On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on R

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on R R. ´ Alvarez-Nodarse∗, M. K. Atakishiyeva†, and N. M. Atakishiyev‡ ∗Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain and Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, E-18071 Granada, Spain E-mail: [email protected] †Facultad de Ciencias, UAEM, Apartado Postal 396-3, CP 62250, Cuernavaca, Morelos, M´exico E-mail: [email protected] ‡Instituto de Matem´aticas, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaca, Morelos, M´exico E-mail: nati[email protected] Abstract We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics, based on a q-extension of the classical Hermite polynomials Hn(x), recently introduced by us in [1]. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line Rwith respect to a positive weight function. A detailed description of the corresponding q-system is carried out. 1 Introduction In [1] we introduced a q-extension of the classical Hermite polynomials Hn(x), which satisfy the following requirements: They are polynomials in the variable x, which obey a three-term recurrence relation; They are orthogonal on the whole real line Rwith respect to a continuous positive weight function; In the limit as q→1 they coincide with the Hermite polynomials Hn(x). Such a family enables one to build a q-deformed version of the linear harmonic oscillator in quantum mechanics, which is still defined on the whole real line Rand enjoys the continuous orthogonality property on Rwith respect to a positive weight function. Let us point out here that there are several publications (see [2]–[10] and references therein) devoted to the study of explicit realizations, which represent q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator) Hn(x)e−x2/2. But none of these realizations satisfies all of the aforementioned requirements: the continuous weight functions in [2, 4, 7] are supported on the finite intervals; the continuous weight functions in [3, 8] are not positive; the q-extensions in [2], [4]–[9] are not expressed in terms of polynomials in the independent variable; and, finally, the orthogonality relations in [5]–[7], [10] are discrete. Our main goal in this paper has been to employ this q-extension of the Hermite polynomials, Hn(x;q), in order to built a q-analogue to the linear harmonic oscillator in quantum mechanics. Section 2 collects those known results from [1] about the polynomials Hn(x;q), which are needed in section 3 to derive an explicit form of the wave functions ψn(x;q) in this q-model and their properties. Section 4 is devoted to explicit construction of the generators of the dynamical symmetry algebra suq(1,1) in terms of the lowering and raising q-difference operators a(x;q) and a†(x;q). Concluding section 5 contains a brief discussion of q-coherent states for this q-extension of the quantum harmonic oscillator. 1 2 Definition and properties of the polynomials Hn(x;q) In [1] the following family was introduced H2n(x;q) := (−1)n(q;q)nL(−1/2) n(x2;q) = (−1)n(q1/2;q)n1φ1 q−n q1/2 q;−qn+1/2x2!= (−1)n2φ1 q−n,−x2 0 q;qn+1/2!, H2n+1(x;q) := (−1)n(q;q)nx L(1/2) n(x2;q) = (−1)n(q3/2;q)nx1φ1 q−n q3/2 q;−qn+3/2x2!= (−1)nx2φ1 q−n,−x2 0 q;qn+3/2!, (2.1) where L(α) n(x;q) are q-Laguerre polynomials, 1φ1and 2φ1denote the basic hypergeometric polynomials and (a;q)nis the q-shifted factorial (we employ standard notations of q-analysis, see, for example, [11] or [12]). In (2.1) and throughout the sequel it is assumed that qis a fixed number such that 0 < q < 1. This family is generated by the three-term recurrence relation xHn(x;q) = q−n/2Hn+1(x;q)−(1 −q−n/2)Hn−1(x;q), n = 0,1,2, ... , (2.2) with the initial condition H0(x;q)≡1. The polynomials (2.1) satisfy the continuous orthogonality relation ∞ Z −∞ Hm(x;q)Hn(x;q)dx Eq(x2)=πq−n/2(q1/2;q1/2)n(q1/2;q)1/2δmn (2.3) on the whole real line Rwith respect to the positive weight function w(x) = 1/Eq(x2) = 1/(−x2;q)∞[1]. The polynomials Hn(x;q) constitute a q-extension of the classical Hermite polynomials Hn(x) since these polynomials reduce to the latter in the limit as q→1 , i.e., lim q→1(1 −q)−n/2Hn(p1−q x;q) = 2−nHn(x),(2.4) From the recurrence relation (2.2) it follows that the Hn(x;q) can be expressed in terms of the discrete q-Hermite polynomials ˜ h(x;q) of type II as Hn(x;q2) = qn(n−1)/2˜ hn(x;q) := i−n2φ0 q−n, ix − q;−qn!.(2.5) So from the known q-difference equation for the discrete q-Hermite polynomials ˜ hn(x;q) (see [13], (3.29.5), p.119) one deduces that (1 −qn/2)x2Hn(x;q) = (1 + q1/2+x2)Hn(x;q) −(1 + x2)Hn(q1/2x;q)−q1/2Hn(q−1/2x;q). (2.6) Similarly, one readily verifies that the forward and backward shift operators for the polynomials Hn(x;q) are of the form hq−1 2xd dx −1iHn(x;q) = q−1/2(1 −qn/2)xHn−1(x;q), h(1 + x2)q1 2xd dx −1iHn(x;q) = xHn+1(x;q), (2.7) 2 respectively, where qa x d dx is the dilation operator, i.e., qa x d dx f(x) = f(qax). A Rodrigues-type difference formula for the polynomials Hn(x;q) can be written as Hn(x;q) = (−x)−nEq(x2) (q1 2xd dx ;q−1/2)nE−1 q(x2),(2.8) where we have slightly simplified the n-th power of the q-derivative operator Dq(cf (3.29.10) in [13], p.119) by representing it in the form Dn q≡1 (1 −q)nxn(qxd dx ;q−1)n, n = 0,1,2, ... . (2.9) It is not difficult to prove (2.9) by induction on the power n. Finally, using the generation function for the discrete q-Hermite polynomials ˜ hn(x;q) of type II [13], one finds that (−xt;q1/2)∞ (−t2;q)∞ =∞ X n=0 1 (q1/2;q1/2)nHn(x;q)tn.(2.10) 3 Wave functions ψn(x;q)and their properties We wish to discuss a q-model of the linear harmonic oscillator, which is described by the wave functions of the form ψn(x;q) := d−1 n(q)Hn(x;q)E−1/2 q(x2) (3.1) with the normalization constant dn(q) := q−n/4qπ(q1/2;q)1/2(q1/2;q1/2)n. Then, by continuous orthogonality relation (2.3), these functions are orthonormal on R, that is, ∞ Z −∞ ψm(x;q)ψn(x;q)dx =δmn.(3.2) The wave functions ψn(x;q) are defined by (3.1) in such a way that in the limit as q→1 they coincide with the orthonormalized Hermite functions (or the wave functions of the linear harmonic oscillator in non-relativistic quantum mechanics): lim q→1ψnp1−q ξ;q=1 p√π2nn!Hn(ξ) exp (−ξ2/2) =: ψn(ξ).(3.3) This limit property of ψn(x;q) follows immediately from (2.4) and the well-known fact lim q→1Eq((1 −q)z) = ez(3.4) about the Jackson q-exponential function Eq(z) (see [11] or [12]). From (2.6) and (3.1) one obtains that the wave functions ψn(x;q) are eigenfunctions of the q-Hamiltonian H(x;q), H(x;q)ψn(x;q) = En(q)ψn(x;q), En(q) := 1−qn/2 1−q1/2.(3.5) By equation (2.6), the explicit form of this self-adjoint q-difference operator is H(x;q) := 1 (1−q1/2)x2h(1 + x2+q1/2)I−p1 + x2q1 2xd dx −q1 2(1−xd dx )p1 + x2i,(3.6) 3 where Iis the identity operator. This expression for H(x;q) in terms of the dilation operators q±1 2xd dx may create an impression that the H(x;q) contains singularity at x= 0 due to the presence of the factor x2in the denominator. To remove this doubt one should take into account that, by definition (3.6), H(x;q)ψn(x;q) = 1 (1−√q)x2(1 + √q+x2)ψn(x;q) −p1 + x2ψn(q1/2x;q)−pq+x2ψn(q−1/2x;q)i(3.7) for all n= 0,1,2, ... . Besides, from (3.1) it is evident that the wave functions ψn(x;q) have regular behavior around x= 0. Now substituting the sum of first two terms c0+c1xfrom the expansion of ψn(x;q) around x= 0 into expression in square brackets in (3.7) and keeping only constant and linear in xterms, one readily verifies that (1 + √q) (c0+c1x)−(c0+c1√q x)−√qc0+c1 √qx= 0 . Consequently, the total combination inside the square brackets in (3.7) behaves like x2in the x→0 limit and the right side of (3.7) therefore assumes a constant value at x= 0. This confirms that there is no singularity at x= 0. We observe also that the eigenvalues En(q) of H(x;q) are bounded from above by the asymptotic value E∞(q) = 1/(1 −q1/2) and, since En+1(q)−En(q) = qn/2, they are not equidistant. From (2.2) it follows that the wave functions ψn(x;q) satisfy the three-term recurrence relation x ψn(x;q) = q−(2n+1)/4q1−q(n+1)/2ψn+1(x;q) + q(1−2n)/4q1−qn/2ψn−1(x;q) (3.8) with the initial condition that the ground state ψ0(x;q) = d−1 0(q)E−1/2 q(x2). Likewise, from the explicit form of the forward and backward shift operators (2.7) it follows that a(x;q)ψn(x;q) = pEn(q)ψn−1(x;q), a†(x;q)ψn(x;q) = pEn+1(q)ψn+1(x;q),(3.9) where the q-difference lowering and raising operators a(x;q) and a†(x;q) are given by a(x;q) = q1/4 √1−q1/2xq−1 2xd dx √1 + x2−I, a†(x;q) = q1/4 √1−q1/2x√1 + x2q1 2xd dx −I, (3.10) respectively. We invite the reader to verify that these operators are indeed mutually adjoint in the Hilbert space L2(R, dx) of square integrable functions f(x) with respect to dx. Similar to the case of the quantum linear harmonic oscillator, the lowering and raising operators (3.10) factorize the Hamiltonian (3.6), that is, H(x;q) = a†(x;q)a(x;q).(3.11) Moreover, it is not difficult to verify, by using (3.10), that their another (i.e., when the operator a(x;q) is right multiplied by its adjoint operator a†(x;q)) product a(x;q)a†(x;q) 4 is equal to I+q1/2H(x;q). This means that the operators a(x;q) and a†(x;q) satisfy the q-commutation relation of the form a(x;q)a†(x;q)−q1/2a†(x;q)a(x;q)≡ha(x;q), a†(x;q)iq1/2=I . (3.12) It should be noted at this point that we have used above the known explicit form of the forward and backward shift operators (2.7) for the polynomials Hn(x;q) in order to find the lowering and raising operators a(x;q) and a†(x;q). But we could have started equivalently with the q-difference equation (3.5) itself and have directly factorized it in terms of the same operators a(x;q) and a†(x;q) (for a more detailed discussion of the factorization of difference equations, see, for example, [15, 16]). So we have established that our q-model is governed by the Hamiltonian (3.6), which admits the factorization (3.11) in terms of the operators a(x;q) and a†(x;q), satisfying the qcommutation relation (3.12). This characteristic property of the Hamiltonian (3.6) is known to reflect the fact that the dynamical symmetry of this q-model is described by the quantum algebra suq(1,1) [14]. In the next section we construct explicitly the generators of this algebra in terms of the lowering and raising operators a(x;q) and a†(x;q). 4 Dynamical symmetry In this section we remind the reader first how one constructs a dynamical symmetry algebra for the linear harmonic oscillator, which is governed in non-relativistic quantum mechanics by the well-known Hamiltonian H(x) := ~ω 2ξ2−d2 dξ2≡~ωN(x) + 1 2,(4.1) where ξ=pmw/~xis a dimensionless coordinate, N(x) is the particle number operator, N(x) := a†(x)a(x),(4.2) and the annihilation and creation operators are defined as usual: a(x) = 1 √2ξ+d dξ , a†(x) = 1 √2ξ−d dξ , a(x), a†(x)≡a(x)a†(x)−a†(x)a(x) = I . (4.3) By using (4.1) and (4.3) one readily verifies that [H(x), a(x) ] = −a(x),hH(x), a†(x)i=a†(x).(4.4) Observe that in the case of the linear harmonic oscillator (4.1) there is no much difference between the Hamiltonian H(x) and the particle number operator N(x): the former operator, divided by the factor ~ω, is equal to the latter one plus a constant term 1/2. So, the particle number operator N(x) satisfies the same commutation relations (4.4) with the annihilation and creation operators a(x) and a†(x). Having factorized the Hamiltonian H(x) (or, equivalently, the particle number operator N(x)) in terms of the annihilation a(x) and creation a†(x) operators, one explicitly constructs the closed Lie algebra su(1,1) with the three generators K0(x) := 1 2~ωH(x)≡1 2N(x) + 1 2, K+(x) := 1 2a†(x)2, K−(x) := 1 2a2(x).(4.5) 5 Indeed, it is not difficult to verify that thus defined generators satisfy the standard commutation relations [K0(x), K±(x)] = ±K±(x),[K−(x), K+(x)] = 2 K0(x),(4.6) of the algebra su(1,1). Unitary irreducible representations of this algebra are known to be characterized by eigenvalues of the invariant (that is, commuting with all three generators (4.5)) Casimir operator C:= K0(x) [ K0(x)−I]−K+(x)K−(x) = s(s−1) I . (4.7) A direct calculation of the Casimir operator (4.7) with the aid of (4.5) shows that the eigenvalue s(s−1) in this particular case is equal to −3/16. This means that the parameter smay be equal to either s1= 1/4 or s2= 3/4. Each of these two values of sdefines a unitary irreducible representation of the algebra su(1,1): D+(1/4) consists of those eigenstates of the Hamiltonian H(x), which correspond to the eigenvalues s1+n=n+1/4 = (2n+1/2)/2, n= 0,1,2, ..., of the generator K0(x) = H(x)/2~ω; whereas D+(3/4) corresponds to the eigenvalues s2+n=n+ 3/4 = (2n+ 1 + 1/2)/2 of the same generator K0(x). So in this way one arrives at the correct spectrum En=~ω(n+ 1/2) of the Hamiltonian H(x), without solving an eigenvalue problem for the appropriate Schr¨odinger equation. Thus eigenstates of H(x) with the eigenvalues E2nform the unitary irreducible representation D+(1/4) and those with E2n+1 form another one, D+(3/4). The Fock space HFof all eigenfunctions {ψn(x)}of the Hamiltonian H(x) splits into two su(1,1)-irreducible subspaces for H(x) is symmetric with respect to the inversion x→ −x. Therefore the inversion operator P,P x =−x, commutes with all three generators (4.5) and HFdecomposes into two irreducible components, HF=H0⊕ H1,(4.8) consisting of the wave functions ψn(x) with even and odd indices n, respectively. The irreducible subspaces H0and H1are characterized by the eigenvalues (−1)ǫof the operator P with ǫ= 0 in H0and ǫ= 1 in H1. It is clear that the subspaces H0and H1correspond to the unitary irreducible representations D+(1/4) and D+(3/4), respectively. Now we are in a position to discuss a dynamical symmetry algebra for the q-model (3.1). To construct it one needs to introduce first the operator [14] N(x;q) := 2 ln qln h1−(1 −q1/2)H(x;q)i.(4.9) Since the wave functions ψn(x;q) are eigenfunctions of the q-Hamiltonian Hn(x;q) with the eigenvalues En= (1 −qn/2)/(1 −q1/2), from the definition (4.9) one deduces that N(x;q)ψn(x;q) = n ψn(x;q),(4.10) that is, N(x;q) is the particle number operator and [N(x;q), a(x;q)] = −a(x;q),hN(x;q), a†(x;q)i=a†(x;q).(4.11) At the next step one defines a new set of the operators b(x;q) := q−N(x;q)/8a(x;q), b†(x;q) := a†(x;q)q−N(x;q)/8,(4.12) which satisfy, according to (4.11), the following commutation relation b(x;q)b†(x;q)−q1/4b†(x;q)b(x;q) = q−N(x;q)/4.(4.13) 6 This is readily verified with the aid of (4.11). The operators b(x;q), b†(x;q), and N(x;q) directly lead to the dynamical algebra suq1/2(1,1) with the generators K+(x;q) := γb†(x;q)2, K−(x;q) := γ b2(x;q), K0(x;q) := 1 2N(x;q) + 1 2, γ= [ 1/2 ]q1/2. (4.14) It is not difficult to check that thus defined generators (4.14) satisfy the standard commutation relations [K0(x;q), K±(x;q)] = ±K±(x;q),[K−(x;q), K+(x;q)] = [2K0(x;q)]q1/2,(4.15) of the quantum algebra suq1/2(1,1). The q-number [ A]qin (4.14) is given by the common expression [A]q:= qA−q−A q−q−1.(4.16) We are interested in the positive discrete series representations of the quantum algebra suq(1,1) with lowest weights. These irreducible representations of suq(1,1) are denoted by T+ l, where lis the lowest weight, which can be any positive number (see, for example, [17]). It is the characteristic property of every T+ lthat the generator K0(x;q) has the eigenvalues l+n,n= 0,1,2, ..., in T+ l. The invariant Casimir operator in the case under discussion is equal to C(q) := [ K0(x;q)−1/2 ]2 q1/2−K+(x;q)K−(x;q)−1 4I=[ 1/4 ]2 q1/2−1/4I. (4.17) This means that two possible values of the parameter sin this case are s1(q) = 1/2−[ 1/4 ]q1/2, s2(q) = 1/2 + [ 1/4 ]q1/2.(4.18) Since [a]q→ain the limit as q→1 by definition of the q-number (4.16), the eigenvalue of C(q) in (4.17) reduces in this limit to the eigenvalue for the Casimir operator in the case of the linear harmonic oscillator (4.7). Evidently, the same happens with the values of s1(q) and s2(q): they coincide in this limit with the corresponding values of the parameter sin (4.7), i.e., lim q→1s1(q) = 1 4,lim q→1s2(q) = 3 4.(4.19) From (4.17) it now follows that the lowest weights in our case are 1/4 and 3/4. Therefore by (4.14) the eigenvalues of the particle number operator N(x;q)≡2K0(x;q)−1/2 are equal to 2nand 2n+1, n= 0,1,2, ..., respectively. Taking into account interrelation (4.9) between the operators N(x;q) and H(x;q), one thus arrives at the correct spectrum (3.5) for the Hamiltonian H(x;q), without solving an eigenvalue problem for H(x;q). So we conclude that the wave functions ψn(x;q), defined in (3.1), form a representation of the quantum algebra suq1/2(1,1) in the Fock space HF. This representation in the space HFis reducible precisely for the same reason as in the case of the linear harmonic oscillator (4.1). Thus HFsplits into two suq1/2(1,1)-irreducible subspaces H0≡T+ 1/4and H1≡T+ 3/4, consisting of the wave functions ψn(x;q) with even and odd indices n, respectively. 7 5q-coherent states As in the case of the non-relativistic linear harmonic oscillator, one can construct q-coherent states for this model as eigenfunctions of the lowering operator a(x;q), that is, a(x;q)ϕζ(x;q) = ζ ϕζ(x;q),(5.1) where ζis some arbitrary number. To find an explicit form of these states ϕζ(x;q), we first note that by (3.8) ψn(x;q) = cn(q)ha†(x;q)inψ0(x;q), cn(q) := s(1 −q1/2)n (q1/2;q1/2)n .(5.2) Consequently, with the aid of (3.8) it is not difficult to verify that the states ϕζ(x;q) := fq(ζ)∞ X n=0 cn(q)ζnψn(x;q),(5.3) where fq(ζ) is some normalization factor (see below), are indeed the eigenstates of the operator a(x;q) with the eigenvalues ζ. They form an overcomplete system in the Hilbert space HFand they are not orthogonal in this space. In fact, one can prove, by using expansion (5.3) and orthogonality relation (3.2), that Z∞ −∞ ϕζ(x;q)ϕζ′(x;q)dx =fq(ζ)fq(ζ′)eq1/2(1 −q1/2)ζ ζ′,(5.4) where eq(z) := ∞ X n=0 zn (q;q)n =1 (z;q)∞ ,|z|<1. The normalization condition that the integral on the left of (5.4) is equal to 1 requires to choose fq(ζ) = qEq1/2−(1 −q1/2)ζ2. Thus, ϕζ(x;q) = qEq1/2−(1 −q1/2)ζ2∞ X n=0 cn(q)ζnψn(x;q),(5.5) Substitute now into expansion (5.5) explicit form of the coefficients cn(q) from (5.2) and the normalization constants dn(q) for the wave functions ψn(x;q) from (3.1) and employ then the generating function (2.10) for the polynomials Hn(x;q). This yields the final form of the normalized q-coherent eigenfunctions of the lowering operator a(x;q): ϕζ(x;q) = sEq1/2−(1 −q1/2)ζ2 π(q1/2;q)1/2Eq(x2) Eq1/2q1/4p1−q1/2xζ  Eqq1/2(1 −q1/2)ζ2.(5.6) Acknowledgments: The research of RAN has been partially supported by the DGES grant BFM 2003-06335-C03-01 and PAI grant FQM-0262. The participation of NMA in this work has been supported in part by the UNAM–DGAPA grant IN102603-3 ´ Optica Matem´atica. 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