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Phenomenological Quantization Scheme in a Nonlinear Schrödinger Equation

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    VOLUME 57 27 OCTOBER 1986 NUMBER 17

    Phenomenological Quantization Scheme in a Nonlinear Schrodinger Equation

    J. N. Maki and T. Kodama Centro Brasileiro de Pesquisas Flsicas-CNPq, 22290-Rio de Janeiro, Rio de Janeiro, Brazil

    (Received 3 April 1986)

    A phenomenological quantization scheme is carried out for the collision process of identical soli-tons of a nonlinear Schrodinger equation. An effective potential is determined from the classical expression for the time delay. The corresponding effective S matrix reproduces the exact S matrix of the quantized nonlinear field as well as the ground-state binding energy for large TV, but it is not adequate to describe the scattering process in the low-energy region.

    PACS numbers: 03.65.Nk

    In this Letter we discuss one possible way of associ-ating the dynamics of classical soliton solutions to col-lective degrees of freedom of the many-body system whose properties are described by a quantized non-linear field. We show that, in contrast to most the-ories for collective motion, the high-energy behavior of the scattering amplitude between two systems of quantum bound states is satisfactorily described in this manner.

    It is well known that there are some classes of non-linear wave equations which are exactly solvable.1

    Although none of them corresponds to the realistic three-dimensional case, the study of these systems may well provide theoretical foundations of the quanti-zation of nonlinear fields. In particular, the one-

    dimensional nonlinear Schrodinger equation (NLSE)


    * 87* + 2 ^ 8* ^ + 2hH2J

    where N is the total number of particles in the system and m is the mass of the particle.

    Equation (1) exhibits a family of soliton solutions.3

    For example, the one-soliton solution is given by

    + i U r ) - - a exp[2iHx-x0)-(2if[/m)(t2--n2)t]

    U2m)1'2 cosh{27j[x-jc


    where r), , x0, and x0 are arbitrary constants. It is easy to see that the two-soliton solution, Eq. (4), splits into two one-soliton solutions in the limit of \t\ * oo. Equation (4) represents the scattering processes of two identical solitary waves in their center-of-mass system with a relative kinetic energy given by4

    = 1 6 l V / m 2 e 2 . (5)

    In this case, the time delay caused by the interaction between the two solitary waves is calculated as


    = - (&/m4)UN)-l\n(l + N2/a2), (6)

    where the new variables a and iVare defined by N'- 4ti2r)/me2, (j) a = 20T2/mAf4)1/2. (8)

    The variable iV, for instance, is related to the normali-zation integral of the system

    2N-C \^2.s\2dx, (9)

    so that it may be regarded as the number of particles contained in each solitary wave packet in the asymptot-ic region.

    In the quantized version, the above two-soliton scattering can be viewed as the collision of bound states of two AT-particle systems with translational kinetic energy K The exact quantum-mechanical S matrix can be explicitly calculated by the method first introduced by McGuire5 and later developed by Yang as


    S2N~0Ut((N' + l'f . . . ,2iV')(l',2\ . . . ,iV')|(l,2, . . .,N)(N+lt . . . ,2N))'


    -n \ia-j 1 ia+J, 2

    ia-N ia + N

    _ ia-N\ ia + N\

    r(ia)r(N-ia) r(-ia)r(N+ia)


    It has been shown that for large JV values the quantum many-body expressions approach the classical ones.9 For example, the amplitude of the Bethe A nsatz for the N-particle bound-state wave function converges to the classical one-soliton solution. In addition, the time delay calculated from the S-matrix equation (10) leads to

    A f = 7 ^ l n 5 - = - * JN 2 +4[-Re4>Oa)+ReiHiV-/a)] j f

    4K3 1

    me 4 aN

    In 1 + *WH*n- (ID which coincides with Eq. (6) to the leading order in the parameters a and N. Thus the classical solution of soliton scattering can be regarded as the collective motion of two iV-particle bound states. Yoon and Negele8 discussed this problem in the framework of the time-dependent Hartree approximation.

    In phenomenological treatments of the collective motion of quantum-mechanical many-body systems, there is often introduced an effective Lagrangean (or Hamiltonian) for selected collective variables. In such treatments the effective Hamiltonian is obtained by starting from the expectation value of the original Hamiltonian of the system:

    Schrodinger equation

    XW(Q,tySt-HdB(Q9t)9 (13)

    HMP)-{*QtP\H\^QtP), (12)

    where \^Q$P) is the state vector of the system parametrized by the suitably chosen collective canoni-cal variables Q and P.

    Once the effective Hamiltonian is defined as a func-tion of Q and P, the quantum mechanics of the collec-tive motion is expected to be properly described by the


  • VOLUME 57, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 27 OCTOBER 1986

    tion" procedure. In particular, it may happen that the quantum dynamics described by such a requantized Hamiltonian has nothing to do with the real quantum-mechanical properties of the original system.

    The NLSE is most appropriate to study this prob-lem, since its exact solution is known. We may then carry out the phenomenological quantization pro-cedure on the NLSE and compare the result with the exact one.

    In order to find the effective Hamiltonian which describes the collective motion (in our case the rela-tive motion between centers of mass of the two soli-tons), we start with the expression of the time delay, Eq. (6). If one writes the classical Hamiltonian as

    where /x = y M = } m i V


    is the reduced mass, x = xi X2, and U^ is the effective potential, the time delay is expressed as

    | l /2

    / : At-M 1 1

    (E-Uerr)l/2 1 / 2

    dx. (15)

    Expanding Eqs. (6) and (15) with respect to l/E and comparing all coefficients, we find7 that the local po-tential

    f/eff(x) - -r)N22/cosh2(>nx) (16)

    r(to)r(fa + r)r(\-to)r(x-fa + 7)co8 eff r ( - ia)T{ - ia + | ) r ( X + ia)T{\ + to + 1 >

    Noting that

    A - J V + I + O U / A O ,

    and with the help of the formula11

    r ( Z + a ) = Z a r ( Z ) [ l + a ( a - l ) / 2 Z + 0 ( l / Z 2 ) ] ,

    is a solution of the integral equations (6) and (15), although we failed to prove its uniqueness.

    According to the phenomenological quantization scheme, we must now consider the following Schrodinger equation:

    ihf-Uxj) at

    2fx a x 2 (x,t)+UQrr(x)(x,t). (17)

    We would now investigate how well this equation describes the quantum-mechanical aspects of the col-lective variable x, which corresponds to the original many-body system, Eq. (2). For this purpose, it is convenient to calculate the S matrix corresponding to Eq. (17). This equation is known to be exactly solv-able for the potential of Eq. (16). In this case we get10

    Sef f T ( 2 t o ) r ( 2 X - 2 t o ) cosTr(X-to)

    T ( - 2 t o ) r ( 2 X + 2to) cos7r(X + to)' ( 1 8 )

    where X = f + f ( l + 16iV2)1/2, with a and TV given by Eqs. (7) and (8), respectively.

    To compare this result with Eq. (10), we first make use of the formula11

    T(2Z) = (27 r ) - 1 / 2 2 2 Z ^ 1 / 2 r (Z ) r (Z4 - \ ) , (19)

    and rewrite Eq. (18) as

    >COSTT(A to)

    >cos7r(A + to) (20)


    we find

    -(N-ia) N+ ia

    r(to)r(iV-to) r(-to)r(;V+to) (22)

    Thus we see that our SefT tends to the exact S matrix S2N of Eq. (10) for large a and N. However, this is not yet quite satisfactory, since such information from Seff in this domain of large a and iVis exactly the same as that furnished by the classical quantity At, Eq. (6).

    To examine quantum effects reflected in Eq. (18), we look for bound-state poles of Seff. These bound-state poles come from positive zeros of cos7r(X + to), i.e.,

    to/ = X / - / ^ 0 . (23)

    The ground-state / = 0 energy is then given by

    Zigs (X -)2mN4/4ti2

    - - (me4N3/4?i2){1 + Oil/N)},

    which should be compared to the exact value

    ^exact=-(^ 4 /4^ 2 )A^ 3 .



    It is interesting to observe that for J V 1 , our phenomenological quantization scheme gives an excel-


  • VOLUME 57, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 27 OCTOBER 1986

    lent estimate of binding energy of the original quantum-mechanical 2iV-body system. This fact and Eq. (22) seem to support the adequacy of the phenomenological quantization proposed here.

    However, the weak point in claiming that Eq. (17) correctly describes the quantum-mechanical property of the collective motion is that our 5efT exhibits too many bound-state poles,10 whereas there is only one for the exact expression. These extra poles spoil the behavior of the scattering amplitude in the low-energy region.12

    We summarize our present investigation as follows: (1) We calculated the effective potential between two identical solitons of the NLSE starting from the classi-cal expression for time delay. (2) Using this effective potential, we studied the quantum mechanics of two solitons, according to the phenomenological quantiza-tion scheme. (3) We have shown that for large N and a values (large Eor weak coupling) our effective S ma-trix reproduces the exact S matrix of the original 2JV system. (4) The ground-state binding energy is also correctly estimated for large N. (5) At low energies the scattering amplitude does not behave correctly even for large N values.

    Most theories of collective motion in nuclear physics are concerned with establishing a Hamiltonian which reproduces low-energy behavior. The model presented here reproduces high-energy behavior. Because of t