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q-deformed angular momentum operator and possible application to triatomic molecules
23 January 1995 PHYSICS
ELSEVIEK
LETTERS
A
Physics Letters A 197 (1995) 221-226
q-deformed angular momentum operator and possible application to triatomic molecules A. Kundu a,1, Y.J. Ng b*2 aFb 17, Mathematik/Informatik, GH-Kassel Universitdt, Holliindische Strasse 36, 34109 Kassel, Germany b Centerfor Theoretical Physics, MIT, Cambridge, MA 02139, USA
Received 26 July 1993; revised manuscript received 6 October 1994; accepted for publication 14 October 1994 Communicated by D.D. Holm
Abstract A q-deformation of the angular momentum given through deformed radius vector and momentum operator is proposed. Such a realization of the SU,(2) algebra in three oscillator modes is different from the standard one and the corresponding Hamiltonian of the deformed oscillators naturally generates anisotropy along the z-axis and the required type of rotation-vibrational coupling for symmetric-top like molecules. Possible application to triatomic molecular spectra is discussed and for the more general asymmetric case a further realization of the deformed angular momentum in anisotropic oscillators with mutually prime frequencies is proposed.
Quantum group and related q-deformed structures, though discovered only recently, are penetrating into different branches of physics at a fast rate [l-8]. Realization of the SU,(2) quantum group through single mode [ 91 as well as two [ IO,1 1 ] qoscillators also opened up various possible applications and relevant studies [ 121. There have been several interesting attempts to apply such q-deformations to molecular spectroscopy, nuclear physics, quantum chemistry, chemical physics etc. Among them, the coincidence (with high accuracy) of experimentally observed vibrational and rotational spectra of some diatomic molecules with the q-oscillator model is indeed a surprising result [ 1,2]. Nevertheless, in spite of all these important achievements, the
’ Permanent address: Saha Institute of Nuclear Physics, AF/l Bidhannagar, Calcutta 700 064, India. * Permanent address: Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA.
question raised in Ref. [ 13 1, viz. how the well known angular momentum operator z=rxp,
(1)
satisfying the algebra [1,, /*I= +1* ,
[l,,
I-1=21,,
(2)
should be deformed such that the corresponding L would satisfy the quantum algebra
[b,Ll=*L,
[~+,Ll=[2&1,,
(3)
seems not to have been answered properly yet. (Throughout this paper we use the notation [xl,= (qX-q-X)/(q-q-‘).) Therefore our aim here is to focus on this problem and attempt to find a consistent deformation of r and p operators, so that L would satisfy algebra (3 ). Naturally it would mean realizing the SU,(2) algebra through three different q-oscillator modes (altematively one may possibly look directly for a realization
03759601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SsDrO375-9601(94)00858-2
A. Kundu. Y.J. Ng /Physics Letters A 197 (1995) 221-226
222
of SO,( 3 ) ) some of which must be deformed. Consequently, exploiting the idea of application to diatomic molecules [ 1,2 1, one may hope and speculate to achieve here similar results for triatomic molecules. For this purpose we rewrite relation ( 1) in a form more suitable for deformation, 1, =Ti(rkp3-pkr3), &=$i(r+p--r-p+),
(4)
where the standard definition xLt-=x1 *ix, for operators has been used. Introducing further three independent oscillators as II, =f(r+
+ip,),
6=(1IJZ)(++iP3),
(5)
we express (4) in the form 1, =Jz
(a-al -at+a,) ,
I_ =fi
(aTa
l,=n+
-n_
-a+af)
,
,
(6)
where n, =a1 u+ . Note that the canonical commutation relations [r,, p,] = iS, would induce the oscillator algebra
[a*,at+l=l,
[4,41=1,
[n,,at,l=at+,
[hafl=aI,
(7)
and their complex conjugates, the other relations being trivial. Using (7) it is easy to check that the I operator ( 6 ) indeed satisfies the standard SU ( 2 ) algebra (2). Now to get generators L of SU,( 2) we propose to deform 1given by (6 ) in the form L, =Jz
(A-A$-ADA,)
,
L_ =fi
(ALA,-A+Af)
, (8)
where the A are some deformed oscillators satisfying the relations
[n3,Afl=AI,
[A+,At,l=(n++11f2(n+-n-) -n+f2(n+
where f2(x) = [ [x+l+ 1 ] I,[ [l-x] I4 (for fixed values of I) with the notation [ [x] lq= [x]Jx. It is not difficult to check that due to the deformed oscillator algebra( 9)) expressions (8) indeed satisfy the known quantum algebra (3 ). Note, however, that though here the generators L satisfy the known quantum algebra SU,( 2), the realization (8) expressed through three independent deformed oscillator modes as well as the corresponding oscillator algebra (9) differ from the known realizations through standard q-oscillators [IO,1 1 ] or other deformations studied in Ref. [ 12 1. Here out of three oscillator modes only two, viz. A *, are practically deformed, while A3 = a3 remains unchanged. Interesting, due to such deformation the A+ and A_ oscillators no longer remain independent and their mutual commutation relation is given by the last expression in ( 9 ) . However, since at q-+ 1,f( x) -+ 1, one gets back in this limit the usual oscillators satisfying algebra (7) and a+, a_ become independent again with vanishing commutator between them. Therefore (8 ), in fact, represents a realization of SU,( 2) through three oscillator modes, two of them being deformed. We also observe that the q-deformed oscillators introduced here define a simple deforming map to the usual oscillators (7) as
A+ =f(n+ -n_)a+
,
A- =a_f(n+ -n_)
A3 =a3,
, (10)
which is more like the deformations discussed in Ref. [ 121 than that of the known q-oscillators [ 111. This gives yet another realization for the deformed angular momentum operator expressed now through un-
[n+,Ail=A$,
L, =fi
(a-al-a$a3)f(n+
L- =fif( L3=n+ -n_
-n_)
,
n, -n_)(ata3-a+a3>, .
(11)
One may observe that mapping tines the deformation
( 10) due to (5 ) deof radius vector r, as
-n_ -1))
[A_,A~]=(n_+1)f2(n+-n--l) -n-f*(n+
(9)
f2(n+-n_-1)A-A+’
deformed oscillators as
L3=l,=n+-n_,
[A,,A!il=l,
A+A_ = f2(n+ -n-I
-n-j.
R+ =r+f(13) ,
R- =f(bk
,
R3=r3,
and gives similar expressions for the deformed momentum operator P,
A. Kunda, Y.J. Ng /Physics Letters A 197 (1995) 221-226
p+ =P+f(l,)
9 p3 =P3.
9 p- =flb)P-
Therefore the deformed angular momentum operators may finally be expressed through the deformed canonical operators R and P as L+ =Ti(R+P3-RJPk),
L3=13,
(4’ )
with the q-commutation relations R,R_=fl(lx)R_R+,
P+P_=fl(l,)P_P+,
R+P_=fl(Z,)P_R++2if*(l,-1)) P+R_=fl(lj)R_P+-2if2(lJ-l),
(9’)
223
with KQJ. h, is the vibrational part of the Hamiltonian, which for the triatomic molecules in the general anisotropic case may be given by [ 14 ]
oi being different vibrational frequencies of the normal coordinates. However, it is known that Hamiltonian ( 12) is not enough for describing the observed molecular spectra and suitably chosen rotation-vibrational coupling terms must be added to it. Such an interaction term is usually introduced by formally replacing the coefficients A,, B, in ( 13)
wheref l(6) =f *(13- 1 )/f *(lJ), supplemented by
by
[R3, PJI =i ,
A,,,=A,-
c a:(n,+$c&) I
,
B,,, =B,-
C af(n,+idi)
,
[&,&l=[&,P,l=O,
[R,,P,l=[P,,R,l=[P,,P,l=O,
(15)
and [I,,R,]=kR,,
with d, being the degeneracy. This is equivalent to adding an interacting term
[I,,P,l=kP,,
14, R,l= [I,, PJI =0 , to have a closed algebra. We note that L3 can also be written in a more symmetric form in terms of the deformed canonical operator as [2L3],=i(R+P_ -R_ P+), which may be shown using the Wigner-Eckart theorem. Therefore this gives the expressions of the deformed angular momentum operators through deformed radius vector and deformed momentum operators, answering thus our major question raised above. The above realization of SU,(2) generators (11) recovers via (6) the mapping between deformed and undeformed angular momentum: L+=l+f(l,), L_=f(l,)l_, L,=lJ.Afterfindingthis q-deformation of angular momentum through three oscillators, we try to look for its possible applications to triatomic molecules. The Hamiltonian of polyatomic molecules may be given by [ 141 h=h,+h,.
(12)
h, represents the rotational part of the Hamiltonian, h,=B,J(J+l)+(A,-B,)K*+(e,+e,,i,)
,
(13)
where A,, Be are the equilibrium values of the coeffrcients and K is the quantum number of the component of the angular momentum along the figure axis
II,=-
c cr;B(ni+$d,)J(J+l)
F (4
-&‘)(n~+~d~)K*,
(16)
to the Hamiltonian resulting as h=hr+h,+hi.
(17)
In the isotropic case, i.e. when all the frequencies wi are equal, h, becomes completely degenerate, h,=o(n,
+n*+&+$)
=w(2(n+
+n_)+n3+$),
and invariant under SU (2 ), i.e. commuting with the angular momentum operators 1 (6). However, if we deform h, using our deformed oscillators as H,=o((At,,A+}+{At,A-}+t{Af,A,}),
(18)
something interesting happens. Firstly, using mapping ( 10) one may express ( 18) in the form H,=o(g(l,l,)(n+
+n_ +l)+ns+f)
)
(19)
where g(1, ls)=f *(&-l)+f *(l,), 13=n+--n_. Comparison with the isotropic harmonic oscillator shows clearly that such a deformation brings simultaneously the anisotropy and a coupling between vibrational and rotational modes into the picture. Con-
224
A. Kundu, Y.J. Ng /Physics
Letters A 197 (1995) 221-226
sidering the deformation q=e” to be small and retaining up to the first nontrivial contribution, we get from ( 19) the corresponding energy spectrum as
A,,,-B,,,=A,-B,-fwa~(n,+n2+1),
E,=eo+e,,,+~~++~:),+e~Z?,,
A,=A,+wa;,
(20)
B,,, =B,-fwc&n,
+nz+l)
)
&Be--fwcu;.
where e, is the same as the original harmonic oscillator term h,,
Since (23) is in the familiar form of rotational-vibrational spectra of triatomic molecules [ 141, we can consider the absorption band for the transition
introduces anisotropy and a correction &= )wcr2(1(1+ 1) +I:) to the rotational energy. On the other hand,
An=+l,
e$:!=focu21(l+l)(n,+n2+l),
(21)
generates interaction between vibrational and rotational modes, while e~2,),=fwa2(n,+n2+l)1:,
(22)
accounts for the interaction with explicit anisotropy along the z-direction, which may be considered as the direction of the figure axis giving K=l,. Therefore starting from the noninteracting Hamiltonian ( 12) and deforming it by our deformed oscillators we may generate in a natural way the required kind of higher order interacting terms (apart from some corrections to the original Hamiltonian) in the Hamiltonian. Indeed, identifying
and
where (Y=&, we see that for molecules with o$ = 2@, we may obtain the needed vibration-rotation interaction terms ( 16) by adjusting the deformation parameter CLNote that usually such interaction terms are required to be added in an ad hoc manner to the original Hamiltonian to achieve a better fit with experimental results. To consider the effect of these additional terms in the case of triatomic molecules, we rewrite the deformed Hamiltonian (for small values of (Yand renaming J= 1) in the form H=h,+H,
=4v,J(J+l)+(A,v, where
--4v,W2 >
(23)
AJ=O,kl,
AK=o,
corresponding to the parallel band of symmetric-top like molecules. Note that this gives from (23) the conventional P, Q and R bands as follows. P-band (for AJ= - 1) : Y=v,,-~B~,~J-Av(J,
K),
(24a)
R-band (for AJ= + 1): v=v,,+~B~,~(J+~)-Av(J,K),
(24b)
and Q-band (for AJ= 0): v= v0 -Av(J,
K) ,
(24~)
where
=$wa$(J(J+1)+K2),
(24d)
is responsible for the nonconstant spacing and asymmetry of bands observed due to the rotation-vibrational coupling. To have a qualitative picture of the absorption band obtained through our deformation, we take some representative values for Be, vo, and chose n’ , 4wt.x; appropriately. It is interesting to observe that the graphical pictures (Fig. 1) described by (24 ) resemble clearly the typical parallel band spectra with distinct P, Q, R bands observed for symmetric-top like polyatomic molecules [ 141. The intensity distributions are based on the formulas given in Ref. [ 141. It should be fair, however, to note that the spectral perturbations represented by these or similar additional terms can be postulated a priori independently of the deformation mechanism. Our proposed application to triatomic molecules would indeed be farfetched were it not for the curious success with which the q-oscillator has been applied to the diatomic molecules as shown in Ref. [ 1 ] and [ 2 1. But one should
A. Kunda. Y.J. Ng /Physics Letters A 197 (1995) 221-226
225
I
I
I
'spectra.dat" -
K-4
1
1
I
I
I
I
1
I
K-3
a) K-2
K=l
K=O
b)
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
Fig. 1. Typical parallel band absorption spectra of symmetriotop molecules simulated from (24) with n’ =O, vO= 13 10 cm-‘, B,= 10 cm- ‘, fox-x+$ ~0. I cm- ’ . The heights of the spectral lines correspond to their intensities. Sub-bands for different K values (KG J) in (a) are directly superimposed in (b). The characteristic features of such polyatomic molecular spectra are [ 141 (i) appearance of distinct P, Q and R bands, (ii) absence of Q band for K= 0, (iii) nonconstant spacing and asymmetry in the distribution of P and R bands due to the rotation-vibrational coupling, (iv) shifting of the origin of sub-bands and (v) diminishing spectral intensity of P and R sub-bands relative to Q with increasing K.
keep in mind the speculative nature of this proposed application. For application to asymmetric triatomic molecules, we have to consider possibly the more general anisotropic case, when all three oi are different. In general the corresponding Schrbdinger equation for the polyatomic molecule is given by
For n= 3 this corresponds to triatomic molecules with three degrees of freedom and the anisotropic Hamiltonian being given by ( 14). Unfortunately, for such a general anisotropic case, the oscillator realization of SU( 2) or SU,(2) is not known. We stress again that ( 6 ) and ( 8 ) are useful only when all Wiare equal. Nevertheless an interesting representation in the oscillator mode can still be constructed [ 15,161, when the frequenciesw+, o3 have integer ratios. Using this
idea we can write for this anisotropic case an expression similar to (8 ) r+ =J2
(Liif
-l?t,a”3)
)
r_
(lcd3
-c?+Zf)
)
=Jz
r, =fi+
-n’_
.
(26)
The operators Zi here are generalized bosons [ 15 ] with the usual commutation relations (7) but with the number operator fii being related to the original number ni as rii= [nil/&], where [ ] denotes the integral part and the integers ki represent the anisotropy of the model as
It is not difficult to check that such bosons may be expressed through the standard oscillators as [ 15 ]
226
c+=
A. Kundu. Y. J. Ng /Physics
(iiF)“*
and C. Rasco for useful discussions, while the other author (AK) acknowledges the support of the Alexander von Humboldt-Stiftung.
ta+)k.
The roperators through these generalized bosons satisfy the SU (2) algebra and commute with the anisotropic Hamiltonian h++c+l;;.
n
n3
(27)
We now try to do the same trick with the deformed algebras, i.e. try to find an expression similar to (S), but realized through some generalized deformed oscillators related to the anisotropic case [ 16 1, such that the resultant 2 would satisfy again the quantized algebra %I,( 2 ). Such generalized deformed oscillators A’may be expressed through the corresponding undeformed oscillators d exactly in the same way as (lo), Al+=f(n”+ -L)d+ 23 ‘53 .
, A’_ =r?_f(fi+
Letters A 197 (1995) 22 I-226
-ii_)
) (28)
As a result of the deformations in the anisotropic case we may get again expressions similar in form to (9 ), ( 19)-( 22) etc. In all these expressions the number operator ni will be replaced now by its anisotropic counter part: Ai=ni/ki+ri/ki (O< ricki), ri being integer valued objects depending on the anisotropic parameters k,. Consideration of these terms generated through deformation in the presence of anisotropy and representing rotation-vibrational spectra with couplings for asymmetric triatomic molecules, is a problem worth attention. Finally we mention about some recent attempts in this direction concerning application to polyatomic molecules [ 171. One of us (YJN) thanks the US Department of Energy for partial support of this work under Grant No. DE-FG05-85ER-40219; he thanks K.A. Milton
References [ 1 ] Z. Chang, Y.H. Guo and H. Yan, Phys. Lett. A 156 ( 199 1) 192; Z. Changand H. Yan, Phys. Lett. A 158 (1991) 242. [2] D. Bonatsos, P. Raychev and R. Roussev, Chem. Phys. Lelt. 175 (1990) 300. [ 31 P. Raychev, R. Roussev and Yu.F. Smimov, J. Phys. G 16 (1990) L137. [4] M. Chaichian, D. Ellinas and P. Kulish, Phys. Rev. Lett. 65 ( 1990) 980. [5] M. Kibler et al., J. Phys. A 24 (1991) 5283. ] A. Alekseev, L.D. Faddeev and Volkov, preprint CERN TH5081/91. ] C. Ramirez, H. Ruegg and H. Ruiz-Altaba, The quantum group and conformal field theory, in Nonperturbative methods in low-dimensional QFT, eds. G. Domikos et al. (World Scientific, Singapore, 199 1). ] A. Kundu and B. Basumallick, Mod. Phys. Lett. 7 ( 1992) 61. [9] A. Kundu and B. Basumallick, Phys. Lett. A 156 (1991) 175. lo] A.J. MacFarlane, J. Phys. A 22 (1989) 4581; L.C. Biedenham, J. Phys. A 22 ( 1989) L873; C.P. Sun and H.C. Fu, J. Phys. A 22 (1989) L983. ll]X.C.Song,J.Phys.A23(1990)L821. [ 121 T.L. Curtright and C.K. Zachos, Phys. Lett. B 243 (1990) 237. [ 131 Y.J. Ng, J. Phys. A 23 (1989) 1023. [ 141 G.M. Barrow, Introduction to molecular spectroscopy (McGraw-Hill, New York, 1964); G. Herzberg, Molecular spectra and molecular structure. II Infrared and Raman spectra of polyatomic molecules (Van Nostrand, New York, 1951). [ 15 ] J. Katriel and A. Solomon, J. Phys. A 24 ( 199 1) 2093. [ 161 A. Ghosh, A. Kundu and P. Mitra, Symmetries of anisotropic oscillator and q-oscillator models, Saha Inst preprint (1992). [ 171 Bonatsos and C. Daskaloyannis, Chem. Phys. Lett. 203 (1993) 150.
ELSEVIEK
LETTERS
A
Physics Letters A 197 (1995) 221-226
q-deformed angular momentum operator and possible application to triatomic molecules A. Kundu a,1, Y.J. Ng b*2 aFb 17, Mathematik/Informatik, GH-Kassel Universitdt, Holliindische Strasse 36, 34109 Kassel, Germany b Centerfor Theoretical Physics, MIT, Cambridge, MA 02139, USA
Received 26 July 1993; revised manuscript received 6 October 1994; accepted for publication 14 October 1994 Communicated by D.D. Holm
Abstract A q-deformation of the angular momentum given through deformed radius vector and momentum operator is proposed. Such a realization of the SU,(2) algebra in three oscillator modes is different from the standard one and the corresponding Hamiltonian of the deformed oscillators naturally generates anisotropy along the z-axis and the required type of rotation-vibrational coupling for symmetric-top like molecules. Possible application to triatomic molecular spectra is discussed and for the more general asymmetric case a further realization of the deformed angular momentum in anisotropic oscillators with mutually prime frequencies is proposed.
Quantum group and related q-deformed structures, though discovered only recently, are penetrating into different branches of physics at a fast rate [l-8]. Realization of the SU,(2) quantum group through single mode [ 91 as well as two [ IO,1 1 ] qoscillators also opened up various possible applications and relevant studies [ 121. There have been several interesting attempts to apply such q-deformations to molecular spectroscopy, nuclear physics, quantum chemistry, chemical physics etc. Among them, the coincidence (with high accuracy) of experimentally observed vibrational and rotational spectra of some diatomic molecules with the q-oscillator model is indeed a surprising result [ 1,2]. Nevertheless, in spite of all these important achievements, the
’ Permanent address: Saha Institute of Nuclear Physics, AF/l Bidhannagar, Calcutta 700 064, India. * Permanent address: Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA.
question raised in Ref. [ 13 1, viz. how the well known angular momentum operator z=rxp,
(1)
satisfying the algebra [1,, /*I= +1* ,
[l,,
I-1=21,,
(2)
should be deformed such that the corresponding L would satisfy the quantum algebra
[b,Ll=*L,
[~+,Ll=[2&1,,
(3)
seems not to have been answered properly yet. (Throughout this paper we use the notation [xl,= (qX-q-X)/(q-q-‘).) Therefore our aim here is to focus on this problem and attempt to find a consistent deformation of r and p operators, so that L would satisfy algebra (3 ). Naturally it would mean realizing the SU,(2) algebra through three different q-oscillator modes (altematively one may possibly look directly for a realization
03759601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SsDrO375-9601(94)00858-2
A. Kundu. Y.J. Ng /Physics Letters A 197 (1995) 221-226
222
of SO,( 3 ) ) some of which must be deformed. Consequently, exploiting the idea of application to diatomic molecules [ 1,2 1, one may hope and speculate to achieve here similar results for triatomic molecules. For this purpose we rewrite relation ( 1) in a form more suitable for deformation, 1, =Ti(rkp3-pkr3), &=$i(r+p--r-p+),
(4)
where the standard definition xLt-=x1 *ix, for operators has been used. Introducing further three independent oscillators as II, =f(r+
+ip,),
6=(1IJZ)(++iP3),
(5)
we express (4) in the form 1, =Jz
(a-al -at+a,) ,
I_ =fi
(aTa
l,=n+
-n_
-a+af)
,
,
(6)
where n, =a1 u+ . Note that the canonical commutation relations [r,, p,] = iS, would induce the oscillator algebra
[a*,at+l=l,
[4,41=1,
[n,,at,l=at+,
[hafl=aI,
(7)
and their complex conjugates, the other relations being trivial. Using (7) it is easy to check that the I operator ( 6 ) indeed satisfies the standard SU ( 2 ) algebra (2). Now to get generators L of SU,( 2) we propose to deform 1given by (6 ) in the form L, =Jz
(A-A$-ADA,)
,
L_ =fi
(ALA,-A+Af)
, (8)
where the A are some deformed oscillators satisfying the relations
[n3,Afl=AI,
[A+,At,l=(n++11f2(n+-n-) -n+f2(n+
where f2(x) = [ [x+l+ 1 ] I,[ [l-x] I4 (for fixed values of I) with the notation [ [x] lq= [x]Jx. It is not difficult to check that due to the deformed oscillator algebra( 9)) expressions (8) indeed satisfy the known quantum algebra (3 ). Note, however, that though here the generators L satisfy the known quantum algebra SU,( 2), the realization (8) expressed through three independent deformed oscillator modes as well as the corresponding oscillator algebra (9) differ from the known realizations through standard q-oscillators [IO,1 1 ] or other deformations studied in Ref. [ 12 1. Here out of three oscillator modes only two, viz. A *, are practically deformed, while A3 = a3 remains unchanged. Interesting, due to such deformation the A+ and A_ oscillators no longer remain independent and their mutual commutation relation is given by the last expression in ( 9 ) . However, since at q-+ 1,f( x) -+ 1, one gets back in this limit the usual oscillators satisfying algebra (7) and a+, a_ become independent again with vanishing commutator between them. Therefore (8 ), in fact, represents a realization of SU,( 2) through three oscillator modes, two of them being deformed. We also observe that the q-deformed oscillators introduced here define a simple deforming map to the usual oscillators (7) as
A+ =f(n+ -n_)a+
,
A- =a_f(n+ -n_)
A3 =a3,
, (10)
which is more like the deformations discussed in Ref. [ 121 than that of the known q-oscillators [ 111. This gives yet another realization for the deformed angular momentum operator expressed now through un-
[n+,Ail=A$,
L, =fi
(a-al-a$a3)f(n+
L- =fif( L3=n+ -n_
-n_)
,
n, -n_)(ata3-a+a3>, .
(11)
One may observe that mapping tines the deformation
( 10) due to (5 ) deof radius vector r, as
-n_ -1))
[A_,A~]=(n_+1)f2(n+-n--l) -n-f*(n+
(9)
f2(n+-n_-1)A-A+’
deformed oscillators as
L3=l,=n+-n_,
[A,,A!il=l,
A+A_ = f2(n+ -n-I
-n-j.
R+ =r+f(13) ,
R- =f(bk
,
R3=r3,
and gives similar expressions for the deformed momentum operator P,
A. Kunda, Y.J. Ng /Physics Letters A 197 (1995) 221-226
p+ =P+f(l,)
9 p3 =P3.
9 p- =flb)P-
Therefore the deformed angular momentum operators may finally be expressed through the deformed canonical operators R and P as L+ =Ti(R+P3-RJPk),
L3=13,
(4’ )
with the q-commutation relations R,R_=fl(lx)R_R+,
P+P_=fl(l,)P_P+,
R+P_=fl(Z,)P_R++2if*(l,-1)) P+R_=fl(lj)R_P+-2if2(lJ-l),
(9’)
223
with KQJ. h, is the vibrational part of the Hamiltonian, which for the triatomic molecules in the general anisotropic case may be given by [ 14 ]
oi being different vibrational frequencies of the normal coordinates. However, it is known that Hamiltonian ( 12) is not enough for describing the observed molecular spectra and suitably chosen rotation-vibrational coupling terms must be added to it. Such an interaction term is usually introduced by formally replacing the coefficients A,, B, in ( 13)
wheref l(6) =f *(13- 1 )/f *(lJ), supplemented by
by
[R3, PJI =i ,
A,,,=A,-
c a:(n,+$c&) I
,
B,,, =B,-
C af(n,+idi)
,
[&,&l=[&,P,l=O,
[R,,P,l=[P,,R,l=[P,,P,l=O,
(15)
and [I,,R,]=kR,,
with d, being the degeneracy. This is equivalent to adding an interacting term
[I,,P,l=kP,,
14, R,l= [I,, PJI =0 , to have a closed algebra. We note that L3 can also be written in a more symmetric form in terms of the deformed canonical operator as [2L3],=i(R+P_ -R_ P+), which may be shown using the Wigner-Eckart theorem. Therefore this gives the expressions of the deformed angular momentum operators through deformed radius vector and deformed momentum operators, answering thus our major question raised above. The above realization of SU,(2) generators (11) recovers via (6) the mapping between deformed and undeformed angular momentum: L+=l+f(l,), L_=f(l,)l_, L,=lJ.Afterfindingthis q-deformation of angular momentum through three oscillators, we try to look for its possible applications to triatomic molecules. The Hamiltonian of polyatomic molecules may be given by [ 141 h=h,+h,.
(12)
h, represents the rotational part of the Hamiltonian, h,=B,J(J+l)+(A,-B,)K*+(e,+e,,i,)
,
(13)
where A,, Be are the equilibrium values of the coeffrcients and K is the quantum number of the component of the angular momentum along the figure axis
II,=-
c cr;B(ni+$d,)J(J+l)
F (4
-&‘)(n~+~d~)K*,
(16)
to the Hamiltonian resulting as h=hr+h,+hi.
(17)
In the isotropic case, i.e. when all the frequencies wi are equal, h, becomes completely degenerate, h,=o(n,
+n*+&+$)
=w(2(n+
+n_)+n3+$),
and invariant under SU (2 ), i.e. commuting with the angular momentum operators 1 (6). However, if we deform h, using our deformed oscillators as H,=o((At,,A+}+{At,A-}+t{Af,A,}),
(18)
something interesting happens. Firstly, using mapping ( 10) one may express ( 18) in the form H,=o(g(l,l,)(n+
+n_ +l)+ns+f)
)
(19)
where g(1, ls)=f *(&-l)+f *(l,), 13=n+--n_. Comparison with the isotropic harmonic oscillator shows clearly that such a deformation brings simultaneously the anisotropy and a coupling between vibrational and rotational modes into the picture. Con-
224
A. Kundu, Y.J. Ng /Physics
Letters A 197 (1995) 221-226
sidering the deformation q=e” to be small and retaining up to the first nontrivial contribution, we get from ( 19) the corresponding energy spectrum as
A,,,-B,,,=A,-B,-fwa~(n,+n2+1),
E,=eo+e,,,+~~++~:),+e~Z?,,
A,=A,+wa;,
(20)
B,,, =B,-fwc&n,
+nz+l)
)
&Be--fwcu;.
where e, is the same as the original harmonic oscillator term h,,
Since (23) is in the familiar form of rotational-vibrational spectra of triatomic molecules [ 141, we can consider the absorption band for the transition
introduces anisotropy and a correction &= )wcr2(1(1+ 1) +I:) to the rotational energy. On the other hand,
An=+l,
e$:!=focu21(l+l)(n,+n2+l),
(21)
generates interaction between vibrational and rotational modes, while e~2,),=fwa2(n,+n2+l)1:,
(22)
accounts for the interaction with explicit anisotropy along the z-direction, which may be considered as the direction of the figure axis giving K=l,. Therefore starting from the noninteracting Hamiltonian ( 12) and deforming it by our deformed oscillators we may generate in a natural way the required kind of higher order interacting terms (apart from some corrections to the original Hamiltonian) in the Hamiltonian. Indeed, identifying
and
where (Y=&, we see that for molecules with o$ = 2@, we may obtain the needed vibration-rotation interaction terms ( 16) by adjusting the deformation parameter CLNote that usually such interaction terms are required to be added in an ad hoc manner to the original Hamiltonian to achieve a better fit with experimental results. To consider the effect of these additional terms in the case of triatomic molecules, we rewrite the deformed Hamiltonian (for small values of (Yand renaming J= 1) in the form H=h,+H,
=4v,J(J+l)+(A,v, where
--4v,W2 >
(23)
AJ=O,kl,
AK=o,
corresponding to the parallel band of symmetric-top like molecules. Note that this gives from (23) the conventional P, Q and R bands as follows. P-band (for AJ= - 1) : Y=v,,-~B~,~J-Av(J,
K),
(24a)
R-band (for AJ= + 1): v=v,,+~B~,~(J+~)-Av(J,K),
(24b)
and Q-band (for AJ= 0): v= v0 -Av(J,
K) ,
(24~)
where
=$wa$(J(J+1)+K2),
(24d)
is responsible for the nonconstant spacing and asymmetry of bands observed due to the rotation-vibrational coupling. To have a qualitative picture of the absorption band obtained through our deformation, we take some representative values for Be, vo, and chose n’ , 4wt.x; appropriately. It is interesting to observe that the graphical pictures (Fig. 1) described by (24 ) resemble clearly the typical parallel band spectra with distinct P, Q, R bands observed for symmetric-top like polyatomic molecules [ 141. The intensity distributions are based on the formulas given in Ref. [ 141. It should be fair, however, to note that the spectral perturbations represented by these or similar additional terms can be postulated a priori independently of the deformation mechanism. Our proposed application to triatomic molecules would indeed be farfetched were it not for the curious success with which the q-oscillator has been applied to the diatomic molecules as shown in Ref. [ 1 ] and [ 2 1. But one should
A. Kunda. Y.J. Ng /Physics Letters A 197 (1995) 221-226
225
I
I
I
'spectra.dat" -
K-4
1
1
I
I
I
I
1
I
K-3
a) K-2
K=l
K=O
b)
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
Fig. 1. Typical parallel band absorption spectra of symmetriotop molecules simulated from (24) with n’ =O, vO= 13 10 cm-‘, B,= 10 cm- ‘, fox-x+$ ~0. I cm- ’ . The heights of the spectral lines correspond to their intensities. Sub-bands for different K values (KG J) in (a) are directly superimposed in (b). The characteristic features of such polyatomic molecular spectra are [ 141 (i) appearance of distinct P, Q and R bands, (ii) absence of Q band for K= 0, (iii) nonconstant spacing and asymmetry in the distribution of P and R bands due to the rotation-vibrational coupling, (iv) shifting of the origin of sub-bands and (v) diminishing spectral intensity of P and R sub-bands relative to Q with increasing K.
keep in mind the speculative nature of this proposed application. For application to asymmetric triatomic molecules, we have to consider possibly the more general anisotropic case, when all three oi are different. In general the corresponding Schrbdinger equation for the polyatomic molecule is given by
For n= 3 this corresponds to triatomic molecules with three degrees of freedom and the anisotropic Hamiltonian being given by ( 14). Unfortunately, for such a general anisotropic case, the oscillator realization of SU( 2) or SU,(2) is not known. We stress again that ( 6 ) and ( 8 ) are useful only when all Wiare equal. Nevertheless an interesting representation in the oscillator mode can still be constructed [ 15,161, when the frequenciesw+, o3 have integer ratios. Using this
idea we can write for this anisotropic case an expression similar to (8 ) r+ =J2
(Liif
-l?t,a”3)
)
r_
(lcd3
-c?+Zf)
)
=Jz
r, =fi+
-n’_
.
(26)
The operators Zi here are generalized bosons [ 15 ] with the usual commutation relations (7) but with the number operator fii being related to the original number ni as rii= [nil/&], where [ ] denotes the integral part and the integers ki represent the anisotropy of the model as
It is not difficult to check that such bosons may be expressed through the standard oscillators as [ 15 ]
226
c+=
A. Kundu. Y. J. Ng /Physics
(iiF)“*
and C. Rasco for useful discussions, while the other author (AK) acknowledges the support of the Alexander von Humboldt-Stiftung.
ta+)k.
The roperators through these generalized bosons satisfy the SU (2) algebra and commute with the anisotropic Hamiltonian h++c+l;;.
n
n3
(27)
We now try to do the same trick with the deformed algebras, i.e. try to find an expression similar to (S), but realized through some generalized deformed oscillators related to the anisotropic case [ 16 1, such that the resultant 2 would satisfy again the quantized algebra %I,( 2 ). Such generalized deformed oscillators A’may be expressed through the corresponding undeformed oscillators d exactly in the same way as (lo), Al+=f(n”+ -L)d+ 23 ‘53 .
, A’_ =r?_f(fi+
Letters A 197 (1995) 22 I-226
-ii_)
) (28)
As a result of the deformations in the anisotropic case we may get again expressions similar in form to (9 ), ( 19)-( 22) etc. In all these expressions the number operator ni will be replaced now by its anisotropic counter part: Ai=ni/ki+ri/ki (O< ricki), ri being integer valued objects depending on the anisotropic parameters k,. Consideration of these terms generated through deformation in the presence of anisotropy and representing rotation-vibrational spectra with couplings for asymmetric triatomic molecules, is a problem worth attention. Finally we mention about some recent attempts in this direction concerning application to polyatomic molecules [ 171. One of us (YJN) thanks the US Department of Energy for partial support of this work under Grant No. DE-FG05-85ER-40219; he thanks K.A. Milton
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