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Deformed oscillator for the Morse potential
Volume 203, number 2,3
CHEMICAL PHYSICS LETTERS
19 February 1993
Deformed oscillator for the Morse potential Dennis Bonatsos a and C. Daskaloyannis b ’ Institute ofNuclear Physics,NCSR “Demokritos”,15310Aghia Paraskevi,Attiki, Greece b Departmentof Theoretical Physics, Universityof Thessaloniki, 54006 Thessaloniki, Greece Received 16 September 1992;in final form 25 November 1992
Using quantum algebraic techniques we construct a generalized deformed oscillator giving the Same spectrum as the Morse potential exactly. The generalized deformed bosons used for this purpose behave, up to first order approximation, as the Qbosons already considered in the literature. The relations between the deformed creation and annihilation operators constructed here and the ones introduced earlier by Levine and by Nieto and Simmons are also found.
1. Introduction
Quantum algebras (also called quantum groups) [ l-41 are receiving much attention in physics. From
the mathematical point of view they are g-deformations of the universal enveloping algebras of the corresponding Lie algebras. When the deformation parameter q is set equal to 1, the usual Lie algebras are obtained. Quantum algebras are concrete examples of Hopf algebras [ 3,4]. Initially used for solving the quantum Yang-Baxter equation [ 5 1, they are now finding applications in several branches of physics, especially after the introduction of the q-deformed harmonic oscillator [ 6,7]. Applications in conformal field theory [ 8 1, quantum gravity 191, quantum optics [ 10 1, as well as in the description of spin chains [ 111 have already appeared, while in atomic physics attention has been focused on the hydrogen atom [ 12,13 1. In nuclear physics it has been found that the pairing correlations in a single-j shell can be described in terms of the usual q-deformed harmonic oscillators approximately [ 141, or in terms of a generalized q-deformed harmonic oscillator exactly [ 151, the deformation parameter r (where q=eir), which is serving as a small parameter, being related to the inverse of the size of the shell. Rotational spectra of diatomic molecules [ 16,171, deformed nuclei [ 18,191, and superdeformed nuclei [20] have been described in terms of the q-deformed rotator having the symmetry SU,(2), the de150
formation parameter r* (with q=eir) being found [ 191 to correspond to the softness parameter of the variable moment of inertia (VMI) model. The implications of the SU,(2) symmetry on the electric transition probabilities connecting the rotational levels have also been studied [ 2 11. Vibrational spectra of diatomic molecules have been described in terms of q-deformed harmonic [22] and anharmanic [ 23,241 oscillators having the SU,( 1, 1) symmetry. WKB equivalent potentials for these oscillators have been determined [25,26], the WKB equivalent potential corresponding to the SU,( 1, 1) symmetry was found [ 261 to be a deformation of the modified Pbschl-Teller potential, which is connected to the Morse potential [27] by a known transformation [ 281. The Morse potential [ 271 has been the subject of several algebraic investigations. Levine [ 29,301 represented the Morse Hamiltonian by a quadratic Hamiltonian using the operator algebra of two coupled oscillators. This representation corresponds to special values of the potential strength, constrained by the maximum number of the energy eigenvalues. Corder0 and Hojman [ 3 I], after expressing the generators of the SU ( 1, 1) algebra by a combination of the potential, the energy and the momentum operators, obtained the energy spectrum using the discrete eigenvalues of the SU( 1, 1) Casimir operators. The SU( 1, 1) symmetry of the Morse potential has been used by several authors [ 32-371. A spectrum
0009-26141931606.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 203, number2,3
CHEMICAL PHYSICS LETTERS
similar to the one of the Morse potential is also obtained in the 0 (4)limit of the vibron model for diatomic molecules, which has a U(4) overall symmetry [38]. It has also been used recently in the description of vibrations of polyatomic molecules by representing each bond in the molecule by a Morse potential [ 39,401. It is well understood that the Morse potential corresponds to an anharmonic oscillator. Therefore it should be possible to construct creation and annihilation operators describing a Morse oscillator in a way similar to the description of the usual harmonic oscillator in the occupation number representation (“second quantization” ). Some attempts in this direction already exist in the literature. Nieto andSimmons [ 4 1] investigated the raising and lowering operator formalism for the Morse potential by using the recurrence relations connecting the known eigenfunctions of the Schrijdinger equation. This method was applied [42-451 to nearly all known solvable potentials for which the solution of the Schriidinger equation is known analytically and the recurrence relations of the eigenfunctions are explicitly known. Levine [ 29,301 used the abovementioned method of two coupled oscillators, while some creation and annihilation operators for highly anharmonic oscillators have been introduced by Kellmann [46]. In this Letter, we show how the construction of suitable creation and annihilation operators can be achieved directly from the spectrum through the use of quantum algebraic techniques. The relation between the bosons which are going to be obtained for the Morse potential and the usual Q-bosons will also be studied. Furthermore, the relations between the operators which will be introduced in the present approach and those introduced earlier by Levine [29,30] and Nieto and Simmons [41-451 will be demonstrated. Solving the Schriidinger equation for the Morse potential [ 47-49 ] V(x) =D(
1-e-aX)2
(1)
where
(2)
20 and W=(Y ;. J
(3)
The anharmonicity constant x~, defined by Cooper [48], is a perturbational parameter for the Morse potential, measuring the deviation from the harmonic oscillator limit, which is obtained at x,-+0.
2. The generalized deformed oscillator Our first aim is to construct a deformed oscillator giving the same spectrum as in eq. (2). Deformed oscillators giving the spectrum of a Piischl-Teller potential have already been studied in ref. [SO], where the method of ref. [ 5 11 was applied. This method gives the deformed oscillator directly from the energy spectrum. The PGschl-Teller spectrum coincides with the spectrum of the Morse potential up to an energy shift, therefore the oscillator algebra of [50] can be transferred to the case of the Morse potential. A general deformation of the harmonic oscillator can be given by the basic relation [ 5 1 ] ~(aa+)-f(a+a)=1,
(4)
where a + (a) are creation (annihilation) operators and f(x) is a real analytic function defined on the real positive axis. In the case of the usual harmonic oscillator one has f(x) =x, which leads to the usual boson commutation relation ]a, a ’ ] = 1. The number operator N satisfies, by definition, the commutation relations [u,N]=u
and
[u+,N]=-a+.
(5)
It can be shown [ 5 1] that N=f(a+u)
.
(6)
If (4) holds, then it is also true that uu+=g(u+u),
(7)
where the function g(x) is defined by g(x)=F(ltf(x)),
one obtains the energy spectrum E(n)=~w[(n+t)-x,(n+f)2],
1 Rar xe= ?W
19February1993
F=f-’ .
(8)
If 1a) is a basis of eigenvectors of the number operator N, Nla)=cyla),
(9) 151
CHEMICAL
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19 February
PHYSICS LETTERS
1993
then from eq. (5 ) one has (20)
aI@=JGiP-I1,
afla)=J{a+l]
ja!+l),
(10)
where [a] is a function of (Y.Furthermore from eq. (8) one has
(11) from which we conclude that [ 5 1] [a] =F(a)
*
(12)
corresponding to the eigenvalues E,=Q(
[ntl]
t [n])
=+A(F(n+ l)+F(n))
.
(21)
If the spectrum is given by a definite real function of the number n+f, that is, E,=~AH(n+:),
(22)
The eigenvector /0)) corresponding to the zero eigenvalue of the number operator N, satisfies the following relation:
then
if&‘(O)=0 (orf(O)=O)
The specific properties of the generalized deformed oscillator formulated above are fixed by the structure function F(n). By choosing
thena/O)=O.
(13)
In this Letter, we assume that the function F(x) is zero when x=0. The number operator N defined by eq. (6) satisfies eqs. ( 5). Other useful identities can be calculated easily and they are listed below: u+u=F(N)=
H(xtf)=f(F(x+l)$F(x)).
F(n)=n
(23)
(24)
one obtains the usual harmonic oscillator. For
[N],
aa+=F(N+l)=[N+l],
(14)
F(n)=
4” -4 =inI q_y:ln
(25)
and
{a,a+}=[N+l]+[N].
obtains the q-deformed harmonic oscillator of Biedenharn [ 61 and Macfarlane [ 71, while by selecting
one
[a,af]=[N+l]-[N], (15)
Let h(z) be an entire function, then the following properties are true: h(N)(a+)~=(u+)~h(Ntm),
(16)
h(N+m)(a)“=(a)“h(N).
(17)
The eigenvectors of the number operator N are generated by the formula
In>=
J+qj(u+)“lo>,
(18)
where
[El! = kfiI [kl= )j F(k) .
(19)
These eigenvectors are also eigenvectors of the energy operator 152
(26) the Q-deformed oscillator of Arik and Coon [ 521 can be generated. As can be seen using eq. (26), the Qdeformed oscillator satisfies the commutation relation:
au+-Qu+u= 1 .
(27)
For Q> 1 the spectrum of this oscillator increases more rapidly than the equidistant spectrum of the usual harmonic oscillator, while for Q< 1 its spectrum increases less rapidly than equidistant, i.e. it is compressed. It is worth mentioning that for Q< 1 this oscillator has similarities to the Morse oscillator for values of Q near to unity, as we shall show later.
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19 February
CHEMICAL PHYSICS LETTERS
3. A generalizeddeformedoscillator for the Morse potential
1993
a(n) = [n]Q =n+;T(n2-n)+hT2(2N3-3N2tl)
We now look for a structure function which will give a spectrum similar to that of eq. (2). Since eq. (2) contains a polynomial quadratic in n and in addition we wish to keep F(0) = 0, the structure function should be of the form P(n)=a,n+b,nZ.
(28)
Using eqs. (2), (23) and (27) we easily conclude that a solution to the problem is provided by the structure function F(n)=n-X$2,
(29)
F(n)+F(n+l) 2
=~w{(n+f)-x,(n+f)2-fXe}.
(30)
Then eq. (2) can be written in the form E(n)=E’(n)-
$$,
Keeping terms up to order of T2 the energy eigenvalues of the Q-deformed oscillator can be written as
=(n+;)(l-$T+gT2)+(,tf)‘(tT-tT2) +(~z+;)~;T~+(~T+&T~).
(34)
We remark that to lowest order in T this equation coincides with eq. (30) with x,=-;T.
which leads to a spectrum E’(n)&u
(33)
+$T3(N4-2N3+N2)+..._
(31)
i.e. the spectrum of the generalized deformed oscillator constructed here is the same as the spectrum of the Morse potential up to an additive constant. The operators a and a+ satisfy the commutation relation [a,a+]=F(N+l)-F(N)=l-x,(2N+l). (32) It is then clear that a and a+ are not pure bosons. However, their behavior comes closer to the pure bosonic behavior for small values of the anharmonicity constant x,, as expected.
4. The Q-deformedoscillator The relation between the generalized deformed oscillator for the Morse potential, which we have just constructed, and the Q-deformed oscillator is worth considering. Defining Q = eT with T real and assuming T to be small, eq. (26) can be written as a Taylor expansion in the form
(35)
The next order terms (as T* (n+ f )3), however, can cause sizeable deviations between eqs. (30) and (34) for large values of n. Furthermore, from eqs. (26) and (27) one easily obtains for the Q-deformed boson operators that
=1tTn+‘T*n2t’ T3n3t 2 6
... .
(36)
We remark then that the commutation relation between Q-deformed boson operators resembles the one of the operators suitable for the Morse potential (given in eq. (32 ) ) to lowest order in T with the identification x,= - t T. We conclude therefore that the generalized deformed oscillator corresponding to the Morse potential behaves, to first order approximation, as the Q-deformed oscillator already considered in the literature.
5. The formalismof Levine Furthermore, it is worth checking the relation between the present formalism and that introduced earlier by Levine [29,30] through the use of two coupled oscillators. In refs. [ 29,301 the Hamiltonian has the form H&zw&4+A_ tfz()) .
(37)
The operators A,, I,, are defined as 153
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A, =N,~=a+/l,
A_=N,:!2P+a,
Zo[~-,~+l,
(38)
where (Y,p satisfy the usual harmonic oscillator commutation relations and N,, is a number characteristic of the chosen coupling, the quantum number of the highest bound state being 4 (N,,- 1) . After a little algebra one can prove that the operators defined above are related to the deformed oscillator operators by the following relations: A +=
19 February 1993
CHEMICAL PHYSICS LETTERS
’
J
“‘l)’ (41)
This operator can be connected to the creation and number operators of the generalized deformed oscillator as follows: A+(N)=&
Nmf 1 + NQ 2 max J
(n+l)(l-x,(n+l))
(1-x,(2n+l))(l-xC(2n+3))
U-l(N)L(N)a+U(N)
,
(42)
where (39)
U(N)=Jl-xJ2Ntl)
and
r,(N)=(I-2Nx,)-‘. the anharmonicity parameter of the deformed oscillator being given by
(43)
The corresponding lowering operator (eq. (2.17) in ref. [ 4 I] ) can be written as
1 x”=N,,,. The fact that N,, is a positive integer imposes restrictions on the possible values of the potential depth, as one can see from eq. (3). When these restrictions are met, the algebra of the deformed oscillator accepts a finite dimensional representation, and, furthermore, there is a polynomial representation of the algebra [ 501. It is worth remarking at this point that in Levine’s formalism [29,30] the two coupled bosomsgenerate the algebra of the Morse deformed oscillator. This is also true in the case of two coupledfermions in a single-j nuclear shell, when the coupled pair has zero angular momentum [ 14,15 1.
A_(N)In)=&
l-lx=$;;l) e
x
n( 1-x,n)
J
(1-x,(2ntl))(l-x,(2n-1))
‘n-1>* (44)
The lowering operator can be connected to the annihilation and number operators of the deformed oscillator as follows: A_(N)=&
U-‘(N)aL(N)U(N).
(45)
The lowering and raising operators have n-dependent representations using the position (x) and momentum (d/d-x) operators (see eq. (2.15) of ref. [41] ) as follows:
[A-W)-A+(N)1 6. The formalism of Nieto and Simmons Finally, it is interesting to examine the relation between the generalized deformed oscillator introduced here and the formalism of Nieto and Simmons [ 4 1-45 1. Initially, Nieto and Simmons [ 4 1 ] introduced n-dependent raising and lowering operators for the Morse potential, using the recurrence relations of the Laguerre polynomials. The raising operator (see eq. (2.17) in ref. [ 4 1 ] ) can be written in our formalism as 154
U-'(N)[uL(N)-L(N)a+]U(N), (46)
where Z=CXX.A more complicated formula can be derived for exp (z) : exp(z)=& +L(NS
U-‘(N)[aL(N)+L(N)a+]U-‘(N) l)L(N)
.
(47)
Furthermore, in ref. [ 441 Nieto and Simmons in-
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CHEMICAL PHYSICS LEmERS
traduced for the Morse potential the n-independent raising and lowering operators J&. By similar calculations one can find the following relations between these operators and the creation and annihilation operators of the present generalized deformed oscillator: &+ =fie
U-l(N)a+U(N),
JCL=,/&
U-'(N)aU(N)
,
(48)
while the Hamiltonian is given by H=ffiw(&+&-S&_rBZ+ t1,.
(49)
Therefore there is an equivalence between the formalism of Nieto and Simmons [ 42-45 ] and the formalism of the generalized deformed oscillator proposed here. The difference is that the generalized deformed oscillator is uniquely defined from the structure function F(n), which satisfies eq. (23), i.e. it is uniquely determined directly from the energy spectrum alone, while the lowering and raising operators in refs. [ 41-451 are determined by the potential appearing in the Schrijdinger equation and by the recurrence relations of the eigenfunctions.
19Febmary1993
SchrGdinger equation for the hydrogen atom has also been studied recently [ 131. It should be of interest to construct the q-deformed Schriidinger equation for the Morse potential and check what changes the qdeformation inflicts on the energy eigenvalues. It should also be of interest to compare the spectra resulting from the q-deformed Schriidinger equation for the modified PBschl-Teller potential to those of the q-deformed anharmonic oscillator with SU,( 1, 1) symmetry of ref. [ 241, for which the WKB equivalent potential has been found in ref. [ 261 to be a qdeformation of the usual modified P8schl-Teller potential. The q-deformed version of the vibron model [ 381 of molecular structure, as well as of the interacting boson model of nuclear structure ( [ 541, see refs. [55,56] for recent overviews) are also of interest, a step in that direction already taken in ref. [ 571. Work in these directions is in progress.
Acknowledgement Support from the Greek Ministry of Research and Technology is gratefully acknowledged by one of the authors ( DB )
7. Discussion In conclusion, we have shown in this Letter how quantum algebraic techniques can be used for constructing a generalized deformed oscillator giving the same spectrum as the Morse potential. The resulting oscillator behaves, up to first order approximation, as the Q-deformed oscillator already considered in the literature. The creation and annihilation operators introduced earlier by Levine [29,30] and by Nieto and Simmons [41-451 are found to be related to the deformed boson operators introduced here, thus exhibiting the unifying power of the present method. The generalized deformed bosons introduced here can be helpful in calculations in which Morse oscillators are involved, since they allow for the use of occupation number representation techniques in a simple way. The spectrum of the q-deformed harmonic oscillator can also be obtained from solving the q-deformed SchrGdinger equation [ 531 for the usual harmonic oscillator potential. The q-deformed
References [ 1] P.P. Kulish and N.Yu. Reshetikhin, Zapiski Semenarov LOMI 101 (1981) 101. [2] E.K. Sklyanin, Funct. Anal. Appl. 16 (1982) 262. [ 31 V.G. Drinfeld, in: Proceedings of the International Congress of Mathematicians, ed. A.M. Gleason (American Mathematical Society, Providence, RI, 1986) p. 798. [4] M. Jimbo, Letters Math. Phys. 11 (1986) 247. [ 5 ] M. Jimbo, in: Braid group, knot theory and statistical mechanics, eds. C.N. Yang and M.L. Ge (World Scientific, Singapore, 1989) p. 111, [6] L.C. Biedenham, J. Phys. A 22 (1989) L873. [7] A.J. Macfarlane, J. Phys. A 22 (1989) 4581. [8] V. Pasquier and H. Saleur, Nucl. Phys. B 330 (1990) 523. [9] S. Mizoguchi and T. Tada, Phys. Rev. Letters 68 (1992) 1795. [ IO] V. Buzek, I. Mod. Optics 38 ( 1991) 801. [ 111P.P.Kulish and E.K. Sklyanin, J. Phys. A 24 (1991) L435. [ 121M. Kibler and T. NCgadi,J. Phys. A 24 (1991) 5283. [ 131X.C. Song and L. Liao, J. Phys. A 25 (1992) 623. [ 141D. Bonatsos, J. Phys. A25 (1992) LlOl. [ 151D. Bonatsos and C. Daskaloyannis, Phys. Letters B 278 (1992) 1.
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[ 161D. Bonatsos, P.P. Raychev, R.P. Roussev and Yu.F. Smirnov, Chem. Phys. Letters 175 ( 1990) 300. [ 171Z. Changand H. Yan, Phys. Letters A 154 (1991) 254. [ IS] P.P. Raychev, R.P. Roussev and Yu.F. Smimov, J. Phys. G 16 (1990) L137. [ 191D. Bonatsos, E.N. Argyres, S.B. Drenska, P.P. Raychev, R.P. Roussev and Yu.F. Smimov, Phys. Letters B 251 (1990) 477. [ 201 D. Bonatsos, S.B. Drenska, P.P. Raychev, R.P. Roussev and Yu.F. Smirnov, J. Phys. G 17 ( 1991) L67. [ 211 D. Bonatsos, A. Faessler, P.P. Raychev, R.P. Roussev and Yu.F. Smimov, J. Phys. A 25 (1992) 3275. [ 221 Z. Chang and H. Yan, Phys. Rev. A 43 ( 1991) 6043. (231 D. Bonatsos, P.P. Raychev and A. Faessler, Chem. Phys. Letters 178 (1991) 221. [ 241 D. Bonatsos, E.N. Argyres and P.P. Raychev, J. Phys. A 24 (1991) L403. [25] D. Bonatsos, C. Daskaloyannis and K. Kokkotas, J. Phys. A24 (1991) L795. [ 261 D. Bonatsos, C. Daskaloyannis and K. Kokkotas, J. Math. Phys. 33 (1992) 2958. [27] P. Morse, Phys. Rev. 34 (1929) 57. 1281Y. Alhassid, F. Giirsey and F. Iachello, Ann. Phys. 148 (1983) 346. [ 291 R.D. Levine, Chem. Phys. Letters 95 (1983) 87. [30] R.D. Levine, J. Phys. Chem. 89 (1985) 2122. 1311P. Corder0 and S. Hojman, Lettere Nuovo Cimento 24 (1970) 1123. [ 321 M. Berrondo and A. Palma, in; Group theoretical methods in physics, eds. J. Ehlers, K. Hepp, R. Kippenhahn, H.A. Weidenmiillerand I. Zittartz (Springer, Berlin, 1980) p. 3. [ 331 S. Kais and R.D. Levine, Phys. Rev. A 41 (1990) 2301. [ 341 C.E. Wulfman and R.D. Levine, Chem. Phys. Letters 97 (1983) 361.
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[ 351 Y. Alhassid, F. Iachello and F. Giirsey, Chem. Phys. Letters 99 (1983) 27. [ 361 C.E. Wulfman and R.D. Levine, Chem. Phys. Letters 104 (1984) 9. [ 371 D.J. Lee, K.J. Shin and SK. Kim, Chem. Phys. Letters 175 (1990) 87. [ 381 F. Iachello and R.D. Levine, J. Chem. Phys. 77 ( 1982) 3046. [ 391 F. Iachello and S. Oss, Phys. Rev. Letters 66 ( 1991) 2976. [401 F. Iachello and S. Oss, Chem. Phys. Letters 187 ( I99 I ) 500. [4 1 ] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. A 19 ( 1979) 438. [ 421 M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D 20 (1979) 1321. [43] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D 20 (1979) 1332. [441 M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D 20 ( 1979) 1342. [45] M.M. Nieto, Phys. Rev. D 22 (1980) 391. [46] M.E. Kellman, J. Chem. Phys. 82 (1985) 3300. (471 S. Fhigge, Practical quantum mechanics (Springer, Berlin, 1974). [48] I.L. Cooper, Chem. Phys. 112 (1987) 67. [ 491 I.L. Cooper, J. Phys. A 25 ( 1992) 1671. [SO] C. Daskaloyannis, J. Phys. A 25 ( 1992) 2261. [ 51] C. Daskaloyannis, J. Phys. A 24 ( 1991) L789. [ 521 M. Arik and D.D. Coon, J. Math. Phys. 17 ( 1976) 524. [ 531 J.A. Minahan, Mod. Phys. Letters A 5 (1990) 2625. [ 541 A. Arima and F. Iachello, Ann. Phys. 99 (1976) 253; 111 (1978) 201; 123 (1979) 468. [55] F. Iachello and A. Arima, The interacting boson model (Cambridge Univ. Press, Cambridge, 1987). [ 561 D. Bonatsos, Interacting boson models of nuclear structure (Clarendon Press, Oxford, 1988). [SS] D. Bonatsos, A. Faessler, P.P. Raychev, R.P. Roussev and Yu.F. Smirnov, J. Phys. A 25 (1992) L267.
CHEMICAL PHYSICS LETTERS
19 February 1993
Deformed oscillator for the Morse potential Dennis Bonatsos a and C. Daskaloyannis b ’ Institute ofNuclear Physics,NCSR “Demokritos”,15310Aghia Paraskevi,Attiki, Greece b Departmentof Theoretical Physics, Universityof Thessaloniki, 54006 Thessaloniki, Greece Received 16 September 1992;in final form 25 November 1992
Using quantum algebraic techniques we construct a generalized deformed oscillator giving the Same spectrum as the Morse potential exactly. The generalized deformed bosons used for this purpose behave, up to first order approximation, as the Qbosons already considered in the literature. The relations between the deformed creation and annihilation operators constructed here and the ones introduced earlier by Levine and by Nieto and Simmons are also found.
1. Introduction
Quantum algebras (also called quantum groups) [ l-41 are receiving much attention in physics. From
the mathematical point of view they are g-deformations of the universal enveloping algebras of the corresponding Lie algebras. When the deformation parameter q is set equal to 1, the usual Lie algebras are obtained. Quantum algebras are concrete examples of Hopf algebras [ 3,4]. Initially used for solving the quantum Yang-Baxter equation [ 5 1, they are now finding applications in several branches of physics, especially after the introduction of the q-deformed harmonic oscillator [ 6,7]. Applications in conformal field theory [ 8 1, quantum gravity 191, quantum optics [ 10 1, as well as in the description of spin chains [ 111 have already appeared, while in atomic physics attention has been focused on the hydrogen atom [ 12,13 1. In nuclear physics it has been found that the pairing correlations in a single-j shell can be described in terms of the usual q-deformed harmonic oscillators approximately [ 141, or in terms of a generalized q-deformed harmonic oscillator exactly [ 151, the deformation parameter r (where q=eir), which is serving as a small parameter, being related to the inverse of the size of the shell. Rotational spectra of diatomic molecules [ 16,171, deformed nuclei [ 18,191, and superdeformed nuclei [20] have been described in terms of the q-deformed rotator having the symmetry SU,(2), the de150
formation parameter r* (with q=eir) being found [ 191 to correspond to the softness parameter of the variable moment of inertia (VMI) model. The implications of the SU,(2) symmetry on the electric transition probabilities connecting the rotational levels have also been studied [ 2 11. Vibrational spectra of diatomic molecules have been described in terms of q-deformed harmonic [22] and anharmanic [ 23,241 oscillators having the SU,( 1, 1) symmetry. WKB equivalent potentials for these oscillators have been determined [25,26], the WKB equivalent potential corresponding to the SU,( 1, 1) symmetry was found [ 261 to be a deformation of the modified Pbschl-Teller potential, which is connected to the Morse potential [27] by a known transformation [ 281. The Morse potential [ 271 has been the subject of several algebraic investigations. Levine [ 29,301 represented the Morse Hamiltonian by a quadratic Hamiltonian using the operator algebra of two coupled oscillators. This representation corresponds to special values of the potential strength, constrained by the maximum number of the energy eigenvalues. Corder0 and Hojman [ 3 I], after expressing the generators of the SU ( 1, 1) algebra by a combination of the potential, the energy and the momentum operators, obtained the energy spectrum using the discrete eigenvalues of the SU( 1, 1) Casimir operators. The SU( 1, 1) symmetry of the Morse potential has been used by several authors [ 32-371. A spectrum
0009-26141931606.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 203, number2,3
CHEMICAL PHYSICS LETTERS
similar to the one of the Morse potential is also obtained in the 0 (4)limit of the vibron model for diatomic molecules, which has a U(4) overall symmetry [38]. It has also been used recently in the description of vibrations of polyatomic molecules by representing each bond in the molecule by a Morse potential [ 39,401. It is well understood that the Morse potential corresponds to an anharmonic oscillator. Therefore it should be possible to construct creation and annihilation operators describing a Morse oscillator in a way similar to the description of the usual harmonic oscillator in the occupation number representation (“second quantization” ). Some attempts in this direction already exist in the literature. Nieto andSimmons [ 4 1] investigated the raising and lowering operator formalism for the Morse potential by using the recurrence relations connecting the known eigenfunctions of the Schrijdinger equation. This method was applied [42-451 to nearly all known solvable potentials for which the solution of the Schriidinger equation is known analytically and the recurrence relations of the eigenfunctions are explicitly known. Levine [ 29,301 used the abovementioned method of two coupled oscillators, while some creation and annihilation operators for highly anharmonic oscillators have been introduced by Kellmann [46]. In this Letter, we show how the construction of suitable creation and annihilation operators can be achieved directly from the spectrum through the use of quantum algebraic techniques. The relation between the bosons which are going to be obtained for the Morse potential and the usual Q-bosons will also be studied. Furthermore, the relations between the operators which will be introduced in the present approach and those introduced earlier by Levine [29,30] and Nieto and Simmons [41-451 will be demonstrated. Solving the Schriidinger equation for the Morse potential [ 47-49 ] V(x) =D(
1-e-aX)2
(1)
where
(2)
20 and W=(Y ;. J
(3)
The anharmonicity constant x~, defined by Cooper [48], is a perturbational parameter for the Morse potential, measuring the deviation from the harmonic oscillator limit, which is obtained at x,-+0.
2. The generalized deformed oscillator Our first aim is to construct a deformed oscillator giving the same spectrum as in eq. (2). Deformed oscillators giving the spectrum of a Piischl-Teller potential have already been studied in ref. [SO], where the method of ref. [ 5 11 was applied. This method gives the deformed oscillator directly from the energy spectrum. The PGschl-Teller spectrum coincides with the spectrum of the Morse potential up to an energy shift, therefore the oscillator algebra of [50] can be transferred to the case of the Morse potential. A general deformation of the harmonic oscillator can be given by the basic relation [ 5 1 ] ~(aa+)-f(a+a)=1,
(4)
where a + (a) are creation (annihilation) operators and f(x) is a real analytic function defined on the real positive axis. In the case of the usual harmonic oscillator one has f(x) =x, which leads to the usual boson commutation relation ]a, a ’ ] = 1. The number operator N satisfies, by definition, the commutation relations [u,N]=u
and
[u+,N]=-a+.
(5)
It can be shown [ 5 1] that N=f(a+u)
.
(6)
If (4) holds, then it is also true that uu+=g(u+u),
(7)
where the function g(x) is defined by g(x)=F(ltf(x)),
one obtains the energy spectrum E(n)=~w[(n+t)-x,(n+f)2],
1 Rar xe= ?W
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F=f-’ .
(8)
If 1a) is a basis of eigenvectors of the number operator N, Nla)=cyla),
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1993
then from eq. (5 ) one has (20)
aI@=JGiP-I1,
afla)=J{a+l]
ja!+l),
(10)
where [a] is a function of (Y.Furthermore from eq. (8) one has
(11) from which we conclude that [ 5 1] [a] =F(a)
*
(12)
corresponding to the eigenvalues E,=Q(
[ntl]
t [n])
=+A(F(n+ l)+F(n))
.
(21)
If the spectrum is given by a definite real function of the number n+f, that is, E,=~AH(n+:),
(22)
The eigenvector /0)) corresponding to the zero eigenvalue of the number operator N, satisfies the following relation:
then
if&‘(O)=0 (orf(O)=O)
The specific properties of the generalized deformed oscillator formulated above are fixed by the structure function F(n). By choosing
thena/O)=O.
(13)
In this Letter, we assume that the function F(x) is zero when x=0. The number operator N defined by eq. (6) satisfies eqs. ( 5). Other useful identities can be calculated easily and they are listed below: u+u=F(N)=
H(xtf)=f(F(x+l)$F(x)).
F(n)=n
(23)
(24)
one obtains the usual harmonic oscillator. For
[N],
aa+=F(N+l)=[N+l],
(14)
F(n)=
4” -4 =inI q_y:ln
(25)
and
{a,a+}=[N+l]+[N].
obtains the q-deformed harmonic oscillator of Biedenharn [ 61 and Macfarlane [ 71, while by selecting
one
[a,af]=[N+l]-[N], (15)
Let h(z) be an entire function, then the following properties are true: h(N)(a+)~=(u+)~h(Ntm),
(16)
h(N+m)(a)“=(a)“h(N).
(17)
The eigenvectors of the number operator N are generated by the formula
In>=
J+qj(u+)“lo>,
(18)
where
[El! = kfiI [kl= )j F(k) .
(19)
These eigenvectors are also eigenvectors of the energy operator 152
(26) the Q-deformed oscillator of Arik and Coon [ 521 can be generated. As can be seen using eq. (26), the Qdeformed oscillator satisfies the commutation relation:
au+-Qu+u= 1 .
(27)
For Q> 1 the spectrum of this oscillator increases more rapidly than the equidistant spectrum of the usual harmonic oscillator, while for Q< 1 its spectrum increases less rapidly than equidistant, i.e. it is compressed. It is worth mentioning that for Q< 1 this oscillator has similarities to the Morse oscillator for values of Q near to unity, as we shall show later.
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3. A generalizeddeformedoscillator for the Morse potential
1993
a(n) = [n]Q =n+;T(n2-n)+hT2(2N3-3N2tl)
We now look for a structure function which will give a spectrum similar to that of eq. (2). Since eq. (2) contains a polynomial quadratic in n and in addition we wish to keep F(0) = 0, the structure function should be of the form P(n)=a,n+b,nZ.
(28)
Using eqs. (2), (23) and (27) we easily conclude that a solution to the problem is provided by the structure function F(n)=n-X$2,
(29)
F(n)+F(n+l) 2
=~w{(n+f)-x,(n+f)2-fXe}.
(30)
Then eq. (2) can be written in the form E(n)=E’(n)-
$$,
Keeping terms up to order of T2 the energy eigenvalues of the Q-deformed oscillator can be written as
=(n+;)(l-$T+gT2)+(,tf)‘(tT-tT2) +(~z+;)~;T~+(~T+&T~).
(34)
We remark that to lowest order in T this equation coincides with eq. (30) with x,=-;T.
which leads to a spectrum E’(n)&u
(33)
+$T3(N4-2N3+N2)+..._
(31)
i.e. the spectrum of the generalized deformed oscillator constructed here is the same as the spectrum of the Morse potential up to an additive constant. The operators a and a+ satisfy the commutation relation [a,a+]=F(N+l)-F(N)=l-x,(2N+l). (32) It is then clear that a and a+ are not pure bosons. However, their behavior comes closer to the pure bosonic behavior for small values of the anharmonicity constant x,, as expected.
4. The Q-deformedoscillator The relation between the generalized deformed oscillator for the Morse potential, which we have just constructed, and the Q-deformed oscillator is worth considering. Defining Q = eT with T real and assuming T to be small, eq. (26) can be written as a Taylor expansion in the form
(35)
The next order terms (as T* (n+ f )3), however, can cause sizeable deviations between eqs. (30) and (34) for large values of n. Furthermore, from eqs. (26) and (27) one easily obtains for the Q-deformed boson operators that
=1tTn+‘T*n2t’ T3n3t 2 6
... .
(36)
We remark then that the commutation relation between Q-deformed boson operators resembles the one of the operators suitable for the Morse potential (given in eq. (32 ) ) to lowest order in T with the identification x,= - t T. We conclude therefore that the generalized deformed oscillator corresponding to the Morse potential behaves, to first order approximation, as the Q-deformed oscillator already considered in the literature.
5. The formalismof Levine Furthermore, it is worth checking the relation between the present formalism and that introduced earlier by Levine [29,30] through the use of two coupled oscillators. In refs. [ 29,301 the Hamiltonian has the form H&zw&4+A_ tfz()) .
(37)
The operators A,, I,, are defined as 153
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number2,3
A, =N,~=a+/l,
A_=N,:!2P+a,
Zo[~-,~+l,
(38)
where (Y,p satisfy the usual harmonic oscillator commutation relations and N,, is a number characteristic of the chosen coupling, the quantum number of the highest bound state being 4 (N,,- 1) . After a little algebra one can prove that the operators defined above are related to the deformed oscillator operators by the following relations: A +=
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CHEMICAL PHYSICS LETTERS
’
J
“‘l)’ (41)
This operator can be connected to the creation and number operators of the generalized deformed oscillator as follows: A+(N)=&
Nmf 1 + NQ 2 max J
(n+l)(l-x,(n+l))
(1-x,(2n+l))(l-xC(2n+3))
U-l(N)L(N)a+U(N)
,
(42)
where (39)
U(N)=Jl-xJ2Ntl)
and
r,(N)=(I-2Nx,)-‘. the anharmonicity parameter of the deformed oscillator being given by
(43)
The corresponding lowering operator (eq. (2.17) in ref. [ 4 I] ) can be written as
1 x”=N,,,. The fact that N,, is a positive integer imposes restrictions on the possible values of the potential depth, as one can see from eq. (3). When these restrictions are met, the algebra of the deformed oscillator accepts a finite dimensional representation, and, furthermore, there is a polynomial representation of the algebra [ 501. It is worth remarking at this point that in Levine’s formalism [29,30] the two coupled bosomsgenerate the algebra of the Morse deformed oscillator. This is also true in the case of two coupledfermions in a single-j nuclear shell, when the coupled pair has zero angular momentum [ 14,15 1.
A_(N)In)=&
l-lx=$;;l) e
x
n( 1-x,n)
J
(1-x,(2ntl))(l-x,(2n-1))
‘n-1>* (44)
The lowering operator can be connected to the annihilation and number operators of the deformed oscillator as follows: A_(N)=&
U-‘(N)aL(N)U(N).
(45)
The lowering and raising operators have n-dependent representations using the position (x) and momentum (d/d-x) operators (see eq. (2.15) of ref. [41] ) as follows:
[A-W)-A+(N)1 6. The formalism of Nieto and Simmons Finally, it is interesting to examine the relation between the generalized deformed oscillator introduced here and the formalism of Nieto and Simmons [ 4 1-45 1. Initially, Nieto and Simmons [ 4 1 ] introduced n-dependent raising and lowering operators for the Morse potential, using the recurrence relations of the Laguerre polynomials. The raising operator (see eq. (2.17) in ref. [ 4 1 ] ) can be written in our formalism as 154
U-'(N)[uL(N)-L(N)a+]U(N), (46)
where Z=CXX.A more complicated formula can be derived for exp (z) : exp(z)=& +L(NS
U-‘(N)[aL(N)+L(N)a+]U-‘(N) l)L(N)
.
(47)
Furthermore, in ref. [ 441 Nieto and Simmons in-
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traduced for the Morse potential the n-independent raising and lowering operators J&. By similar calculations one can find the following relations between these operators and the creation and annihilation operators of the present generalized deformed oscillator: &+ =fie
U-l(N)a+U(N),
JCL=,/&
U-'(N)aU(N)
,
(48)
while the Hamiltonian is given by H=ffiw(&+&-S&_rBZ+ t1,.
(49)
Therefore there is an equivalence between the formalism of Nieto and Simmons [ 42-45 ] and the formalism of the generalized deformed oscillator proposed here. The difference is that the generalized deformed oscillator is uniquely defined from the structure function F(n), which satisfies eq. (23), i.e. it is uniquely determined directly from the energy spectrum alone, while the lowering and raising operators in refs. [ 41-451 are determined by the potential appearing in the Schrijdinger equation and by the recurrence relations of the eigenfunctions.
19Febmary1993
SchrGdinger equation for the hydrogen atom has also been studied recently [ 131. It should be of interest to construct the q-deformed Schriidinger equation for the Morse potential and check what changes the qdeformation inflicts on the energy eigenvalues. It should also be of interest to compare the spectra resulting from the q-deformed Schriidinger equation for the modified PBschl-Teller potential to those of the q-deformed anharmonic oscillator with SU,( 1, 1) symmetry of ref. [ 241, for which the WKB equivalent potential has been found in ref. [ 261 to be a qdeformation of the usual modified P8schl-Teller potential. The q-deformed version of the vibron model [ 381 of molecular structure, as well as of the interacting boson model of nuclear structure ( [ 541, see refs. [55,56] for recent overviews) are also of interest, a step in that direction already taken in ref. [ 571. Work in these directions is in progress.
Acknowledgement Support from the Greek Ministry of Research and Technology is gratefully acknowledged by one of the authors ( DB )
7. Discussion In conclusion, we have shown in this Letter how quantum algebraic techniques can be used for constructing a generalized deformed oscillator giving the same spectrum as the Morse potential. The resulting oscillator behaves, up to first order approximation, as the Q-deformed oscillator already considered in the literature. The creation and annihilation operators introduced earlier by Levine [29,30] and by Nieto and Simmons [41-451 are found to be related to the deformed boson operators introduced here, thus exhibiting the unifying power of the present method. The generalized deformed bosons introduced here can be helpful in calculations in which Morse oscillators are involved, since they allow for the use of occupation number representation techniques in a simple way. The spectrum of the q-deformed harmonic oscillator can also be obtained from solving the q-deformed SchrGdinger equation [ 531 for the usual harmonic oscillator potential. The q-deformed
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