Factorization and Algebra Let us re-consider 21) which is d 2 φ dψ 2 + jj +1) ǫ sin 2 ψ +2Acotψ ) φ = 0, 41) where we used C = ǫ and A = µrav 0. If we perform the changes of variable and 2 also of function, tan ψ 2 = 1 e iy π 2, φz) = χ 42) coshz in 41), we get χ y) ǫ 1 +2iAcosy 4 sin 2 χy)+j + 1 )χy) = 0, 43) y 2 whichisknownastypeaoperatorsinthebookbymiller [27]. In[27], typeafactorization tells us about a linear second-order differential equation like 41) can be factorized if 41) is written as A + j +1)A j +1)Yǫ,j) = ǫ Rj +1))Yǫ,j) 44) where A j)a + j)yǫ,j) = ǫ Rj))Yǫ,j) 45) Here, A ± are known as ladder operators which read and satisfy A + y 1,y 2 ) = y 1,A y 2 ). Let py,j) be Then, plugging 48) into 45) and 46), we have A ± = ± d +py,j). 46) dy Yǫ,j ±1) = A Yǫ,j) 47) py,j) = j +s)coty + t siny. 48) A A + Y = Y + j +s)j +s+1)+t2 +2tj +s+1/2)cosy sin 2 Y +ǫy = 0 49) y where we use s,t are real constants. And we may give the ladder operators and Rj) as A ± = ± d t +j +s)coty + dy siny 50) 7

8 Rj) = j +s) 2. 51) Square integrability of the solutions requires j = 0,1,2,...l, ǫ = Rl). To define the unknown parameters s and t, we can compare 49) and 43) as follows: t 2 +j +s)j +s+1) = ǫ ) Using 53) and substituting it into 52), we obtain If we consider 41), ǫ is given by [27], tj +s+1/2) = ia. 53) ǫ = A 2 /t 2 +t 2. 54) ǫ = Rj +1) = j +1) 2 A2 j +1) 2. 55) Hence, we have two ǫ expressions, if we equate 54) and 55), we get A t = ±l+1), j +s+1/2 = ±i l+1 A t = ±i, j +s+1/2 = ±l+1). 57) l+1 Now, we may consider the Lie algebra generators which we call X 1, X 2, O 1, O 2. The new variables y 1,y 2 can be used in the eigenfunction which is [18] 56) Φψ,y 1,y 2 ) e iνy 1 φψ)e iκy 2 58) where ν = ± A, κ = ±l+1). 59) l+1 The operators X 1,X 2 act on the eigenfunctions as given below X 1 Φ ν,κ = νφ ν,κ, X 2 Φ ν,κ = κφ ν,κ 60) where X 1 = i y 1, X 2 = i y 2. 61) 8

9 One can map each ladder operator into the new one by taking 42) into the consideration as Ā ± = isinψ d +ij +s±1/2)cosψ +tsinψ 62) dψ which lead to get generators O 1, O 2 as O 1 ± = e±iy 1 ±sinψ ψ icosψ sinψ ) y 1 y 2 O 2 ± = e±iy 2 ±sinψ ψ icosψ sinψ ) y 2 y 1 where we use 63) 64) O 1 ± = ie ±iy 1 Ā 1, t = ±l+1), s+t+1/2 = ±i A l+1 65) O 2 ± = ie ±iy 2 Ā A 2, t = ±i, j +s+1/2 = ±l+1). l+1 66) One can look at the commutation relations satisfied by the generators above [X 1,O ± ] = ±O ±, [O +,O ] = 2X 1 67) and Casimir operators whose action on the Φ ν,κ is given by CΦ ν,κ = jj +1)Φ ν,κ 68) equal to C = O + 1 O 1 +X 1 X 1 1) = O + 2 O 2 +X 2 X 2 1). 69) 5 Conclusion We have studied the Mie potential in spherical curved spaces with constant positive curvature through both analytical and algebraic approaches including the Infeld factorization. It is seen that, Mie potential is transformed into a Rosen- Morse I-like potential in spherical spaces. We have obtained spectrum and eigenfunctions of the system using polynomial solutions. Our results agree with [22] if the limit R is used for the solutions. Then, we have used factorization method to determine the algebra for the system. We remind that the conserved quantity, eigenvalue of the Casimir operator is the potential parameter. Using ladder operators which we obtained with factorization method, we have constructed 9

10 the so2,1) algebra for the Mie potential in spherical spaces. In the study of [28], N dimensional Mie potential is studied and su1, 1) algebra generators were represented by infinite-dimensional Hilbert subspaces of the radial quantum states. Hence we have extended the system to so2, 1) algebra through a more general procedure. We give graphs of the potentials Vr) and Vψ) for some specific molecules CH,NO,N 2 and the data for the potential parameters are given in [29]. Figure 1 shows that the Mie potential which is in flat space, a standard Coulombic potential energy plus electronic kinetic energy for each molecule, and figure 2 shows that the Mie potential in spherical spaces with constant positive curvature which may be interesting that the potential takes the form of well with two minima. Finally, relativistic equations and factorization procedure in spherical spaces within the construction of the algebra may be studied in the future. V r ev N 2 NO CH r Figure 1: Graph of Vr) in 10). V Ψ ev CH N 2 NO Ψ Figure 2: Graph of Vψ) in 12). References [1] J. F. Cariñena, M. F. Rañada, J. Phys. A: Math. Theor

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