acta physica slovaca vol. 56 No. 4, August 2006 P-FUNCTION OF THE PSEUDOHARMONIC OSCILLATOR IN TERMS OF KLAUDER-PERELOMOV COHERENT STATES

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1 acta physica slovaca vol. 56 No. 4, August 26 P-FUNCTION OF THE PSEUDOHARMONIC OSCILLATOR IN TERMS OF KLAUDER-PERELOMOV COHERENT STATES D. Popov,a, D.M. Davidović b, D. Arsenović c, V. Sajfert d a University Politehnica of Timişoara, Department of Physics, Piaţa Regina Maria No., 34 Timişoara, Romania b The Vinča Institute of Nuclear Sciences, P.O. Box 522, Belgrade, Serbia and Montenegro c Institute of Physics, Pregrevica 8, P.O. Box 57, Belgrade, Serbia and Montenegro d Technical Faculty M. Pupin, University of Novi Sad, Novi Sad, Serbia and Montenegro Received 28 July 25, in final form 6 April 26, accepted April 26 Using the Klauder-Perelomov coherent states of the pseudoharmonic oscillator Hamiltonian, corresponding diagonal P -representation for the density matrix of the gas of pseudoharmonic oscillators in thermal equilibrium is constructed. With a help of this new result, some corresponding characteristic thermal expectation values are obtained. PACS: 3.65.Ca, 3.65.Fd, 5.3.-d, 5.3.Ch Introduction The pseudoharmonic oscillator PHO) potential [, 2] is an anharmonic potential, which, like the harmonic oscillator HO) potential, also allows an exact mathematical treatment. This potential may be considered in a certain sense as an intermediate potential between the harmonic oscillator potential an ideal model) and anharmonic potentials which are more realistic). A comparative analysis of potentials HO-3D 3-dimensional harmonic oscillator potential) and PHO is performed in [2]. In the present work, we examine some properties of the Klauder-Perelomov coherent states of the pseudoharmonic oscillator. Using properties of these states we construct, in Section 3, the diagonal P -representation of the density operator for the PHO quantum canonical gas. This is the main new result of this work. Using it, we also derive expressions for some significant expectation values for this physical system. 2 Klauder-Perelomov coherent states for PHO The effective potential of the PHO is V J r) = mω2 r 8 r2 r ) 2 + h2 r r 2m J J + ) r 2, ) address: dusan popov@yahoo.co.uk /6 c Institute of Physics, SAS, Bratislava, Slovakia 445

2 446 D. Popov et al. where r is the equilibrium distance between the diatomic molecule nuclei and J =,, 2,... is the rotational quantum number. This potential has been treated in detail previously so we will only recall some formulae that will be need. Using Molski s techniques [3] for the Morse oscillator) in our previous paper [4], we have rewritten the PHO effective potential as follows: where V J r) = mω2 8 r2 J λ = r r J r J r ) 2 + mω2 4 r2 J r2 ), 2) J + ) 2 mω ) h r2 3) and r J = 2 h mω λ 2 ) 2. 4) 4 By using the substitution ω = 2ω and the dimensionless variable y = mω ) 2 h r, the corresponding rovibrational Schrödinger equation for the reduced radial function u λ v r) and the dimensionless Hamiltonian is [4] [ d 2 2 dy y2 + λ 2 ) ] 2v + λ + ) u λ 2 4 y2 v y) =. 5) In the previous paper [4], we have demonstrated that the SU, ) is the natural dynamical group associated with the bounded states of the PHO. The discrete representations of the SU, ) group are given by K 2 v, k = kk ) v, k, 6) K + v, k = v + )v + 2k) v +, k, 7) K v, k = vv + 2k ) v, k, 8) where v is the vibrational quantum number, k = 2 λ + ) > 2 and the PHO realization of the raising and lowering operators K ± is K ± = ±y d 2 dy ± ) 2 y2 + 2v + λ +. 9) Generally, the Klauder-Perelomov coherent states are obtained if the generalized displacement unitary operator exp αk + α K ) on the lowest state of the quantum system v =, k are applied [5 7]: z, k = exp αk + α K ), k = e zk+ e ΓK3 e z K, k, )

3 P-function of the PHO 447 where α = 2 θ e iϕ, z = α α tanh α = tanh θ 2 e iϕ and where the group generator K 3 is K 3 = 2 [K, K + ], Γ = ln z 2 ). ) The parameters θ, ) and ϕ [, 2π] are group parameters similar to the Euler angles. The condition z < shows that the SU, ) KP-CSs z, k are defined in the interior of the unit disc. In terms of the basis vectors v, k, using the equations 7) and 8) and the equation K 3 v, k = k + v) v, k, 2) the KP-CSs of the PHO may be expanded as where z, k = N z 2 ) ρv; k) = v= Γv + )Γ2k) Γv + 2k) z v ρv; k) v, k, 3) = 2k )Bv + ; 2k ) 4) and Ba; b) is the Euler beta function. We note here that in agreement with the choices of ρv; k), there are many different families of coherent states, as illustrated in a series of recent works [8 ]. The normalization constant N z 2 ) is obtained from the normalization condition z, k z, k =, so that [ N z 2 ) ] 2 z 2 ) v = 2k) v = F 2k; z 2 ) = v! z 2, 5) ) 2k v= is the Poch- where p F q...) is the generalized hypergeometric function, while a) v = Γa+v) Γa) hammer symbol [2]. Finally, the KP-CSs of the PHO are z, k = z 2 ) k v= z v ρv; k) v, k. 6) The overlap of two KP-CSs of the PHO the scalar product) is σ, k z, k = σ 2 ) k z 2 ) k σ z) 2k, 7) where σ and z are the complex numbers. We need to calculate the convergence radius R [9] v R = lim ρv; k). 8) v In view of the limit [2] lim x Γx + a) Γx)x a = 9)

4 448 D. Popov et al. it is not difficult to prove that, for the KP-CSs of the PHO, we have R =. Then dµz, k) z, k z, k = I, 2) with the integration measure dµz, k) = d2 z π W k z 2 ), d 2 z = drez)dimz) = dϕ dr r, 2) where z = r exp iϕ), r [, ], ϕ [, 2π]. In order to determine the unknown weight function, after the angular integration, we obtain v= v, k v, k ρv; k) R dr r 2v+ W k r 2 ) r 2 ) 2k = I, 22) from which the following infinite set of equations results: R dr r 2v+ W k r 2 ) r 2 ) 2k = ρv; k), v =,, 2,..., R <. 23) The quantities ρv; k) > are then the power moments of the new unknown function h k r 2 ) = r 2 ) 2k W kr 2 ) 24) and the problem stated in 23) is the Hausdorff R < ) moment problem [9]. After the variable change x = r 2, we have R dx x v Γv + ) h k x) = Γ2k) Γv + 2k). 25) In the above Hausdorff moment problem we extend the integer values of v to the complex values s, so that v s and rewrite it as R dx x s Γs) h k x) = Γ2k) Γ2k + s). 26) As is usual in such a problem [8, ], it is convenient to define { } hk x), x R g k x) =, R < x < and to interpret 26) as the Mellin transform g k s) of g kx 2 ): dx x s g k x) gks) def Γs) = M [g k x); s] = Γ2k) Γ2k + s). 28) The explicit formula for obtaining g k x) from gk s) is given by g k x) = 2πi c+i c i 27) ds x s gk def s) = M [gk s); x], 29)

5 P-function of the PHO 449 which denotes the inverse Mellin transform. Using the definition of the Meijer s G-function and Mellin inversion theorem it follows that [3]: dx x s G m,n p,q αx a,..., a n, a n+,..., a p b,..., b m, b m+,..., b q = α s m j= Γb j + s) n j= Γ a j s) q j=m+ Γ b j s) p j=n+ Γa j + s). ) = 3) Comparing equations 26) and 3), we obtain that [3] ) h k x) = Γ2k)G, 2k, x = 2k ) x) 2k 2, 3) so that the weight function W k z 2 ) becomes W k z 2 ) = 2k ) z 2 ) 2. 32) This is a positive function for z < and k > 2. Therefore, the integration measure 2) finally is dµz, k) = 2k π d 2 z z 2 ) 2. 33) The sufficient condition for the solution 32) to be unique is given by the Carleman condition [7, 8]: if the solution of the problem 26) exists then S def = v= [ρv; k)] 2v = {, the solution is unique <, non-unique solutions exist. 34) It is not difficult to prove using the mathematical analysis test methods e.g. the logarithmic or d Alembert test), for the function ρv; k) defined by 4), that the sum S diverges. So, the weight function for the KP-CSs of the PHO is unique. 3 P -representation for PHO canonical gas density matrix We consider a quantum canonical gas of PHOs in thermodynamical equilibrium with a thermostat at temperature T = k B β), where k B is the Boltzmann s constant and β-the well-known temperature parameter. By using the equation 3) and taking into account that the Bargmann index is k = 2 λ + )), the corresponding normalized canonical density operator for a fixed rotational quantum number J or, equivalently, for a fixed number k), is ρ J ρ k = Z k v= e βevj v, k v, k, 35)

6 45 D. Popov et al. where Z J = Z k is the normalization constant, i.e. the partition function for a certain rotational state J and E vj = hω 2k mω 2 r2 + 2 hω v E,J + 2 hω v. 36) The average value of normalized canonical density operator in the KP-CS representation, i.e. the Husimi s distribution function, is ) 2k z, k ρ k z, k = z 2 ) e k z 2 ) k βe,j Z k z z. 37) e β hω By normalizing the density operator to unity, i.e. requiring that T rρ k = dµz, k) z, k ρ k z, k = 38) we obtain the expression for the partition function of the PHO for the fixed rotational state J): Z k = e βmω2 r2 β hω2k ) 2 sinh β hω, 39) where the exponential represents the contribution of the anharmonicity. Using these results, we can write the Husimi distribution function as follows: z, k ρ J z, k ρ k z,k = e β hω) z 2 ) 2k z 2. 4) e β hω Let us now perform the diagonal expansion of the density operator in the KP-CSs: ρ k = dµz, k) z, k P k z 2 ) z, k. 4) In order to find the quasi-probability distribution function P k z 2 ) let us note that the equation f ρ k g = dµz, k) f z, k P k z 2 ) z, k g 42) must be fulfilled for any arbitrary vectors f and g from the Hilbert space. Using equation 35), the left-hand side of 42) becomes LHS Z k v= e βevj f v, k v, k g, 43) while, after the angular integration and the variable change x = r 2, the right-hand side may be represented in the form f v, k v, k g RHS 2k ) dx x v x) 2k 2 P k x). 44) ρv; k) v= Comparing the LHS and the RHS, we conclude that dx x v x) 2k 2 P k x) = e βe,j ρv; k) e 2β hωv, 45) 2k Z k

7 P-function of the PHO 45 which, after the function change, P k x) = Z k e βe,j x) 2k 2 g kx), 46) leads to the integral equation dx x v g k x) = e 2β hω) v Bv + ; 2k ) Γv + ) = Γ2k ) e 2β hω ) v Γv + 2k). 47) The method of solving this equation is the same as the one for obtaining the weight function W k z 2 ) 32), i.e. the solving of the Hausdorff moment problem. Using equation 3) and the tables of the Meijer s G-functions [3], we obtain g k x) = e 2β hω Γ2k )G,, x 2k ) = e 2β hω e 2β hω x ) 2k 2. 48) We can finally write for the P-distribution function P k z 2 ) = e 2β hω ) e 2β hω z 2 ) 2k 2 z 2. 49) Using the tabular integrals it is not difficult to prove that this function satisfies the normalization condition dµz, k) P k z) =. 5) In this way, the diagonal representation of the normalized density operator of the PHO in KS-CSs is ρ k = 2k ) e 2β hω ) d 2 z e 2β hω z 2 ) 2k 2 π z 2 ) 2 z 2 z, k z, k, 5) which fulfills the normalization condition 38). The thermal expectation value the thermal average) of an observable A concerning the PHO is given by A k = T rρ k A) = 2k ) e 2β hω ) 52) d 2 z e 2β hω z 2 ) 2k 2 π z 2 ) 2 z 2 z, k A z, k. In many cases e.g. with the aim of calculating the thermal averages of the powers of the number operator N) it is necessary to solve the integrals of the following kind [2]: I lm = x l x) 2k m Ax) 2 2k 53) Γl + )Γ 2k m) Γ2 2k + l m) 2F 2 2k, l + ; 2 2k + l m; A)

8 452 D. Popov et al. and, besides that, to use the following property of the hypergeometric function 2 F...) [4] 2F a, b; b n; z) = z) a+n n k= n) k b a n) k z k b n) k k!. 54) By using these results, we obtain the thermal expectation values of the number operator N and the square of the number operator N 2 respectively N k = 2k2k ) e 2β hω ) I = N = n 55) e 2β hω N 2 k = 2k2k ) e 2β hω ) I 2 + 2kI 22 ) = 56) = e 2β hω + 2 e 2β hω ) 2 N 2 = n + 2n). These expectation values are independent of the index k. The thermal expectation of the number operator N is the same as the expression of the Bose-Einstein thermal distribution thermal mean occupancy n) n = e 2β hω, 57) which shows that the PHO is suitable for associating it with a boson e.g. a photon or phonon). We can now calculate the thermal second-order correlation function g 2 ) k. g 2 ) k N 2 N N 2 = g 2 ) = 2 58) and the thermal analogue of the Mandel parameter also Q k [5, 6]: Q k N [ g 2 ) ] = N = n. 59) The total normalized density operator which characterizes the quantum gas of pseudoharmonic oscillators is ρ = 2J + )Z J ρ J, 6) Z J where ρ J ρ k is the diagonal representation of the density operator for the rotational state J see equation 5)). Consequently, the total thermal expectation value of an observable A is A = T raρ = 2J + )Z J T raρ J, 6) Z J where T raρ J = A J = A k is the expectation value for the rotational state J. Similarly, the total partition function is Z = J 2J + ) v e βevj = J 2J + )Z J, 62)

9 P-function of the PHO 453 where Z J = Z k is the partition function for the rotational state J see equation 39)). Let us note here that the HO-3D can be considered as an appropriate limit of the PHO. This limit is called the harmonic limit of the PHO and for a certain physical observable A is defined as [7] lim ω 2ω F P HO) lim F P HO) HO = F HO 3D), 63) r α J+ 2 where the quantities with the superscript P HO) corresponds to the PHO with the angular frequency ω, while the same quantities with the superscript HO 3D) corresponds to the HO-3D with the frequency ω ). By applying to the harmonic limit 63), all results and equations obtained in the present paper for PHO lead to the corresponding results and equations for the HO-3D. This fact may be considered as a good check of the correctness of our results. To conclude, our new results for PHO obtained in terms of Klauder-Perelomov coherent states are encouraging, and we believe that these results may contribute to a better understanding of the behaviour and properties of the PHO. Acknowledgement: One of the authors D. P.) wish to acknowledge the Romanian National Council of Scientific Research for financial support by the Grant CNCSIS A647. References [] M. Sage: Chem. Phys ) 43 [2] M. Sage, M. J. Goodisman: Am. J. Phys ) 35 [3] M. Molski: Acta Phys. Polonica A ) 47 [4] D. Popov: J. Phys. A: Math. Gen. 26 2) 6 [5] A. M. Perelomov: Generalized Coherent States and their Applications, Springer, Berlin 986 [6] C. C. Gerry: Phys. Rev. A ) 246 [7] C. Brif, A. Vourdas, A. Mann: J. Phys. A: Math. Gen ) 5873 [8] J. -M. Sixdeniers, K. A. Penson, A. I. Solomon: J. Phys. A: Math. Gen ) 7543 [9] J. R. Klauder, K. A. Penson, J. -M. Sixdeniers: Phys. Rev. A 64 2) 387 [] C. Quesne: Phys. Lett. A 288 2) 24 [] H. Fakhri: Phys. Lett. A 33 23) 243 [2] I. S. Gradshteyn, I. M. Ryzshik: Table of Integrals, Series and Products, 4th ed., in Russian, Fizmatgiz, Moscow 962 [3] A. M. Mathai, R. K. Saxena: Lecture Notes in Mathematics, Vol. 348: Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin-Heidelberg- New York 973 [4] W. Magnus, F. Oberhettinger, R. P. Sony: Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, New York 966 [5] A. I. Solomon: Phys. Lett. A ) 29 [6] L. Mandel, E. Wolf: Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge 995 [7] D. Popov: Acta Phys. Slovaca ) 557

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