ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS

Transcription

1 ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS L. MICU Hoia Hulubei National Institute fo Physics and Nuclea Engineeing, P.O. Box MG-6, RO-0775 Buchaest-Maguele, Romania, (Received August 30, 008) Following the epesentation of a two-body system in classical mechanics, we build up a quantum pictue which is fee of spuious effects and etains the intinsic featues of the intenal bodies. In the coodinate space the system is epesented by the eal paticles, individually bound to a cente of foces which in a cetain limit coincides with the cente of mass and the wave function wites as poduct of the individual wave functions with coelated aguments. Key wods: bound systems, cente of mass states, constained systems.. THE CENTER OF MASS PROBLEM The usual appoaches to the two body poblem in classical and quantum mechanics (see, fo instance, [, ] pp ) stat with the eplacement of the paticle position vectos, and momenta p, p by the cente of mass vaiables m m R= + m+ m m+ m P = p+ p () and the elative ones = m m p = p p m m m m + + whee m and m ae the paticle masses. If the inteaction potential depends only on the elative distance, the eplacement esults in the sepaation of the Hamiltonian in two independent pats: () whee H = H + H (3) int int CM = p + V() m (4) Rom. Joun. Phys., Vol. 53, Nos. 9 0, P. 7 76, Buchaest, 008

2 7 L. Micu and HCM = M P (5) which may be seen as the Hamiltonians of two fictious paticles: a paticle with the educed mass m = mm/ ( m+ m) in the extenal potential V() and a fee paticle with the mass M = m+ m having the momentum P and the position vecto R of the cente of mass. Following the usual quantization pocedue, the wave function of the two-body system in stationay state is the poduct of the eigenfunctions of H int and H CM and the enegy is the sum of the coesponding eigenvalues. As it is well known, a poblem aises in connection with the eigenfunctions of H CM, since the attempt to localize the cente of mass by imposing adequate bounday conditions violates the invaiance at tanslations and geneates an infinite seies of spuious states with abitay high enegy levels. These states have no classical analog and ae not the esult of a dynamical mechanism. They ae spuious and hence have to be sepaated out fom the physical solutions. The poblem has been extensively studied in nuclea many body models [3], whee the achievement of a tanslational invaiant, independent paticle pictue equies a clea sepaation of the cente of mass enegy and states fom the physical ones. The solution poposed by Lipkin, de Shalit and Talmi [4] is to intoduce in the intenal Hamiltonian the contibution of the edundant (o supefluous) coodinates which assue the independent teatment of nucleons. These ones geneate new spuious states and a new tem in the intenal Hamiltonian which compensates the kinetic enegy of the cente of mass in the appoximation of equal masses. Finally its contibution has to be subtacted fom E int and the spuious solutions have to be sepaated out. The sepaation equies special technics which, excepting the case of the hamonic oscillato, is athe had to be done (see, e.g., Ref. [3] (pa..3), [4 6]). The existence of spuious states seem to confim the deeply ooted opinion that the cente of mass poblem eflects the uncetainty elations and is chaacteistic to quantum mechanics. Contay to this belief, we show that the poblem may be solved by popely defining the classical system which has to be quantized. We demonstate that the spuious effects can be eliminated if the est condition of the bound system is explicitly witten as a constaint on the position of the cente of mass. Specifically, we conside the bound system in the est fame whee the cente of mass position is fixed, i.e. R() t = R0 and follow Diac s Hamilton method of teating the constained systems (see [7], [8]). Accoding to Diac, wheneve the naive Hamiltonian (in ou case the naive Hamiltonian is H CM ) is incompatible with the constaints, it has to be

3 3 On the two-body poblem in quantum mechanics 73 eplaced by a ight one. To find it, one has to build up the whole algeba geneated with the aid of Poisson backets fom the naive Hamiltonian and the initial constaints and look fo its cente, i.e. fo the elements whose Poisson backets with evey element of the algeba ae zeo. If the algeba is closed, thee is at least an element with this popety. This is eithe a Casimi invaiant of the algeba o anothe element of its cente and epesents the ight Hamiltonian which guaantees the pesevation of constaints, inclusively of the naive Hamiltonian. In the cente of mass poblem the algeba is geneated by H CM, the constaint R R0 = 0 and the esults of the Poisson backets { H } CM, R = P (6) M and { R, P} =δ. (7) i j ij Fom the above it esults that this algeba is closed and, besides the vaiables R and P, it contains also the identity. The latte and its multiples fom the cente of this algeba and ae the only elements whose Poisson backets with evey element of the algeba ae zeo. Accoding to ou pesciption, one of these multiples having the dimension of enegy is the ight cente of mass Hamiltonian. It esults that H CM has to be eplaced by a constant supposed to be the sum of masses and hence the ight Hamiltonian of a two body system at est eads H m m = + + p + V () m. Obviously, the Poisson backets of the Hamiltonian with all the elements of the above algeba ae zeo. Closing the discussion on the cente of mass poblem, we emak that, unlike the pevious appoaches to the bound state poblem, the cente of mass vaiables ae absent fom the Hamiltonian. This is explicitly due to the est condition R R0 = 0 and has impotant consequences in the quantization pocess. (8). THE PROBLEM OF THE INTERNAL MOTION Accoding to the esults quoted in the peceding section, the dynamics of the bound system at est is uled by and the solution of the equations of motion expesses in tems of the elative vecto which has a fixed point: the cente of mass.

4 74 L. Micu 4 Intoducing the notations μ = m/ ( m+ m), μ = m/ ( m+ m) and denoting by and the position vectos of the intenal paticles with espect to the cente of mass we find: μ +μ = 0 (9) = (0) =μ = ( μ) () = μ = ( μ ) () and daw the following epesentation of the two-body system: the paticles ae at the ends of a segment of vaiable length, (t), otating aound the cente of mass which divides it in two pats, (t) and (t), whose length is invese popotional to the masses m and m. On the othe hand, we notice that consideing the intenal paticles as distinct physical systems and using the pais of individual canonical conjugate vaiables, p and, p whee and satisfy () and (), the Hamiltonian wites as sum of two individual Hamiltonians whee = H I + H I (3) H = m + ( μ ) H int p = p ( ) = μ ; = ( μ) ( μ ) p V m = m + + ( ) (4) H = m + ( μ ) H int p = p ( ) = μ ; = ( μ) ( μ ) = m + p + V( ) m and I and I ae the identity opeatos on the phase space of the fist and second paticle espectively. In the new fom, appeas as the Hamiltonian of a system made of two paticles with the masses m and m individually bound to the same cente of foces which coincides with the cente of mass ( = = = 0) though the extenal potentials V( ) = ( μ ) V( ) = ( μ ) and V ( ) = ( μ ) V ( ) = ( μ). Fo consistency, the position vectos and with espect to the cente of foces have to satisfy () and (). This guaantees that the eal paticles have collinea instantaneous position vectos with espect to the cente of mass and move on simila tajectoies. (5)

5 5 On the two-body poblem in quantum mechanics 75 Moeove, it can be easily checked that, thanks to the popotionality elations () and (), the sum of the kinetic and potential enegy of the fist and second paticle in the potentials V ( ) and V ( ) espectively is equal to the total enegy of the paticle with the educed mass in the potential V(). Also, the angula momentum of the paticle with the educed mass L = p is equal to L+ L whee L = p and L = p ae the angula momenta of the intenal paticles with espect to the cente of mass. This is the classical epesentation of a two-body system which is subject to the quantization pocedue. Fist we emak that, due to the absence of cente of mass vaiables fom the Hamiltonian, thee ae no cente of mass opeatos and hence no spuious cente of mass states. Accoding to the quantization pocedue, the individual vaiables, p and, p in H and H ae eplaced by the canonical conjugated opeatos i, and, i and the eal paticles ae assigned the individual wave functions Ψ ( ν ) and Ψ ( ν ) which satisfy the time-independent Schoedinge equations: ( μ ) m m + V ( ) E ψ ( ) = 0 ( μ ) m ν m + V ( ) E ψ ( ) = 0. ν (6) (7) Then, assuming that, just as in the classical epesentation, the coodinates of the eal paticles ae elated by () and () the wave function of the bound system wites as follows: Ψ ( ) =Ψ ( ) Ψ ( ). (8) ν = ( μ ) ν = ( μ ) In the geneal case whee and ae the position vectos with espect to an abitay point in the est fame, Ψ( ) wites as 3 ( ) ( (3) Ψ = d ρψν ρ) ψν ( ρ) δ ( ρ CM ) (9) whee CM =μ +μ. Futhe, noticing that the coodinates of the intenal paticles as well as H and H ae elated by scale tansfomations, it esults that Ψ ( ) and Ψ ( ) ν ν ae also elated by scale tansfomations, just as the paticle tajectoies in

6 76 L. Micu 6 classical mechanics. In quantum mechanics this means that the main, the obital and the magnetic quantum numbes of the individual and of the elative wave functions satisfy the following elations n = n ; l = l ; m = m. Closing, we notice that (8) satisfies all the equiements imposed to the bound state wave function in the independent paticle pictue: it depends only on the elative vecto and hence it is invaiant at tanslations; in the same time, it is in ageement with the classical pictue of a bound system and, as poduct of the individual wave functions, it etains the intinsic popeties of the intenal paticles. Acknowledgements. The autho acknowledges the financial suppot fom the Romanian Ministy of Education and Reseach unde the contact CEEX 05-D-49/ REFERENCES. H. Goldstein, Mechanics Adison-Wesley Publishing Company Inc. Reading, Mass., 96.. A. Messiah, Mécanique Quantique, IX- IX-3, Dunod, Pais, P. Ring, P. Schuch, The Nuclea Many-Body Poblem, Spinge Velag, N. Y. 980, pp H. J. Lipkin, A. de Shalit, I. Talmi, Nuovo Cim., 773 (959). 5. H. J. Lipkin, Phys. Rev. 0, 395 (958). 6. R. E. Peiels, D. J. Thouless, Nucl. Phys. 38, 4 (96). 7. P. A. M. Diac, Lectues on Quantum Mechanics, Dove Publications, N.Y P. Diþã, Phys. Rev. A (997); P. Diþã Romanian Joun. Phys. 47, 40 (00).