Discrete quantum model of the harmonic oscillator
Natig M. Atakishiyev, Anatoliy U. Klimyk, Kurt Bernardo Wolf
aa r X i v : . [ m a t h - ph ] N ov Discrete quantum modelof the harmonic oscillator
Natig M. Atakishiyev , Instituto de Matem´aticasUniversidad Nacional Aut´onoma de M´exicoAv. Universidad s/n, Cuernavaca, Morelos 62251, M´exico,
Anatoliy U. Klimyk , Bogolyubov Institute for Theoretical PhysicsMetrologichna 14b, Kiev 03143, Ukraine,
Kurt Bernardo Wolf , Instituto de Ciencias F´ısicasUniversidad Nacional Aut´onoma de M´exicoAv. Universidad s/n, Cuernavaca, Morelos 62251, M´exico.
Abstract
We construct a new model of the quantum oscillator, whose energyspectrum is equally-spaced and lower-bound, whereas the spectra ofposition and of momentum are a denumerable non-degenerate set ofpoints in [ − ,
1] that depends on the deformation parameter q ∈ (0 , q -Hermite polynomials. Webuild a Hilbert space with a unique measure, where an analogue ofthe fractional Fourier transform is defined in order to govern the timeevolution of this discrete oscillator. In the limit when q → − onerecovers the ordinary quantum harmonic oscillator. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Introduction
Several algebraic constructions have been proposed in the literature to de-scribe various extensions of the quantum harmonic oscillator. These con-structions are based on various deformations of the standard oscillator Liealgebra, or different associative algebras. In most of these models it is dif-ficult to construct a theory for such oscillators, which is as complete as thewell-known treatment of the standard harmonic oscillator in quantum me-chanics. Namely: a canonical complementarity between position and mo-mentum, explicit forms for the wavefunctions, and a coherent description oftime evolution.The earliest model, generally called the q -oscillator, was proposed byMacfarlane [1] and Biedenharn [2] on the basis of raising and lowering eigen-states of a Hamiltonian with the q -deformed commutator a + q a q − q a q a + q . Atheory of this oscillator has been elaborated (see, e.g. [3]–[6]); yet, it has notbeen clear how to construct position and momentum operators satisfying thebasic commutation relations with a Hamiltonian to characterize infinitesimalharmonic motion. This may be one of the reasons why this q -oscillator hasnot proven attractive for many physicists.The postulates we use to define oscillator models are the following [7, 8]: There exists an essentially self-adjoint position operator Q , whose spec-trum X is the set of positions { x } of the system. There exists a self-adjoint and compact
Hamiltonian operator H , whosecommutator with position defines the momentum operator P ,[ H, Q ] =: − i P, (1)and corresponds to the first Hamilton equation (i = √− H, P ] = i Q, (2)that corresponds with the second Hamilton equation, and which characterizesthe oscillator dynamics. Equivalent to (1)–(2), one can propose the Newton-Lie equation as [ H, [ H, Q ]] = Q . The set of momentum values of the systemis the spectrum of P , which is equal to that of Q because (1)–(2) generate arotation between these two operators (to be written below).2 . The three operators, Q , P and H , close into an associative algebra , i.e. ,they satisfy the Jacobi identity,[ P, [ H, Q ]] + [ Q, [ P, H ]] + [ H, [ Q, P ]] = 0 . (3)We note that the basic commutator [ Q, P ] has not been defined. Due to(1)–(2), the only restriction imposed by the associativity condition (3), sincethe first two summands are identically zero, is that [
Q, P ] must commutewith H , and thus be constant under the oscillator motion. This indicatesthat each distinct choice of the basic commutator [ Q, P ] will yield a dis-tinct model for the oscillator. If the choice is the Heisenberg commutator[
Q, P ] = ~ ˆ1, one has the standard four-generator oscillator Lie algebra H =span { H, Q, P, ˆ1 } of quantum mechanics (containing the Heisenberg algebra H = span { Q, P, ˆ1 } ). In one previous work [9], with the purpose of endowing Q with a finite set of position eigenvalues x ∈ {− j, − j +1 , . . . , j } (we write x | j − j ), the basic commutator was taken to be [ Q, P ] = i [ H − ( j + )ˆ1] =: i J ,in a matrix representation of the Lie algebra so(3) = su(2) = span { Q, P, J } of spin j , so the three operators have the same finite spectrum x | j − j . Thismodel, called the su(2) oscillator, has been applied to study the parallel pro-cessing of finite signals and pixellated images [10]. The general conditions toinclude oscillator dynamics in associative algebras were given in [7].In a previous paper of the same authors [8], a q -algebraic associative struc-ture was proposed on the basis of the quantum algebra su q (2), the Hamilto-nian having a lower-bound equally-spaced spectrum. Using a non-standardbasis to define position and momentum operators that allowed analytic ex-pressions, their spectra was determined to be a finite set of non-equally spacedpoints x s = sinh( sκ ) / sinh κ , with s | j − j and q = e − κ ∈ (0 , q -Kravchuk polynomials, related by a fractional finiteFourier- q -Kravchuk matrix transform, and a natural representation in a suigeneris phase space. The present paper is a continuation of the research in[11] that constructed quantum oscillators with continuous bounded spectrafor the position and momentum operators.In this paper we build an oscillator model on the basis of the Fock repre-sentation of a quadratic associative algebra which is a q -deformation of thestandard oscillator algebra; we denote this by DH , which will be definedin Section 2, while the physical interpretation of the participant operators3hat characterizes this model are set forth in Section 3. The spectrum ofthe Hamiltonian in the algebra is lower-bound and equally spaced, as in itsstandard counterpart. The position spectrum and wavefunctions, orthonor-mal under a specific scalar product over positions, are obtained explicitlyin Section 4, and the momentum wavefunctions in Section 5. This modelcan be characterized for having a space of positions given by an infinite non-degenerate point set contained in the interval [ − , q -orthogonal polynomials (see, for example, [12]), andwe assume throughout that q is a fixed real number in (0 , D q H D q H as the associative algebra generated by a vectorbasis of elements I + , I − , I , satisfying the following commutation relations:[ I , I ± ] = ± I ± , [ I + , I − ] = q I − (1 + q ) q I . (4)Equivalently, introducing I := I + + I − and I := i( I + − I − ), we can charac-terize this algebra by[ I , I ] = − i I , [ I , I ] = i I , [ I , I ] = − q I − (1 + q ) q I ) . (5)The first relation in (4) can be written in the form q I I ± q − I = q ± I ± . (6)This relation and the fact that both q I and q I appear in the second rela-tion of (4) shows that D q H = span { I , I , q I , q I } is a quadratic associa-tive algebra. This is a q -deformation of the oscillator algebra H becauselim q → − D q H = H . Indeed, in the limit lim q → − we obtain from (5) therelations [ I , I ] = − i I , [ I , I ] = i I , [ I , I ] = 2i , (7)4hich are equivalent to the defining relations of H .We are interested in the Fock representation of the algebra D q H ; thisis an irreducible representation constructed on a Hilbert space with the or-thonormal basis of vectors e n , n ∈ { , , , · · · } ( i.e. , n | ∞ ). In this represen-tation, using the ‘box’ q -number [ a ] q := (1 − q a ) / (1 − q ), the operators of thealgebra act by raising and lowering the number n of e n , I + e n = q q n +1 [ n +1] q e n +1 , I − e n = q q n [ n ] q e n − , (8) I e n = n e n , i.e., q I e n = q n e n , (9)and the hermiticity conditions I ∗ + = I − and I ∗ = I are satisfied.In order to have a functional realization of this representation, we considerthe space P of all polynomials in one supplementary variable y , and introduceits basis of monomials e n ↔ e n ( y ) := c n y n , c n = q n ( n − / ( q ; q ) / n , n | ∞ , (10)where ( a ; q ) n := (1 − a )(1 − aq ) . . . (1 − aq n − ) and ( a ; q ) = 1. Acting onanalytic functions f ( y ) ∈ P , the Fock representation can be written in termsof the scale T a and q -difference D q operators, I + = r q − q y T q , q I = T q , I − = [ q (1 − q )] / D q ; (11) T a f ( y ) = f ( ay ) , D q f ( y ) = f ( y ) − f ( qy )1 − q . (12)This realization of the algebra is equivalent to that in (8) –(9), with thefunctions e n ( y ) playing the role of the basis elements e n as eigenfunctions ofthe weight operator I .Let us now introduce a scalar product into the space of polynomials P .This scalar product is of a Fisher-type scalar product and is given by theformula h f ( y ) , f ( y ) i = f ( e D q ) f † ( y ) | y =0 , where e D q := (1 − q ) T q − D q , f and f are polynomials, and f † denotes thepolynomial f , whose coefficients are replaced by their complex conjugate5nes. In this formula, we have the action of the difference operator upon thepolynomial f † . Then it is directly verified that h e n , e n ′ i = δ n,n ′ , n, n ′ | ∞ . (13)Closing the space P with respect to this scalar product we obtain a Hilbertspace that we denote by H . The space H consists of functions f ( y ) = ∞ X n =0 b n e n ( y ) = ∞ X n =0 b n c n y n = ∞ X n =0 a n y n , (14)where a n = b n c n , and c n are determined by (10). Since h e n , e n ′ i = δ n,n ′ bydefinition, for f ( y ) = P ∞ n =0 a n y n and f ( y ) = P ∞ n =0 a ′ n y n we have h f , f i = ∞ X n =0 a n a ′ n | c n | . (15)This means that the Hilbert space H consists of analytic functions f ( y ) = P ∞ n =0 a n y n such that k f k := ∞ X n =0 (cid:12)(cid:12)(cid:12)(cid:12) a n c n (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . (16) The discrete oscillator is a class of oscillator models that depend on theparameter q ∈ (0 , D q H , where the physical observables are assigned to the spectra ofself-adjoint generators of the algebra in the following way:position: Q := p (1 − q ) /q I , (17)momentum: P := p (1 − q ) /q I , (18)Hamiltonian: H := I + ˆ1 . (19)Then, due to (5), H exhibits the Hamiltonian oscillator commutation rela-tions (1)–(2) with Q and P , and determines the basic commutator [ Q, P ],through [
H, Q ] = − i P, [ H, P ] = i Q, (20)[ Q, P ] = 2i(1 − q − ) (cid:2) q H − / − (1 + q ) q H − (cid:3) =: i F ( H ) . (21)6he operator F ( H ) defined in (21) commutes with the Hamiltonian H andis therefore also diagonal in the Fock basis { e n } ∞ n =0 in (10), F ( H ) e n = 2(1 − q − ) (cid:2) q n − (1 + q ) q n (cid:3) e n . (22)This basis of H thus consists of eigenfunctions of a Hamiltonian with equally-spaced eigenvalues, H e n = ( n + ) e n , n | ∞ , (23)coinciding with the energy spectrum of the standard quantum harmonic os-cillator.From (20), the time evolution of the discrete oscillator position and mo-mentum operators is produced byexp(i τ H ) = e i τ/ exp(i τ I ) , (24)and results in the harmonic motion (cid:18) Q ( τ ) P ( τ ) (cid:19) = e i τH (cid:18) QP (cid:19) e − i τH = (cid:18) cos τ sin τ − sin τ cos τ (cid:19) (cid:18) QP (cid:19) , (25)which for τ ∈ [0 , π ) forms a group U(1) of inner automorphisms of thepair of operators Q and P , that we interpret as rotations of a phase planearound its origin (still to be studied for this model, see [8]). The phase e i τ/ is due to the energy of the ground state e , while exp(i τ I ) is the discretecounterpart of the fractional Fourier transform for this model, to be seen inSection 8 below.The associative algebra D q H is a q -deformation of the standard Heisen-berg-Weyl algebra, where I ± in (8) are recognized as the raising and loweringoperators a + q := I + = r q − q ( Q − i P ) , a q := I − = r q − q ( Q + i P ) . (26)From (4) it then follows that[ lim q → − a q , lim q → − a + q ] = lim q → − (cid:2) (1+ q ) q I − q I (cid:3) = 1 , (27)so we recover the standard oscillator. This is the place to emphasize thedifference between the Macfarlane–Biedenharn q -oscillator [1]–[6], which isdefined in terms of q -commutators a q a + q − q a + q a q , and our discrete oscillator,which is formulated exclusively with ordinary commutators.7 Spectrum and eigenfunctions of the posi-tion operator
A direct calculation using (8) shows that in the Fock eigenbasis { e n } ∞ n =0 ofthe Hamiltonian H , the position operator Q = p q − − I acts as Q e n = p q n (1 − q n +1 ) e n +1 + p q n − (1 − q n ) e n − . (28)Since | q n (1 − q n +1 ) | ≤ n | ∞ , the norm k Q k of Q does not exceed 1,and hence it is a bounded operator whose eigenvalues { x } will lie in the realinterval [ − , To find the eigenvectors ψ x ( y ) and the spectrum { x } = X of the positionoperator Q , Q ψ x ( y ) = x ψ x ( y ) , x ∈ X , (29)we represent ψ x ( y ) in the form of a linear combination of the monomials (10), ψ x ( y ) = ∞ X n =0 p n ( x ) e n ( y ) , (30)where p n ( x ) are coefficients depending on the points of the spectrum x ∈ X .When we substitute the expansion (30) into the equation (29), we obtain ∞ X n =0 p n ( x ) p q n (1 − q n +1 ) e n +1 + p n ( x ) p q n − (1 − q n ) e n − = x ∞ X n =0 p n ( x ) e n . (31)From here we find the following three-term recurrence relation for the coef-ficients p n ( x ) in (30), x p n ( x ) = p q n (1 − q n +1 ) p n +1 ( x ) + p q n − (1 − q n ) p n − ( x ) , (32)starting with p − ( x ) = 0 and p ( x ) := 1 setting the common constant factor.8e see from (32) that the coefficients p n ( x ) in (30) are polynomials in x ofdegree n , which can be evaluated uniquely. To solve the recurrence relation,we make the substitution p n ( x ) = ( q ; q ) − / n q − n ( n − / e p n ( x ) , (33)which turns (32) into x e p n ( x ) = e p n +1 ( x ) + q n − (1 − q n ) e p n − ( x ) . (34)Comparing this with the recurrence relation for the discrete q -Hermite poly-nomials of type I, given by φ basic hypergeometric polynomials [13, Eq.(3.28.3)], h n ( z ; q ) := q n ( n − / φ ( q − n , z − ; 0; q ; − q z ) , (35) z h n ( z ; q ) = h n +1 ( z ; q ) + q n − (1 − q n ) h n − ( z ; q ) , (36)we establish that e p n ( x ) = h n ( x ; q ). We can thus write the coefficient polyno-mials in (30) as p n ( x ) = ( q ; q ) − / n q − n ( n − / h n ( x ; q ) . (37)From (32) follows that these polynomials have definite parity: p n ( x ) =( − n p n ( x ).Collecting these results we write the eigenfunctions ψ x ( y ) of the positionoperator Q as ψ x ( y ) = ∞ X n =0 ( q ; q ) − / n q − n ( n − / h n ( x ; q ) e n ( y ) (38)= ∞ X n =0 ( q ; q ) − n h n ( x ; q ) y n , (39)= ( y ; q ) ∞ ( xy ; q ) ∞ , (40)where in the last expression we use the symbol ( a ; q ) ∞ := Q ∞ n =0 (1 − aq n ) andthe summation formula in [13, Eq. (3.28.11)]. Because of the convergence of( y ; q ) ∞ in (40) for the basis { ψ x ( y ) } x ∈X , we must restrict the domain ofdefinition of functions f ( y ) ∈ H to the open disk | y | <
1. Then the condition( xy ; q ) ∞ < x ∈ X of Q are contained in the interval [ − , .2 The spectrum of position The spectrum of the self-adjoint position operator Q ∼ I = I + + I − can befound from the series (38); from (28) we see that in the basis { e n ( y ) } ∞ n =0 theoperator Q is a self-adjoint Jacobi tridiagonal matrix of the form Q = b a · · · a b a · · · a b a · · · a b a · · · ... ... ... ... ... . . . , a n = 0 . (41)We can now use the theory of these matrices from [14, Chap. VII], (see also[6]) to connect their spectra with the corresponding measures for orthogo-nal polynomials. In this vein, we note that in the Fock basis the positioneigenfunctions ψ x ( y ) are expanded in terms of the basis elements { e n ( y ) } ∞ with the polynomial coefficients p n ( x ) in (30), which are given in terms ofdiscrete q -Hermite polynomials of type I in (37). According to the resultsin [14, Chap. VII], these polynomials are then orthogonal with respect to aspectral measure d µ ( x ) of the operator, which is unique up to a constantfactor, on a set X ⊂ ℜ that is the simple spectrum of Q .In finding the spectrum of the position operator Q , we recall that thediscrete q -Hermite polynomials h n ( x ; q ) obey the orthogonality relation Z − ( q x ; q ) ∞ h k ( x ; q ) h m ( x ; q ) d q x = δ k,m (1 − q )( q ; q ) ∞ ( − q ) ∞ ( q ; q ) m q m ( m − / = 2 δ k,m (1 − q )( q ; q ) ∞ ( − q ; q ) ∞ ( q ; q ) m q m ( m − / , (42)where R − f ( x ) d q x is the symbol of the q -integral (see [13, Eq. (3.28.2)]).This orthogonality relation can be written in the form of a sum [12], ∞ X n =0 ( q n +2 ; q ) ∞ q n (cid:16) h k ( q n ; q ) h m ( q n ; q ) + h k ( − q n ; q ) h m ( − q n ; q ) (cid:17) = 2 δ k,m ( q ; q ) ∞ ( − q ; q ) ∞ ( q ; q ) m q m ( m − / . (43)This means that the spectrum X of Q is the simple set of points X = { q n , − q n ; n | ∞ } , (44)10nd that the corresponding eigenfunctions are ψ q n ( y ) , ψ − q n ( y ) , n | ∞ , (45)given by (38)–(40). The spectrum of Q is discrete, which means that theeigenfunctions ψ ± q n ( y ) form a denumerable orthogonal basis in the Hilbertspace H ; we note that X ⊂ [ − ,
1] has a unique accumulation point 0 thatdoes not belong to the set.
The eigenfunctions of Q were determined only up to constant factors, so weproceed to normalize the eigenfunctions { ψ ± q s ( y ) } ∞ in their form (39). From(38) and the orthogonality of the basis { e n ( y ) } ∞ n =0 we obtain h ψ x ( y ) , ψ x ′ ( y ) i H = δ x,x ′ ∞ X n =0 q − n ( n − / ( q ; q ) n h n ( x ; q ) h n ( x ′ ; q ) , (46)where x and x ′ take values in X = { q s , − q s ; s | ∞ } . We can calculate thissum as follows: we build the functions e h n ( q s ; q ) := s ( q s +2 ; q ) ∞ q s q ; q ) ∞ ( − q ; q ) ∞ ( q ; q ) n q n ( n − / h n ( q s ; q ) , (47) e h n ( − q s ; q ) := s ( q s +2 ; q ) ∞ q s q ; q ) ∞ ( − q ; q ) ∞ ( q ; q ) n q n ( n − / h n ( − q s ; q ) . (48)We can see the e h n ( ± q s ; q ) as the elements ( n, ± s ) of a matrix of numbers(integer n ≥ s ≥ (cid:18) { e h n ( q s ; q ) } ∞ n,s =0 { e h n ( − q s ; q ) } ∞ n,s =0 (cid:19) . (49)Columns of this matrix are orthonormal due to the orthogonality relation(43) for the discrete q -Hermite polynomials h k ( z ; q ). In the infinite dimen-sional case, the orthonormality of columns does not immediately lead to the11rthonormality of rows. But in accordance with the reasoning of Ref. [15],one can state that rows of this matrix are also orthogonal, i.e. , ∞ X n =0 e h n ( q s ; q ) e h n ( q s ′ ; q ) = δ s,s ′ , (50) ∞ X n =0 e h n ( q s ; q ) e h n ( − q s ′ ; q ) = 0 , (51) ∞ X n =0 e h n ( − q s ; q ) e h n ( − q s ′ ; q ) = δ s,s ′ . (52)Substituting (47)–(48) into (50)–(52), we obtain( q s +2 ; q ) ∞ q s q ; q ) ∞ ( − q ; q ) ∞ ∞ X n =0 h n ( ± q s ; q ) h n ( ± q s ′ ; q )( q ; q ) n q n ( n − / = δ s,s ′ , (53)where one has to take only the upper or only the lower signs. Returning tothe scalar product in (46), we find h ψ ± q s ( y ) , ψ ± q s ′ ( y ) i H = δ s,s ′ q ; q ) ∞ ( − q ; q ) ∞ q s ( q s +2 ; q ) ∞ . (54)We thus arrive at the functionsΨ x ( y ) ≡ Ψ ± q s ( y ) := s ( q s +2 ; q ) ∞ q s q ; q ) ∞ ( − q ; q ) ∞ ψ ± q s ( y ) (55)which are orthonormal under the scalar product (46) in H , h Ψ x ( y ) , Ψ x ′ ( y ) i H = δ x,x ′ , x, x ′ ∈ X . (56) The momentum operator P ∼ I = i( I + − I − ) acts on the basis { e n ( y ) } ∞ as P e n = i (cid:16)p q n (1 − q n +1 ) e n +1 − p q n − (1 − q n ) e n − (cid:17) (57)12 cf . (28)]. When we change this basis to another { e e n } ∞ with e e n = i n e n , onecan see that in the new basis the momentum operator P acts as a matrixwith the same coefficient elements as the position operator in Section 4 onthe former basis. This means that the spectrum of momentum P coincideswith the spectrum of position Q , namely, Spec P = X , where X is given in(44). Similarly, the eigenfunctions of momentum P can be found in the sameway as the eigenfunctions of Q , by using the basis { e e n } ∞ .Let φ p ( y ) satisfy P φ p ( y ) = pφ p ( y ), an eigenfunction of P correspondingto the eigenvalue p , with an expansion in the mode eigenbasis { e n ( y ) } ∞ givenby φ p ( y ) = ∞ X n =0 g n ( p ) e n ( y ) , (58)where g n ( p ) are coefficients depending on the momentum p ∈ X . Repeat-ing the process of the previous section, one derives a three-term recurrencerelation for the polynomials g n ( p ) and concludes that g n ( p ) = i n p n ( p ) = i n h n ( p ; q )( q ; q ) / n q n ( n − / , (59)where h n ( z ; q ) are the discrete q -Hermite polynomials of type I from Section4. Hence, the eigenfunctions of momentum are φ p ( y ) = ∞ X n =0 i n h n ( p ; q )( q ; q ) / n q n ( n − / e n ( y ) (60)= ∞ X n =0 (i y ) n ( q ; q ) n h n ( p ; q ) (61)= ( y ; q ) ∞ (i yp ; q ) ∞ , p ∈ X = { q s , − q s ; s | ∞ } . (62)To find the last two expressions we have used the same method as in the caseof eigenfunctions of position in (38)–(40).The normalized eigenfunctions of P areΦ p ( y ) ≡ Φ ± q s ( y ) = s ( q s +2 ; q ) ∞ q s q ; q ) ∞ ( − q ; q ) ∞ φ ± q s ( y ) , (63)satisfying h Φ x ( y ) , Φ x ′ ( y ) i H = δ x,x ′ , x, x ′ ∈ X .13 Coordinate realization of the discrete oscil-lator
In Section 3 we constructed a realization of the discrete oscillator on the spaceof analytic functions in the supplementary variable y with the assignment(10). It is natural to look for a realization of the oscillator on the space offunctions in the position coordinate x ∈ X .Let L ( X ) be the Hilbert space of square-summable functions over x ∈ X (the set of positions of the discrete oscillator), with the scalar product h f , f i L ( X ) := 1( q ; q ) ∞ ( − q ) ∞ ∞ X n =0 ( q n +2 ; q ) ∞ q n × (cid:16) f ( q n ) f ∗ ( q n ) + f ( − q n ) f ∗ ( − q n ) (cid:17) , (64)where ∗ stands for complex conjugation.Since the discrete q -Hermite polynomials are associated with the deter-minate moment problem (see, for example, [6] for the description of thisassociation), the set of polynomials { p n ( x ) } ∞ in (37) constitute a completeset of orthonormal functions in the Hilbert space L ( X ).We construct a one-to-one linear isometry Ω from the Hilbert space H ,onto the Hilbert space L ( X ), given byΩ : H ∋ e ( y ) → f ( x ) = h e ( y ) , ψ x ( y ) i H ∈ L ( X ) , (65)where ψ x ( y ) are eigenfunctions (40) of Q . It follows from (38) that H ∋ e n ( y ) → h e n ( y ) , ψ x ( y ) i H = p n ( x ) . (66)That is, Ω maps the basis { e n ( y ) } of H , which is orthonormal under thescalar product (54), onto the basis { p n ( x ) } of L ( X ), which is orthonormalunder (64); this means that Ω is a one-to-one isometry.In L ( X ), the operator Q acts through multiplication, Q f ( x ) = x f ( x ) . (67)Indeed, since Q ψ x ( y ) = x ψ x ( y ) for Ω e ( y ) = f ( x ) = h e ( y ) , ψ x ( y ) i H , we haveΩ : Q e ( y ) → Q f ( x ) = h Q e ( y ) , ψ x ( y ) i H = h e ( y ) , Q ψ x ( y ) i H = h e ( y ) , x ψ x ( y ) i H = x f ( x ) . (68)14e can find the action of Q , P , and H on the basis elements { p n ( x ) } ∞ n =0 of the Hilbert space L ( X ). According to the recurrence relation (32), whichfollows from the recurrence relation for the discrete q -Hermite polynomials h n ( z ; q ), we have for the position operator Q that Q p n ( x ) = p q n (1 − q n +1 ) p n +1 ( x ) + p q n − (1 − q n ) p n − ( x ) . (69)It follows from formulas (3.28.7) and (3.28.8) in [13] that the momentumoperator P acts on the Hilbert space L ( X ) through P = − i(1 − q ) q H − / (cid:16) D q + 1 q ( q x ; q ) ∞ D q − ( q x ; q ) ∞ (cid:17) , (70)where ( q x ; q ) ∞ is the multiplier in the orthogonality measure in the scalarproduct (64). In particular, P acts on the basis functions { p n ( x ) } ∞ n =0 as P p n ( x ) = i p q n (1 − q n +1 ) p n +1 ( x ) − i p q n − (1 − q n ) p n − ( x ) . (71)Finally, the Hamiltonian H acts on the basis polynomials p n ( x ) of theHilbert space L ( X ) as H p n ( x ) = ( n + ) p n ( x ) . (72)Indeed, according to (23) and (66) we have H p n ( x ) = h He n ( y ) , ψ x ( y ) i H = ( n + ) h e n ( y ) , ψ x ( y ) i H = ( n + ) p n ( x ) . (73)It is interesting to see the lowest discrete oscillator modes p n ( x ) in (37)and (72), both as continuous functions of x ∈ (0 ,
1] and their values at theorthogonality set X , for various values of q . While the continuous functionsexhibit strong oscillations (increasing with n and 1 /q ), their values on X ( q )remain well bounded. As q → − , the corresponding points in X densify,evincing the resemblance of the discrete oscillator with the standard oscillatorwavefunctions. This form of convergence should be studied further, since achange of scale appears necessary as well as a discrete measure that becomes acontinuous Riemann integral in the limit. One analogue for this limit appearsin [16, Fig. 4], where the Meixner functions that describe the discrete modelconverge to the Laguerre-Gauss modes of a radial oscillator; in that casethough, the limit is from hyperboloids to the cone in the three-dimensionalspace of the Lie algebra su(1 , q -deformation.15 Momentum realization of the discrete os-cillator
Consider the Hilbert space L ( P ) of square-integrable functions f ( p ) in themomentum coordinate p in the oscillator with the same scalar product as in(64), where P = X is the spectrum of the momentum operator P , coincidingwith the spectrum of Q . The coefficient polynomials g n ( x ) in formula (59) forthe eigenfunctions of momentum, φ p ( y ) in (58), constitute an orthonormalbasis in L ( P ).To formalize this consideration, we construct, as in the previous Section,a one-to-one linear isometry e Ω from the Hilbert space H onto the Hilbertspace L ( P ), given by e Ω : H ∋ e ( y ) → f ( p ) := h e ( y ) , φ p ( y ) i H ∈ L ( P ) , (74)where φ p ( y ) are the eigenfunctions of momentum P in (60)–(62). [Comparewith (65) requiring the position eigenfunctions ψ x ( y ) in (38)–(40).] Fromhere it is evident that H ∋ e n ( y ) → h e n ( y ) , φ p ( y ) i H = g n ( p ) , (75)that is, e Ω is a one-to-one isometry and maps the orthonormal basis { e n ( y ) } ∞ ∈ H onto the orthonormal basis { g n ( p ) } ∞ ∈ L ( P ).The momentum operator P acts on L ( P ) as a multiplication operatoron all functions of p , P g ( p ) = p g ( p ) . (76)The action of Q , P , and H on the basis of polynomials g n ( p ) can be foundin the form of the recurrence relations Q g n ( p ) = p q n (1 − q n +1 ) g n +1 ( p ) + p q n − (1 − q n ) g n − ( p ) , (77) P g n ( p ) = i p q n (1 − q n +1 ) g n +1 ( p ) − i p q n − (1 − q n ) g n − ( p ) , (78) H g n ( p ) = ( n + ) g n ( p ) . (79) According to (24) and (25), the action of the operator exp(i τ H ) is the timeevolution of the discrete oscillator. On the basis (10) of functions { e ( y ) } ∞ e i τH e n ( y ) = e i τ/ e i nτ e n ( y ) = e i( n +1 / τ e n ( y ) . (80)The operator exp(i τ H ) also acts on the Hilbert space L ( X ), which is char-acterized by the scalar product (64). Now consider the isometry betweenthese two spaces, H ∋ e ( y ) → f ( x ) := h e ( y ) , ψ x ( y ) i H ∈ L ( X ) , (81)that maps functions e ( y ) onto functions f ( x ) of the discrete position coordi-nate x ∈ {− q n , q n } ∞ = X . Then to exp(i τ H ) e ( y ) ∈ H there corresponds afunction exp(i τ H ) f ( ± q s ) of x = ± q s , e i τH f ( ± q s ) = h e i τH e ( y ) , ψ ± q s ( y ) i H = h e ( y ) , e − i τH ψ ± q s ( y ) i H (82)= ∞ X n =0 h e ( y ) , e n i H h e n , e − i τH ψ ± q s ( y ) i H (83)= ∞ X n =0 h e ( y ) , e n i H h e i τH e n , ψ ± q s ( y ) i H (84)= ∞ X n =0 ∞ X m =0 (cid:16) h e ( y ) , Ψ q m ( y ) i H h Ψ q m ( y ) , e n i H + h e ( y ) , Ψ − q m ( y ) i H h Ψ − q m ( y ) , e n i H (cid:17) (85) × e i τ ( n +1 / h e n , ψ ± q s ( y ) i H = ∞ X m =0 (cid:16) K τ ( ± q s , q m ) f ( q m ) + K τ ( ± q s , − q m ) f ( − q m ) (cid:17) ;(86)where, according to (38), K τ ( ± q s , ± q m ) = c s ∞ X n =0 h ψ ± q m ( y ) , e n i H e i τ ( n +1 / h e n , ψ ± q s ( y ) i H (87)= c s e i τ/ ∞ X n =0 q − n ( n − / ( q ; q ) n h n ( ± q m ; q ) e i nτ h n ( ± q s ; q ) (88)and c s := q s ( q s +2 ; q ) ∞ q ; q ) ∞ ( − q ; q ) ∞ . (89)17eing elements of L ( X ), the functions f ( x ), x ∈ X , enter into the scalarproduct (64) as a sum with the weight function ( x q ; q ) ∞ . Because weintend to interpret these as wavefunctions of a quantum oscillator, and dis-play them for comparison with their standard shapes, we should absorb thisweight into the functions, F ( x ) = p ( x q ; q ) ∞ f ( x ) , (90)so that their scalar product acquires, from (64), the ‘standard’ form, h F , F i X := 1( q ; q ) ∞ ( − q ) ∞ X x ∈X | x | F ( x ) F ( x ) ∗ ,x ≡ x ( ± , n ) = ± q n ∈ X , n | ∞ . (91)The matrix elements of operators will be correspondingly rescaled by (90).The oscillator evolution (82)–(86) can be used to define the fractionaldiscrete Fourier transform on the functions F ( x ), x ∈ X , rescaled as in (91).We note that the fractional Fourier integral transform for angle τ differs fromthe standard harmonic oscillator evolution by a phase e i τ/ that is due to theground energy in the oscillator, so the fractional Fourier transform isΦ( τ ) := e − i τ/ exp(i τ H ) . (92)Its action on the rescaled discrete wavefunctions will have the formΦ( τ ) F ( x ) = X x ′ ∈X Φ( x, x ′ ; τ ) F ( x ′ ) , (93)Φ( x, x ′ ; τ ) = e − i τ/ s ( x q ; q ) ∞ ( x ′ q ; q ) ∞ K τ ( x, x ′ ) , (94)where x, x ′ ∈ X , and K τ ( x, x ′ ) is given by (87)–(88) for x = ± q s and x ′ = ± q m . Unfortunately, the bilinear generating function (88) of discrete q -Hermite polynomials of type I could not be summed to a closed form. We constructed a model of the harmonic oscillator that can be realized onbases of coordinate and momentum Hilbert spaces, and its energy modes ex-pressed in terms of discrete q -Hermite polynomials of type I. The spectrum18f the Hamiltonian coincides with that of the standard harmonic oscillatorin quantum mechanics, while the position and momentum operators in thismodel have discrete, denumerably infinite spectra that depend on the exten-sion parameter q contained in the interval [ − , q -Hermite polynomials [17, 18,19], our models (the present one and that in Ref. [8]) fulfill the basic Hamil-ton equations in the form [ H, Q ] = − i P and [ H, P ] = i Q , with standardcommutators — and not q -commutators [1, 2]. Because of this importantcircumstance, the time evolution of the model is a Lie group, which but fora phase is that of fractional Fourier transforms associated with this model[20]. This discrete oscillator is a new and non-trivial deformation of the stan-dard quantum harmonic oscillator; it allows the extension of other standardconcepts of phase space, such as coherent states [8], that will be examinedelsewhere.We believe that the discrete oscillator model can appropriately describediscrete quantum systems on bounded point lattices, and also contributesignificantly to the general theory of special functions. Acknowledgements
This research was supported by the SEP-CONACYT (M´exico) project IN-102603 ´Optica Matem´atica , and Grant 14.01/016 of the State Foundation ofFundamental Research of Ukraine.
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