Finite q-oscillator

0.17 MB·

by  Natig M. Atakishiyev

Finite q-Oscillator
Natig M. Atakishiyev,1 Anatoliy U. Klimyk,1,2
4
0 and Kurt Bernardo Wolf
0
2

n Centro de Ciencias F´ısicas, UNAM, Apartado Postal 48-3, 62251
a
Cuernavaca, Morelos, M´exico
J

9
1

1 Abstract
v
5
The finite q-oscillator is a model that obeys the dynamics of the
3
0 harmonic oscillator, with the operators of position, momentum and
1 Hamiltonian being functions of elements of the q-algebra su (2). The
q
0
spectrum of position in this discrete system, in a fixed representation
4
0 j, consists of 2j+1 ‘sensor’-points xs = 12[2s]q, s ∈ {−j,−j+1,...,j},
/ andsimilarly forthemomentumobservable. Thespectrumofenergies
h
p is finite and equally spaced, so the system supports coherent states.
-
h The wave functions involve dual q-Kravchuk polynomials, which are
t solutions to a finite-difference Schro¨dinger equation. Time evolution
a
m (times a phase) defines the fractional Fourier-q-Kravchuk transform.
: In the classical limit q 1 we recover the finite oscillator Lie algebra,
v →
theN = 2j limitreturnstheMacfarlane–Biedenharnq-oscillator
i
X → ∞
and both limits contract the generators to the standard quantum-
r mechanical harmonic oscillator.
a
PACS numbers: 02.20.Qs, 02.30.Gp, 42.30.Kq, 42.30.Va
1
InstitutodeMatema´ticas,UNAM,ApartadoPostal273-3,62210Cuernavaca,Morelos,
M´exico
2
Bogolyubov Institute for Theoretical Physics, Kiev 03143,Ukraine
1
1. Introduction
Discrete models which are counterparts to well-known continuous sys-
tems, and in particular those which contract to the standard harmonic os-
cillator, are of fundamental interest in theoretical physics [1]–[5]. Moreover,
finite discrete models are also of interest for the parallel processing of sig-
nals through nano-opticaldevices, where the input may involve lasing carbon
tubules, the output being registered by a finite sensor array, and the device
consisting of a shallow planar waveguide—an oscillator which can carry only
a finite number of states [6]. The salient purpose of such a device is to
perform a finite analogue of the fractional Fourier transform [7].
In previous works on the one-dimensional finite oscillator [3, 6, 8], oscilla-
tor systems were characterized in the familiar context of Hilbert spaces and
Lie algebraic theory; in [5] these requirements were formalized into the three
following postulates:
1. There exists an essentially self-adjoint position operator, indicated Q,
whose spectrum Σ(Q) is the set of positions of the system.
2. There exists a self-adjoint and compact Hamiltonian operator, H, which
generatestimeevolutionthroughtheNewton-Lie, orequivalent Hamilton-Lie
equations:
[H,Q] =: iP,
[H,[H,Q]] = Q − (1)
⇐⇒ ( [H,P] = iQ,
where[ , ]isthecommutator. ThefirstHamiltonequationin(1)definesthe
· ·
momentumoperatorP, whilethesecond onecontains theharmonicoscillator
dynamics. The set of momentum values of the system is the spectrum Σ(P)
of P.
3. The three operators, Q, P and H, close into an associative algebra, i.e.,
satisfy the Jacobi identity,
[P,[H,Q]]+[Q,[P,H]]+[H,[Q,P]] = 0. (2)
The second and third postulates determine that [Q,P] must commute
with H, which implies that it can only be of the form [Q,P] = iF(H),
where F is some function of H (including constants) and the i is placed to
make F(H) self-adjoint, but do not otherwise specify this basic commutator
further. For a constant F(H) = h¯ˆ1, one recovers the standard oscillator
2
algebra H = span H,Q,P,ˆ1 , which contains the basic Heisenberg-Weyl
4
{ }
subalgebra W = span Q,P,ˆ1 of quantum mechanics. In our first works
1
{ }
[6, 8] we examined the cases which, in the unitary irreducible representations
of spin j = 1N (N 0,1,... fixed), correspond to the linear function
2 ∈ { }
F(H) = H (j + 1)ˆ1 =: J , and so the operators close into the Lie algebra
− 2 3
so(3) = su(2) = span Q,P,J . The purpose of the present paper is to
3
{ }
study the case when, for q := e−κ, the basic commutator is
[Q,P] = iF (H), H = J +(j + 1)ˆ1, (3)
q 3 2
cosh 1κ cosh(j + 1)κ
F (H) = e−2κJ3 2 e−κJ3 2 (4)
q 2sinh 1κ − 2sinh 1κ
2 2
= 1e−κJ3 e−κJ3cosh 1κ T (cosh 1κ) sinh 1κ, (5)
2 2 − 2j+1 2 2
(cid:16) (cid:17).
where T is the Chebyshev polynomial of the first kind, and q (0,1] or
n
∈
κ [0, ). In particular, F (H) = J returns the previous su(2) case [6].
1 3
∈ ∞
An important ingredient for the postulates of harmonic oscillator dynam-
ics is an unambiguous correspondence between the physical observables of
position, momentum and energy, with the elements of the associative alge-
bra. InSection 2we recall themainrelevant results onthe algebrasu (2) and
q
its standard representation basis. The su (2) nonstandard basis, investigated
q
in [9, 10], is introduced in Section 3 to exhibit our proposed correspondence
explicitly in terms of the generators of su (2). With our postulated choice,
q
the position and energy spectra in the (2j+1)-dimensional representation
j = 1N of su (2) will be
2 q
sinhsκ
Σ(Q) = x = 1[2s] = , s j, j+1,...,j =: s j , (6)
s 2 q 2sinh 1κ ∈ {− − } |−j
2
Σ(H) = E = n+ 1, n 0,1,...,2j =: n 2j. (7)
n 2 ∈ { } |0
We recall the definition of the q-number for q = e−κ:
q1r q−1r sinh 1rκ
2 2
[r]q = [r]q−1 = −[−r]q := q1 −q−1 = sinh21κ . (8)
2 2 2
−
Note that the q-number of an integer r is U (cosh 1κ), the Chebyshev
r−1 2
polynomial of the second kind. The spectrum of momentum is the same
as that of position, Σ(P) = Σ(Q). The classical limit is limq→1[s]q = s,
3
when the q-algebra su (2) becomes the Lie algebra su(2); then, the set of
q
positions become equally spaced and we are back at the previously known
finite oscillator [6]. But for all other values of the deformation parameter
q, the ‘sensor points’ of the system are concentrated towards the center of
the interval, while the endpoints are spread farther apart. Yet the energy
spectrum remains an equally-spaced set, and therefore the system follows
harmonic motion.
Thefiniteq-oscillatorwavefunctionsaretheoverlapsbetweentheposition
and energy eigenbases. They are written out in Section 4 in terms of the dual
q-Kravchuk polynomials, and are orthonormal and complete over the sensor
points of the system. The momentum representation of these wave functions
is addressed in Section 5 with the Fourier-q-Kravchuk transform, and in
Section 6 this transform is fractionalized. The evolution in time of a finite
q-oscillator (or equivalently, the parallel processing of a finite signal along
the axis of a shallow planar waveguide), is the 2-fold cover of the fractional
Fourier-q-Kravchuk transform matrix; the metaplectic sign appears thus for
half-integer values of j, which corresponds to a finite systems of an even
number of points. In Section 7 we introduce the concept of an equivalent
potential for discrete systems which is based, as in the continuous case, on
the existence of a ground state with no zeros. Finally, in Section 8 we verify
that the contraction limits q 1 and N = 2j of the algebra su (2)
q
→ → ∞
reproduce the known results for the finite oscillator and the continuous q-
oscillator. The corresponding limits for the wave functions however, present
further challenge.
2. The algebra su (2) and its standard basis
q
The quantum algebra su (2) is the associative algebra generated by three
q
elements, usually denoted asJ ,J ,J , subject to the commutation relations
+ − 3
[J ,J ] = J , [J ,J ] = [2J ] . (9)
3 ± ± + − 3 q
±
Equivalently, writing J = J iJ , we characterize the algebra su (2) by
± 1 2 q
±
[J ,J ] = iJ , [J ,J ] = iJ , [J ,J ] = i[2J ] . (10)
2 3 1 3 1 2 1 2 2 3 q
The first two commutators in (10) have the structure of the oscillator Hamil-
ton equations (1), while the third one involves the q-number (8), which dis-
tinguishes q-algebras from Lie algebras, the latter corresponding to the case
4
q = 1. The following element in the covering algebra of su (2) commutes
q
with all others,
C := J2 +J2 +[J 1]2 + 1[2J ] 1
q 1 2 3−2 q 2 3 q − 4 (11)
= J J +[J 1]2 1,
+ − 3 − 2 q − 4
and is called its Casimir operator.
It is convenient to have a realization of the su (2) generators in terms
q
of first-degree differential operators, acting on spaces of functions of a
j
H
formal variable x, and depending on the numerical irreducible representation
label j. This is
d
J := J +iJ x 2j x = x[j J ] , (12)
+ 1 2 3 q
↔ − dx q −
h i
1 d 1
J := J iJ x = [j +J ] , (13)
− 1 2 3 q
− ↔ x dx q x
h i
d
J x j, j 0, 1,1,... fixed. (14)
3 ↔ dx − ∈ { 2 }
The set of power monomials xj+m j are eigenfunctions of J and pro-
|m=−j 3
videthestandard basisfortheirreducible space , offinitedimension 2j+1.
j
H
The functions of the basis were chosen in [9, 10] with the following constants:
1/2
2j
fmj (x) := q41(m2−j2) "j+m# xj+m, (15)
q
where the q-binomial coefficient m is defined (using the standard notation
n q
of q-analysis [11]) for m n nonhneigative integers by
≥
m (q;q) (q−m;q)
:= m = ( 1)nqmn−12n(n−1) n, (16)
n (q;q) (q;q) − (q;q)
(cid:20) (cid:21)q n m−n n
n−1
(z;q) := (1 zqk), n = 1,2,3,..., (z;q) = 1. (17)
n 0
−
k=0
Y
For any two complex vectors a,b ,
j
∈ H
j j
a = α fj , b = β fj , (18)
m m m m
m=−j m=−j
X X
5
there is a natural sesquilinear inner product
j
(a,b) := α∗ β , (19)
Hj m m
m=−j
X
with respect to which the standard basis is orthonormal. The action of the
su (2) generators and Casimir operator on the standard basis is well known:
q
J fj = mfj , J fj = [j m+1] [j m] fj , (20)
3 m m ± m ± q ∓ q m±1
C fj = c fj , c := [j+q1]2 1. (21)
q m q m q 2 q − 4
These equations are of course independent of the realization of the basis
vectors fmj by the power monomials fmj (x) in x.
The spectrum of the diagonal generator J [see (14) and (20)] is linear
3
and bounded, as that of a finite version of the quantum harmonic oscillator.
Indeed, this is our choice for the finite q-oscillator Hamiltonian, displaced so
that the ground state has energy 1, namely
2
H = J +j + 1, Hfj = (n+ 1)fj , n := j +m, (22)
3 2 m 2 m
where n 2j is the mode number that counts the number of energy quanta.
|0
At this point we are presented with what would appear as a ‘natural’ as-
signment for the position and momentum operators, Q J and P J ,
1 2
↔ ↔ −
because it would be the simplest generalization of the previously studied
q = 1 case [6, 8]. This choice would bring the first two commutators in
(10) to reproduce correctly the two Hamilton equations in (1), while the
third commutator [Q,P] would have the form (3) with F (H) = 1[2J ] =
q 2 3 q
sinhκ(H j 1)/2sinh 1κ. In this ‘na¨ıve’ model however, the spectra of Q
− − 2 2
and P are not algebraic; they must be computed numerically as roots of a
polynomial equation of degree 2j +1.
3. The nonstandard basis
While we do not discard the model suggested at the end of the previous
Section, we find more attractive to propose a correspondence between the
physical observables of position and momentum, Q,P, and the nonstandard
(also called twisted) operators (see [12]–[14], [9, 10]), which have the virtue of
6
possessing an algebraic spectrum x := 1[2s] , s j . The position (and hence
s 2 q |−j
momentum) observables will be thus identified with the following operators:
Q = J1 := q14J3J1q14J3, (23)
−P = Je2 := q14J3J2q14J3, (24)
while the Hamiltonian H is associaeted to J by (22) as before.
3
Wenotethatwhiletheq-number(8)displayssymmetryunderq-inversions,
q ↔ q−1, [r]q = [r]q−1, the identification of tilded operators in (23)–(24) pre-
serves this symmetry with the concomitant reflection J J . This means
3 3
↔ −
that the ground state of a q < 1 oscillator is the top state of its q−1 > 1
partner.
The commutation relations among the nonstandard operators and J are
3
[J ,Q] = iP, [J ,P] = iQ, (25)
3 3
−
[Q,P] = 2i q21J3(q−12J+J− −q21J−J+)q21J3 =: iFq(Cq,J3) (26)
= i e−κJ3[(C + 1)sinh 1κ+ 1csch 1κ] 1 e−2κJ3 coth 1κ ,
q 4 2 2 2 − 2 2
(cid:16) (cid:17)
where q = e−κ as before. The operator F (C ,J ) defined in (26) commutes
q q 3
with J and is also diagonal in the standard basis; in the irreducible repre-
3
sentation j,
e−2mκcosh 1κ e−mκcosh(j+1)κ
F fj = 2 − 2 fj , (27)
q m 2sinh 1κ m
2
but its spectrum is not a good candidate for an oscillator Hamiltonian, be-
cause it is not equally spaced [unlike (7)], and so the motion would not be
harmonic, but dispersive. In terms ofthe positionand momentum generators
(23)–(24), the Casimir operator (11) acquires the form
C = sech 1κ(Q2 +P2)eκJ3 +D (J ), (28)
q 2 q 3
D (J ) := sech 1κ [J 1]2 1e−κJ3coth 1κ+ 1csch 1κ 1. (29)
q 3 2 3−2 q − 2 2 2 2 − 4
(cid:16) (cid:17)
We recall a previous phase-space picture for the finite oscillator of 2j+1
points, considered in [15], as the (classical) sphere Q2 +P2 +J2 = j(j+1),
3
having circular sections of square radius Q2 +P2 = (j+1)2 (J 1)2 J .
2 − 3−2 − 3
7
For su (2), the corresponding surface now has the section
q
Q2 +P2 = [j+1]2cosh 1κ [J 1]2
2 q 2 − 3−2 q
(cid:16) + 1e−κJ3coth 1κ 1 csch 1κ e−κJ3. (30)
2 2 − 2 2
(cid:17)
Phase space for the finite q-oscillator is suggested thus as q-dependent pear-
shaped sphero¨ıds, tip-up for q < 1 and tip-down for q > 1 (recall the q q−1
↔
symmetry withtheinversion ofJ ). The q-harmonicoscillatorevolution(i.e.,
3
a phase times the so-defined fractional q-Fourier-Kravchuk transform) will
rotate this space around the J vertical symmetry axis of the sphero¨ıd.
3
In this finite q-oscillator model we interpret the eigenvalues x of Q := J
s 1
as the discrete values of the position observable. The eigenfunctions gsj(x)
e
and eigenvalues of this nonstandard operator were found in [9], and they are
of the form
sinhsκ
Qgsj(x) = xsgsj(x), xs = 12[2s]q = 2sinh 1κ = −x−s, s|j−j, (31)
2
gsj(x) = γsj(q41(1−2j)x;q)j−s(−q41(1−2j)x;q)j+s = g−js(−x), (32)
2j 1+q−2s
γj := q12(j+s) . (33)
s v j +s 2( q;q)
uu(cid:20) (cid:21)q2 − 2j
t
They are normalized with respect to the inner product (19), and are orthog-
onal because they correspond to distinct eigenvalues x . This basis of 2j+1
s
functions gsj(x), s|j−j we call the position basis. A signal consisting of 2j +1
values Φ , sensed at the positions x [given in (6)], is
s s
j
Φ = Φ gj , (34)
s s ∈ Hj
s=−j
X
and can be realized either as a function of x, or as a (2j + 1)-dimensional
column vector with components numbered by s j .
|−j
4. Finite q-oscillator mode wave functions
We have now two bases for : the standard basis fj j of mode
Hj { m}m=−j
n = j +m (and energy E = n+ 1), and the nonstandard basis gj j of
n 2 { s}s=−j
position x = 1[2s] . Intherealizationofsu (2)generatorsgivenin(12)–(14),
s 2 q q
8
the mode basis is realized by the power functions in (15), and the position
basis by (32). We can use this realization to find the unitary transformation
between these two orthonormal bases, and thus define the finite q-oscillator
wave functions by the overlap
of mode n = j +m, n 2j,
Φ(2j|q)(x ) := (gj,fj ) |0 (35)
n s s m Hj ( on points xs = 21[2s]q, s|j−j.
By construction, this set of functions is orthonormal and complete under the
inner product (19).
j
H
The overlap (35) is obtained by expanding the function gsj(x) of (32) into
a power series in x, which is then
j j
gsj(x) = Φj(2+jm|q)(xs)∗fmj (x), fmj (x) = Φj(2+jm|q)(xs)gmj (x). (36)
m=−j s=−j
X X
The expansion of gj(x) in x is [9]
s
j 2j 1/2
gsj(x) = γsj q41(j+m)(j+m−1) j+m Kj+m(λ(j−s);−1,2j|q)fmj (x),
m=−j (cid:20) (cid:21)q
X
(37)
expressed in terms of the dual q-Kravchuk polynomials,
q−n, q−ξ, cqξ−2j
K (q−ξ+cqξ−2j;c,2j q) := φ q; q , (38)
n | 3 2 q−2j, 0 (cid:12) !
(cid:12)
(cid:12)
where φ is the basic hypergeometric function defined in (cid:12)[11], and the coef-
3 2
ficients γj are given in (33).
s
Intheparticularcaseofourconcern, theargumentofthedualq-Kravchuk
polynomial is λ(ξ) = q−ξ +cqξ−2j with c = 1 in (37), is given in terms of
−
the positions x = 1[2s] , s j , of the finite q-oscillator by
s 2 q |−j
λ(j s) = q−j+s q−j−s = 2ejκsinhκs
− − −
= 2q−j−21(q −1)xs = −(4ejκsinh 21κ)xs, (39)
and q = e−κ as before. From (37) thus, the finite q-oscillator wave functions
of mode number n = j +m, n 2j, are
|0
9
2j 2j 1+q−2s
Φn(2j|q)(xs) = q12(j+s)+41n(n−1)v j +s n 2( q;q)
uu(cid:20) (cid:21)q2(cid:20) (cid:21)q − 2j
t
Kn(2q−j−21(q 1)xs; 1, 2j q). (40)
× − − |
The explicit expression for the dual q-Kravchuk polynomials in this case is
q−j−m,qs−j, q−j−s
Kj+m(λ(j−s);−1,2j|q) = 3φ2 q−2j, 0− (cid:12)q; q! (41)
(cid:12)
2j (q−j−m;q) (q−j+s;q) ((cid:12) q−j−s;q) qk
k k(cid:12) k
= − ,
(q−2j;q) (q;q)
k=0 k k
X
where (z;q) is defined in (17).
k
The lowest mode of the oscillator is [see (40) for n = j +m = 0],
2j 1+q−2s
Φ0(2j|q)(xs) = q21(j+s)v j +s 2( q;q) = γsj. (42)
uu(cid:20) (cid:21)q2 − 2j
t
The finite q-oscillator wave functions possess definite parity,
Φ(2j|q)( x ) = Φ(2j|q)(x ) = ( 1)nΦ(2j|q)(x ), (43)
n − s n −s − n s
and, as is to be expected, in the limit q 1 return the Kravchuk functions
→
of the finite oscillator [6]
2j 2j
limΦ(2j|q)(x ) = 2−j K (j s; 1,2j), (44)
q→1 n s s j +s n n − 2
(cid:18) (cid:19)(cid:18) (cid:19)
with the classical Kravchuk polynomials, introduced by Kravchuk in [16].
The dual q-Kravchuk polynomials – as all polynomials – are analytic
functions on the complex plane of their argument. As before in the finite
oscillator models [6, 8, 17], this argument is the position coordinate, which
can be analytically continued to real or complex values X, even if the inner
product of the space is only over the point set x , s j. As to the
Hj { s} |j
q-Kravchuk wave functions (40) the factor in front of the polynomial is a
function that is analytic in the argument s within the interval j 1 < s <
− −
10