The u(2) α and su(2) α Hahn Harmonic Oscillators

Transcription

1 Bulg. J. Phys. 40 (013) The u() α and su() α Hahn Harmonic Oscillators E.I. Jafarov 1, N.I. Stoilova, J. Van der Jeugt 3 1 Institute of Physics, Azerbaijan National Academy of Sciences, Javid av. 33, AZ-1143 Baku, Azerbaijan Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 7 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria 3 Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 81-S9, B-9000 Gent, Belgium Abstract. New models for the finite one-dimensional harmonic oscillator are proposed based upon the algebras u() α and su() α. These algebras are deformations of the Lie algebras u() and su() extended by a parity operator, with deformation parameter α. Classes of irreducible unitary representations of u() α and su() α are constructed. It turns out that in these models the spectrum of the position operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. PACS codes: Hk, 0.30.Gp 1 Introduction Discrete and finite quantum systems have attracted much attention in recent years [1]. Such systems have been useful in areas such as quantum computing and quantum optics [, 3]. However, in a finite-dimensional Hilbert space, the canonical commutation relations no longer hold. Several finite oscillator models have been proposed on different defining relations. Our interest comes from models related to a Lie algebra or its deformation. For a one-dimensional finite oscillator, the position operator ˆq, its corresponding momentum operator ˆp and the Hamiltonian Ĥ, which is the generator of time evolution, should satisfy the compatibility of Hamilton s equations with the Heisenberg equations: [Ĥ, ˆq] = iˆp, [Ĥ, ˆp] = iˆq, (1) in units with mass and frequency both equal to 1, and = 1. Furthermore, one requires [4]: Talk at the Second Bulgarian National Congress in Physics, Sofia, September c 013 Heron Press Ltd. 115

2 E.I. Jafarov, N.I. Stoilova, J. Van der Jeugt all operators ˆq, ˆp, Ĥ belong to some (Lie) algebra (or superalgebra) A; the spectrum of Ĥ in (unitary) representations of A is equidistant. The case with A = su() has been investigated in [4 6]. In the common su() representations, labelled by an integer or half-integer j, the Hamiltonian is given by Ĥ = J 0 + j + 1, where J 0 = J z is the diagonal su() operator and its spectrum is n + 1 (n = 0, 1,..., j). ˆq = 1 (J + + J ) = J x and ˆp = i (J + J ) = J y, satisfying the relations (1), have a finite spectrum given by { j, j + 1,...,+j} [4]. The discrete position wavefunctions are given by Krawtchouk functions and their shape is reminiscent of those of the canonical oscillator [4]. Under the limit j they coincide with the canonical wavefunctions in terms of Hermite polynomials [4, 7]. In the present paper we construct two new models for the finite one-dimensional harmonic oscillator based on the algebras u() α and su() α, one-parameter deformations of the Lie algebras u() and su() (see [8, 9] for more details). In section the deformed algebra u() α, its representations and the corresponding model are constructed. The spectrum of the position operator, and its eigenvectors are determined. The structure of these eigenvectors is studied, yielding position wavefunctions. Similarly, in section 3 we consider the model corresponding to su() α. The u() α model Definition 1 Let α be a parameter. The algebra u() α is a unital algebra with basis elements J 0, J +, J, C and P subject to the following relations: C commutes with all basis elements, P is a parity operator satisfying P = 1 and [P, J 0 ] = PJ 0 J 0 P = 0, {P, J ± } = PJ ± + J ± P = 0. () The su() relations are deformed as follows: [J 0, J ± ] = ±J ±, (3) [J +, J ] = J 0 (α + 1) P (α + 1)CP. (4) Proposition Let j be a half-integer (i.e. j is odd), and consider the space V j with basis vectors j, j, j, j + 1,..., j, j. Assume that α > 1. Then 116

3 The u() α and su() α Hahn Harmonic Oscillators the following action turns V j into an irreducible representation space of u() α. C j, m = (j + 1) j, m, (5) P j, m = ( 1) j+m j, m, (6) J 0 j, m = m j, m, (7) (j m)(j + m + 1) j, m + 1, j + m-odd; J + j, m = (j m + α + 1)(j + m + α + ) j, m + 1, j + m-even. (8) (j + m)(j m + 1) j, m 1, j + m-even; J j, m = (j + m + α + 1)(j m + α + ) j, m 1, j + m-odd. (9) Straightforward calculations show that all defining relations from Definition 1 are valid when acting on an arbitrary vector j, m. Irreducibility follows from the fact that (J + ) k j, j is nonzero and proportional to j, j + k for k = 1,,...,j, and similarly (J ) k j, j is nonzero. If ˆq = 1 (J + + J ), ˆp = i (J + J ), Ĥ = J C, (10) it is easy to verify that (1) is satisfied and the spectrum of Ĥ is indeed linear and given by n + 1 (n = 0, 1,..., j). (11) From the actions (8)-(9), one finds that the operator ˆq takes the matrix form 0 M M 0 0 M 1 0 ˆq =. 0 M 1 0.., (1) Mj M j 1 0 with M k = (k + 1)(j k), if k is odd; (k + α + )(j k + α + 1), if k is even. (13) For this matrix, the eigenvalues are known explicitly [10, 11]. Proposition 3 The j + 1 eigenvalues q of the position operator ˆq in the representation V j are given by α j 1, α j + 1,..., α 1; α + 1, α +,...,α + j + 1. (14) 117

4 E.I. Jafarov, N.I. Stoilova, J. Van der Jeugt So in the u() α oscillator model, there is a shift from the origin by α + 1. The next result follows now from [11, Proposition ]: Proposition 4 The orthonormal eigenvector of the position operator ˆq in V j for the eigenvalue q k, denoted by j, q k ), is given by j, q k ) = j m= j U j+m,j+k j, m. (15) Herein, U = (U rs ) 0 r,s j is a (j + 1) (j + 1) matrix with elements U r,j s 1 = U r,j+s+ 1 = ( 1)r Qs (r; α, α + 1, j 1 ), (16) U r+1,j s 1 = U r+1,j+s+ 1 = ( 1)r Qs (r; α + 1, α, j 1 ), (17) where r, s {0, 1,..., j 1 }. The functions Q are normalized Hahn polynomials [1]. Corollary 5 The u() α oscillator wavefunctions are given by: n (q k) = ( 1)n Qk 1 (n; α, α + 1, j 1 ), k = 1, 3,..., j; n+1 (q k) = ( 1)n Qk 1 (n; α + 1, α, j 1 ), k = 1, 3,...,j. 3 The su() α model Definition 6 Let α be a parameter. The algebra su() α is a unital algebra with basis elements J 0, J +, J and P subject to the following relations: P is a parity operator satisfying P = 1 and [P, J 0 ] = PJ 0 J 0 P = 0, {P, J ± } = PJ ± + J ± P = 0. (18) The su() relations are deformed as follows: [J 0, J ± ] = ±J ±, (19) [J +, J ] = J 0 + (α + 1)J 0 P. (0) 118

5 The u() α and su() α Hahn Harmonic Oscillators Proposition 7 Let j be an integer (i.e. j is even), and consider the space W j with basis vectors j, j, j, j +1,..., j, j. Assume that α > 1. Then the following action turns W j into an irreducible representation space of su() α. P j, m = ( 1) j+m j, m, (1) J 0 j, m = m j, m, () (j m)(j + m + α + ) j, m + 1, if j + m is even; J + j, m = (j m + α + 1)(j + m + 1) j, m + 1, if j + m is odd, (3) (j + m + α + 1)(j m + 1) j, m 1, if j + m is odd; J j, m = (j + m)(j m + α + ) j, m 1, if j + m is even. (4) Quite similar as in the su() α deformed case, the position, momentum and Hamiltonian operators defined by: ˆq = 1 (J + + J ), ˆp = i (J + J ), Ĥ = J 0 + j + 1, (5) satisfy (1). The spectrum of the position operator ˆq is not equidistant, it is given by j(α + j + 1), (j 1)(α + j),..., α + ; 0; α +, and the position wavefunctions by..., j(α + j + 1). (6) n (q k) = ( 1)n Qk (n; α, α, j), n = 0, 1,..., j, k = 1,...,j, (7) n+1 (q k) = ( 1)n Qk (n; α +1, α +1, j 1), n = 0, 1,...,j, k = 1,...,j. (8) These discrete wavefunctions, given in terms of Hahn polynomials, have nice properties, and they tend to the parabose wavefunctions when j is large. The analysis shows that for α 1 they tend to the Krawtchouk wavefunctions of the su() model; and for α 1 and j they tend to the canonical oscillator wavefunctions in terms of Hermite polynomials. References [1] A. Vourdas (004) Rep. Progr. Phys [] S.L. Braunstein, V. Buzek and M. Hillery (001) Phys. Rev. A

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