COHERENT STATE ANALYSIS OF ELLIOTT'S MODEL

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COHERENT STATE ANALYSIS OF ELLIOTT’S MODEL

J. Broeckhove, E. Kesteloot, P. van Lewen

To cite this version:

J. Broeckhove, E. Kesteloot, P. van Lewen. COHERENT STATE ANALYSIS OF ELLIOTT’S MODEL. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-347-C2-351.

�10.1051/jphyscol:1987253�. �jpa-00226522�

JOURNAL D E PHYSIQUE

Colloque C2, suppl6ment au n o 6 , Tome 48, juin 1987

COHERENT S T A T E ANALYSIS OF ELLIOTT'S MODEL

J. BROECKHOVE, E. KESTELOOT and P. VAN LEWEN

Rijksuniversitair Centrum Antwerpen, Dienst Theoretische en Wiskundige Natuurkunde, Groenenborgerlaan 171.

B-2020 Antwerpen, Belgium

Abstract: Anatysis of the time evolution of the coherent states for a two-dimensional version of Elliott's rotational model reveals a uniform rotation of the nuclear quacfrupole.

1. INTRODUCTION

The explanation of nuclear collective motion in terms of a miaoscopic description of the many- particle system is a longstanding problem. A remarkable step forward towards its solution has been taken by J. P. Elliott [I] with the introduction of the SU(3) model for nuclear rotations. Elliott has shown that the SU(3) symmetry of the oscillator mean field defines a natural subspace of the shell model in which a semi-realistic nuclear Hamiltonian has a rotational spectrum. However, the model is a static one and the conclusion about rotational properties is based merely on considerations about stationary states. Questions about the nature of rotational motion, such as " what is rotating ? " and " how is it rotating ? " are not answered. Therefore it appears that part of the physical insight is still missing.

In this paper we present a timedependent counterpart of Elliott's model. We investigate the propagation of wavepackets associated with Elliott's subspace. These wavepackets are the Perelomov coherent states for the unitary goup, the time evolution of which is evaluated through the Time Dependent Variational h-inciple (TDVP) [2,3]. The preliminary study we report on in this paper, concerns the two-dimensional version of the model (41.

2. COHERENT STATES FOR THE ELLIOT MODEL

Elliott's model deals with k particles in a degenerate oscillator shell of the isotropic harmonic oscillator potential interacting through a residual two-body force. The single-particle states can be generated with the usual oscillator creation and annihilation operators. For a two-dimensional system one then has the k-particle operatars (j being the particle index) that shift quanta within a shell

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987253

JOURNAL DE PHYSIQUE

with p = x , y

The four operators form an U(2) tensor of rank two and constitute a basis for the u(2) algebra. One of the tensor components is the oscillator Hamiltonian H,, another is proportional to the angular momentum L, (perpendicular to the x-y plane) and two are components of the (traceless) Elliott's quadupole

H, = Cxx + Cw Lz = - i ( c X Y - c Y X j

(21

The latter three define the su(2j algebra. It is also convenient to consider, instead of the above

"physical" basis, the "mathematical" Cartan basis of su(2) with a step up, step down and weight operator

A = Cxy -Cyx , "1, = Cyx , 4 = Cx,

0 (3)

The suitable model space for the nuclear spectrum is according to Elliott's model the carrier space of an irreducible representation of the unitary goup. Such a space can be generated from an extremal state (lowest or highest weight state) by the repeated action of the step up operator. Within that space one also has the coherent states of the representation [2]. They form a manifold of states and are obtained by the action of a general goup operator on the ememal state

[ z > = exp(z11+)10> with z = tg(812) expt-i@) (41

The manifold is parametrized by the complex numbers z. The expectation values

< z l Q , l z > = i c o s e , ~ z l Q 2 1 z > = i c o s $ s i n Q , <zlL#!lz>=jsin$sinO ( 5 ) indicate that each coherent state (c.s.) may be identified with a point on a sphere of radius j , where i is the labet of the irreducible representation that the C.S. belong to (see figure 1).

3. ROTATIONAL MOTION FOR THE COHERENT STATE

The Time Dependent Variational Principle (TDVP) leads to the equations of motion for the c.s parameters [2,3]. These have the Harniltonian form

where the Hamilton function is equal to the expectation value of the quantum Hamilton operator. The Poisson brackets are given by

-1 af ag ag af I af dg ag af ( f , g } = - [

] = - I ---

h j sine dg dB dg d0 hl a@ d(cos8j a@ a(cos0)

I

The time propagation of the quantum coherent states as prescribed by this formalism can be interpreted in terms of a "classical' or "large quantum number" limit, the label j of the representation being the relevant quantum number for this system [5].A typical (8,Q) orbit is shown in figure 2 for the case of a Q'Q residual interaction.The time evolution of the C.S. wave function may be represented by the motion of the point in the phase space i.e the sphere mentioned earlier. The motion proceeds along the circle that is the intersection between the sphere and the plane determined by Q3 = L,f2 = constant ( see figure

Fiaure 1 : The phase space associated with the coherent state parameters.

This is due to the angular mometum conservation law. For a residual interaction that commutes with angular momentum [ V. L, ] = [ H, L, ] = 0 it follows [2] that also { H, L, ) = 0 and hence the classical angular momentum is a conserved quantity, as n of course the energy for evidently one has { H, H I = 0. As must be the case in a two-dimensional phase space, the former yields the same constraint on the traiectory as the latter.

in order to adbess the main feature of our investigation and to bring out mare clearly the nature of of the time propagation of the C.S. we introduce the variables

/-TT ^Q2

4 =

.;

Q,

that define the magnitude and orientation of the quackupole respectively. In fact 2n is the angle over which the coherent state must be rotated in order to bring its quadrupole moments in diagonal form i.e.

C2-350 J O U R N A L D E PHYSIQUE

The time evolution of these variables follows from TDVP equations i.e dqldt ={q, HI and dclldt = {cu.H}.

They must be evaluated for the particular residual interaction that determines H, and for an Elliott's quadupolequadupole interaction

0 ₀

I I I

1 2 3 4 5

F i w e 2. The time evolution of 9 and 4 during five periods of the motion.for j = 1 and LZ = 1. The angles ( 0 solid line, 4 dashed line ) are expressed in radians, the time in periods.

This yields {q, H} = 0 and {cu, H ) = constant, implying a constant magnitude for the quabupole moment and a uniform rotation of the many-body system. The rate of rotation depends on the initial conditions i.e. on the initial angular momentum through the relation

The value for the moment of inertia lM fixed by the relation between o and L is consistent with the value derived from H and L,. In the "large quantumnumbw" limit it is equal to the exact quantum value rl.

4. CONCLUSlONS

In this work we have studied rotational motion in the framework of Elliott's model in two dimensions.

We have applied a coherent state analysis to the model to investigate the time evolution of the

magnitude and orientation of the nuclear quadupole: the magnitude is constant and the orientation indicates uniform rotation of the system. From this we conclude that in a well defined limiting sense Elliott's modelspace supports quantum states with a "rotational motion". The rotational nature of the model goes beyond the spacing of the energy levels and can also be exhibited in its explicit time- evolution.

The TDVP time evolution being an approximation to the exact Schroedinger time evolution of the system. it remains to be investigated how the c.s. loses its coherence property if it is allowed to spread throughout the whole subspace of the irreducible representation.

We conclude that it would be worthwile to pursue the study of rotational motion made in this work and apply it to Elliott's model in three djmensions ; this is now in progess.

Acknowledaement

E. Kesteloot would like to acknowledge support from the llKW Belgium.

References

[ I ] ^J. P. Elliott, Roc. Roy. Soc, A245(1958)128 and 562 ; J. P. Elliott and M. Harvey. Roc. Roy. Soc.

A272(1963)557 ; J. P. Elliott andC. E. Wilsdon, Proc. Roy. Soc. =(1968)509

fl

[4] H. Pilkuhn, Nucl. Phys. g(1959)269

[3] ^J. &oeckhove and P. Van Leuven, Proc. Meeting on Phase Space Approach to Nuclear Dynamics, ICTP Trieste, Italy 1985, World Scientific Publisching Co., 1986, Singapore, M. di Toro et al. eds.,p595

[5] L. G. Yaffe, Physics Today, August 1983,p50