Thomas-Fermi approximation for Bose-Einstein condensates in traps

Thomas-Fermi theory for Bose-Einstein condensates in inhomogeneous traps is revisited. The phase-space distribution function of the condensate in the Thomas-Fermi limit ( \→0) is f 0(R,p)}d(m2Hcl) whereHcl is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. Starting from this distribution function the Thomas-Fermi kinetic energy is calculated for any number of particles. Good agreement between the Gross-Pitaevskii and Thomas-Fermi kinetic energies is found even for low and intermediate particle numbersN. Application of this Thomas-Fermi theory to the attractive case and to the calculation of the frequencies of the monopole and quadrupole excitations in the sum rule approach yields conclusive results as well. The difference with the usual Thomas-Fermi approach to the Bose-Einstein condensates ~large-N limit ! is discussed in detail.


I. INTRODUCTION
The recent discovery of the Bose-Einstein condensation of magnetically trapped atoms has spurred a huge amount of theoretical investigations.Most of them are based on the Gross-Pitaevskii equation ͑GPE͒ ͓1͔, which is the mean-field equation for the condensate wave function ͑order parameter͒.The experimental conditions are such that the atomic gas is at very low density and therefore the mean-field approximation gives indeed excellent results ͓2-5͔.Since the number N of atoms involved is generally large, it is natural that also the Thomas-Fermi ͑TF͒ approximation is applied quite extensively.This has the advantage of yielding in most cases explicit analytical results of great physical transparency.The TF limit in the context of Bose-Einstein condensation in traps has been identified with the limit as the particle number N goes to infinity.However, in the case of Fermi statistics, for which the TF approximation has been invented first ͓6͔, by no means the ប→0 limit ͑that is what the TF approximation really is͒ corresponds to the N→ϱ limit.Rather the situation is such that the TF approach becomes eventually exact in the N→ϱ limit but already for N moderate, i.e., corresponding to the masses of midsized atoms, TF gives very reasonable estimates for various quantities such as ground-state energies, etc.It is the purpose of the present paper to show that also for the case of bosons the N→ϱ limit is not equivalent to the ប→0 limit.Rather the correct ប→0 limit does not neglect the kinetic energy term in the GPE (N→ϱ) and then the TF results are greatly improved for rather small mass numbers like, e.g., NϷ200.We deliberately restrict ourselves here to the TF limit of the GPE at zero temperature.Finite temperature as well as more elaborated theories like the Bogoliubov approach may be the subject of future work.
The paper is organized as follows.In Sec.II the theoretical aspects of the TF approximation to the GPE are presented in detail.In Sec.III the numerical results obtained with the TF method are compared with the ones coming from the GPE.The last section is devoted to discussions and an outlook.

II. GROSS-PITAEVSKII EQUATION. THOMAS-FERMI LIMIT
As mentioned in the Introduction, the basic equation for Bose condensed atoms confined by magnetic traps is, in the low-density limit, given by the GPE for the wave function of the condensate where V ex is the external potential which for simplicity we have considered to be a spherical harmonic oscillator ͑for nonspherical geometry see remarks at the end of the paper͒.The coupling constant is given by gϭ4ប 2 a/m with m the atomic mass and a the scattering length.The chemical potential is identical with the lowest eigenvalue of the selfconsistent potential VϭV ex ϩg͉͉ 2 .͑2͒ It is useful to note that Eq. ͑1͒ can also be rewritten as an equation for the density ϭ͉͉ 2 ⌬ ͬ ϩV ex ϩgϪϭ0.͑3͒ In the large N limit one can drop in Eq. ͑1͒ the kinetic energy or, equivalently, in Eq. ͑3͒ the gradients terms of the density.This leads to ϭ(ϪV ex )/g what is known in the literature ͓3-5͔ as the TF solution of the GPE.However, for moderate particle numbers the kinetic energy is not negligible and there is no reason for dropping it in the proper TF limit (ប →ϱ).Also in the attractive case the kinetic energy is of crucial importance to avoid collapse.
As it is well known from the case of the Fermi systems ͓7͔, the TF-approximation is based on the assumption of a slowly varying potential so that its gradients can be neglected to lowest order.We now will show how this can be exploited in order to derive the zeroth order ប expession for the solution of the GPE.To this end let us write Eq. ͑1͒ in the following way: where Hϭ p 2 /2mϩV ex ϩg is the GPE Hamiltonian and ˆϭ͉͉͗͘ ͑5͒ is the density matrix corresponding to the solution of Eq. ͑1͒.We note that lhs of Eq. ͑4͒ is an operator product of two single particle operators ϪH and ˆ.Taking the Wigner transform ͓7͔ of Eq. ͑4͒ we can use the fact the Wigner transform of a product of two single-particle operators A and B can be written as ͓7͔ ͑ AB͒ Wigner ϭA Wigner exp͑iប⌳ J /2͒B Wigner , ͑6͒ where is the Poisson bracket operator and the upper arrows indicate in which direction the differentation should act.Explicitly to order ប 2 expression ͑6͒ is for example given in Ref. ͓8͔.Since we can symmetrize Eq. ͑4͒ we can write with Eq. ͑6͒ where H cl is the classical counterpart of the Hamiltonian in the GPE and f (R,p) is the Wigner transform of the density matrix ˆof Eq. ͑5͒.To lowest order in ប we obtain from Eq. ͑8͒ what is a c-number equation.With x␦(x)ϭ0 we find for the solution of Eq. ͑9͒ where is the level density and thus 0 is correctly normalized to the particle number N. We would like to point out that Eq. ͑10͒ is not a new result but actually textbook knowledge ͑see for example the monograph by Gutzwiller ͓9͔͒.Let us mention again the fact that Eq. ͑10͒ has been derived under the sole assumption that all the gradients of the potential can be neglected, which is the usual hypothesis of TF theory.Clearly, contrary to what has become known as ''TF approximation'' for trapped atomic bosons ͓3-5͔ , expression ͑10͒ does not neglect the kinetic energy.Equation ͑10͒ also is consistent with the usual TF theory for fermions where not only the lowest level is filled but all levels below the Fermi energy.
One therefore simply has to sum all ␦ functions ͑10͒ over the energies of the levels.Converting the discrete sum into an integral and introducing an extra level density, one obtains where the factor 2 stands for the spin-degeneracy.The lhs of Eq. ͑12͒ is recognized as the standard TF expression for the distribution function of a degenerate Fermi gas ͓7͔.
The main point of this paper is to show that Eq. ͑10͒ leads to a more complex expression for the density (R) ϭ͐dpf 0 (R,p)/(2ប) 3 , which however, reduces in the limit of large numbers N to the usual form ϭ(ϪV ex )/g.This will be the subject of the following paragraphs where it will also be demostrated that Eq. ͑10͒ gives very reasonable results for observables already for modest number of particles, strongly improving over the large N limit.
Before coming to these points we would like to contrast our approach to the one published in Ref. shows that on the level of GPE the approach of those authors reduces to the ususal, i.e., to the large N limit.We therefore see that in the context of condensation of atomic atoms in magnetic traps expression ͑10͒ leads to results turning the TF approach to GPE into a quite reliable one even for very moderate numbers of particles.This we will demostrate in Sec.III.

A. Self-consistent solution. Repulsive case "gÌ0…
Let us first consider the solution of the self-consistent problem at the TF level defined by Eq. ͑10͒ for the repulsive case i.e., gϾ0.From Eq. ͑10͒ we obtain for the density: where the local momentum is given by The self-consistency between Eqs. ͑13͒ and ͑14͒ is easy to solve analytically and we obtain where Kϭ 2m

͑16͒
It is to be noted that (R) is defined only within the classical region limited by ϪV ex (R)ϭ0.It is straightforward to expand in the repulsive case gϾ0 for large values of N It is satisfying to see that to leading order one recovers the result corresponding to the total neglect of kinetic energy in Eq. ͑1͒ ͑see Introduction͒.
The normalization is directly determined from Eq. ͑11͒

͑19͒
Explicitly one obtains from this equation

͑20͒
Using in Eq. ͑19͒ the asymptotic expansion ͑17͒ yields which also can directly be derived in expanding Eq. ͑20͒.
From Eq. ͑20͒ or ͑21͒ we can determine the normalization constant c as a function of .From Eq. ͑21͒ we see that to lowest order in 1/N the normalization constant c drops out and thus in this limit the chemical potential is, as usual, determined by the particle number condition.However, as we will see, via the quantization condition depends on c and thus we can consider Eq. ͑20͒ as determining the normalization in any case.

B. Chemical potential and quantization
The semiclassical density matrix ͑10͒ corresponds to a single wave function.In such a case we have two independent constants to be determined: the normalization ''c'' and the chemical potential , i.e., the lowest eigenvalue of the GPE.Only in the case N→ϱ does the chemical potential coincide with the bottom of the ͑self-consistently deter-mined͒ potential well and no extra determination of is necessary.For finite number of particles a requantization of the lowest order semiclassical theory, i.e., TF approximation in this case, is necessary in order to determine ͑semiclassi-cally͒ the finite gap separating the lowest eigenvalue of the GPE from the bottom of the potential.This procedure is similar to the Wentzel-Kramers-Brillouin ͑WKB͒ requantization.The need for a requantization of the TF theory for individual states has been recognized in Ref. ͓10͔ and in fact our procedure of requantization clearly follows what is proposed there.One may think that the equation for the chemical potential ϭdE/dN yields the right quantization but in fact is equivalent to the particle number condition ͑19͒ and therefore of no help.The standard semiclassical quantization procedure is given by the WKB method.However, in order to have a more explicit formula, we here also apply a slightly simpler method, applicable to the lowest state in a singleparticle potential ͓11͔.To this end we calculate the smooth accumulated level density ͑number of states͒ in TF approximation For a spherical harmonic oscillator ͑HO͒ this gives Taking for E in Eq. ͑23͒ the HO eigenvalues with Kϭ2nϩl and inserting Eqs.͑23͒ and ͑24͒ in the lefthand side of Eq. ͑22͒ yields a semiclassical quantization rule, which becomes exact in the three-dimensional ͑3D͒ spherical HO case.It represents an approximate quantization relation for an arbitrary potential where the quantized energies very well reproduce the centroid of major shells.This has been tested numerically on a potential of Woods-Saxon type of nuclear dimensions ͓11͔.It is evident that in the 1D case the same procedure leads to the exact WKB quantization rule.
For the 3D case this modified quantization prescription is slightly less accurate than WKB for the lowest eigenvalue but has the advantage to be easier and to be readily applicable also to the deformed case ͓11͔.In the present problem, the eigenvalue is then determined by where we have used p 0 (R) from Eq. ͑14͒ with Eq. ͑15͒.
To second order we obtain from Eq. ͑25͒ To leading order in the large N limit we obtain from Eqs.
͑20͒ and ͑21͒ and Eqs.͑25͒ and ͑26͒ This completes the solution in the large N limit.
For later comparison let us also give the standard WKB quantization rule ͓12͔, which we want to evaluate to leading order 2 ϭͱ 2m The classical turning points r 1 and r 2 are determined from the solution of the cubic equation With Eqs.͑29͒ and ͑30͒, Eq. ͑28͒ can be solved for .

C. Kinetic energy
One of the main difficulties with the standard N→ϱ TF limit treated in the literature ͑see Introduction͒ consists in the inability to calculate the kinetic energy ͓3͔.In our approach this does not cause any particular problem and one directly obtains

͑31͒
For example to lowest order we obtain from Eqs. ͑14͒ and ͑17͒ what yields for the kinetic energy per particle, using Eq.͑27͒ This simple result must be contrasted with the usual statment that in the N→ϱ TF limit the kinetic energy cannot be evaluated, since it diverges ͓3͔.Indeed one can write the kinetic energy as

͑34͒
In the N→ϱ limit we have and one can readily verify that with Eq. ͑35͒ E kin of Eq. ͑34͒ diverges logarithmically.This result obviously is in contradiction with Eq. ͑33͒ and we shortly want to elucidate the underlying reason.To this end, we first rewrite Eq. ͑34͒ in a different but obviously equivalent way: with f ˜0 given by the Wigner transform of the density matrix corresponding to Eq. ͑35͒:

͑37͒
Since f ˜0 f 0 we argue that Eq. ͑37͒ is not the correct ប →0 limit of the Wigner phase space function because it is not the solution of the ប→0 limit of the Schro ¨dinger Eq. ͑1͒ given by Eq. ͑9͒.Only ͑10͒ is the correct solution of this equation, which yields for large N One checks that Eq. ͑38͒ upon projection onto r-space gives the correct lowest order expression for the density ͓see Eq. ͑17͔͒.Therefore both Wigner functions ͑37͒ and ͑38͒ yield the same leading order density.However, in spite of being a very suggestive nonlocal generalization of the lowest order local density expression, Eq. ͑37͒ has to be rejected on the above given grounds and the divergency of Eq. ͑34͒ is an artifact.On the contrary the lowest order contribution to the kinetic energy is given by Eq. ͑33͒.Via Eqs.͑21͒ and ͑26͒ it is straightforward to calculate the next to leading order correction to Eq. ͑33͒.It should, however, be remembered that 1/N corrections do not go in parallel with powers in ប and that 1/N corrections also can come from ប 2 corrections to Eq. ͑10͒, which involve second-order gradients of the potential.In any case, the Wigner-Kirkwood expansion of the density matrix is an asymptotic expansion, which in no way can recover the nonanalytic behavior in ប of the quantal solution.
In the present problem, the nonanalyticity in ប of the quantal solution entails a nonanalytic behavior in 1/N ͓see Eq. ͑16͒ of Ref. ͓3͔͔ and therefore a WK-expansion can never recover the quantal behavior in 1/N.It is well known that an asymptotic expansion has to be stopped at a point where the difference to the exact solution is minimal.Afterwards the expansion starts to diverge again.In this paper, we do not intend to develop a systematic expansion simultaneously in ប and 1/N.We rather want to give a complete solution to lowest order in ប, i.e., on the TF level.

D. The attractive case "gË0…
Recently, Bose-Einstein condensation has also been observed for the case of negative scattering length ( 11 Li atoms͒ ͓13͔.For gϽ0 the Gross-Pitaevskii approach leads to metastability for particle numbers Nр1400 ͓2͔.For large particle numbers the system collapses.For the attractive case (a Ͻ0) the correct treatment of the kinetic energy is crucial in the TF limit, since otherwise no stability can be achieved.Formally the TF solution for the density is the same as in Eq. ͑15͒ with, however, the sign of the first member reversed Contrary to the repulsive case no large N expansion is possible here.Therefore, the TF solution has to be considered in full.In the next section, comparison with quantal results will be given.
Let us summarize our findings of this section.It has been pointed out that what in the literature is usually called the Thomas-Fermi approximation of the GPE, i.e., its N→ϱ limit, is not equivalent to the limit ប→0 of the GPE.In an analogy with the case of fermions, where the TF approximation corresponds to the limit ប→0 of the quantal solution, we have also elaborated the ''true'' TF approximation in taking properly the limit ប→0.Taking in addition the large N limit makes this coincide with the ususal N→ϱ limit of the GPE.However, the limit ប→0 of the GPE has the advantage that it gives very good estimates of various quantities ͑including kinetic energy͒ already for rather small number of particles (Nϭ200Ϫ300) and it can be applied to the attractive (gϽ0) case, as will be demostrated in the next section.Of course, as a rue, the (បϭ0) TF approximation works better for large number of particles.

III. NUMERICAL RESULTS
In this section, we proceed to a detailed numerical comparison of the semiclassical approximations with the exact quantum mechanical results.Along this section, energies and lengths are given in harmonic oscillator units: ប and a HO ϭͱប/2m, respectively.First, in Table I we present the chemical potential () and the kinetic (e kin ), harmonic oscillator (e HO ) and the self-interaction (e ) energies per particle calculated quantally and in the full Thomas-Fermi approximation ͑15͒, ͑20͒ and ͑25͒ as a function of the number of atoms enclosed in the trap.We have considered Cs atoms ͑as was done in Ref. ͓14͔͒, the frequency of the harmonic oscillator has been chosen to be ϭ20s Ϫ1 and the scattering length to be aϭ3.2ϫ10Ϫ9 m.
In Table II, we present the results for the chemical potential and the kinetic energy per particle number beyond 20 000.In addition to the quantum mechanical and the full Thomas-Fermi results, we also include the results for the large N limit ͑33͒ and those obtained using the WKB quantization rule ͑in the large N limit͒.Notice that in the large N limit, the WKB chemical potential coincides with the TF one and the kinetic energy is given also by Eq. ͑33͒ but with c 0 replaced by the one calculated from K via Eq.͑28͒.Though globally the semiclassical (បϭ0) TF results reported in Tables I and II are in quite satisfactory agreement with the corresponding quantal values, one nevertheless remarks upon some unexpected features.For instance the kinetic energy in the TF approximation is larger than the exact values for small numbers of particles whereas it undershoots the quantum values quite considerably in the large N limit.To clarify this point we will discuss with some detail the results obtained for a very large number of particles which are collected in Table II.
The numbers presented in this table indicate that the values of the chemical potential obtained quantally and using the (បϭ0) TF approach ͑columns 1 and 2 of Table II͒ only reach its asymptotic TF limit ͑column 3͒ for a very large number of particles (NӍ10 8 ).Although the chemical potential reported in columnn 3 is the large N limit ͑27͒ of our (បϭ0) TF theory, it coincides with the corresponding value obtained in the standard (N→ϱ) TF approach.For a relatively large number of particles the value of obtained with the (បϭ0) TF approach lies halfway in between the limiting (N→ϱ) TF and the quantal results.So even for a very large number of particles the (បϭ0) TF approximation shows a clear superiority over its N→ϱ limit.
Concerning the kinetic energy we first want to point out that with the standard N→ϱ TF theory, as it is well known ͓3͔, it is impossible to calculate the kinetic energy.The rea-son for this shortcoming has been discussed in Sec.II C. Since the (បϭ0) TF theory does not neglect the kinetic energy it can be perfectly calculated ͑column 5 of Table II͒.One can also consider the leading term for large N of the (បϭ0) TF expression for the kinetic energy.This is given by Eq. ͑33͒ and the corresponding values are displayed in column 6 of Table II.We see that the (បϭ0) TF kinetic energy reaches this limiting value for a rather moderate number of particles (NӍ10 5 ).We also realize that the (បϭ0) TF kinetic energy, that is derived with our simplified quantization rule ͑25͒ and ͑26͒, coincides with the value obtained using the WKB quantization rule ͑column 7 of Table II͒ for a very large number of particles (NӍ10 7 Ϫ10 8 ).From Table II it is also seen that even for a very large number of particles TABLE I.Chemical potential (), kinetic energy (e kin ), harmonic oscillator energy (e HO ), and selfinteraction energy (e ) per particle in harmonic oscillator units (ប) calculated quantally ͑QM͒ and in the Thomas-Fermi approximation ͑TF͒ for several numbers of atoms in the traps.The frequency of the harmonic oscillator is ϭ20 s Ϫ1 and the scattering length is aϭ3.2ϫ10Ϫ9 m.

N
(  II.Chemical potential () and kinetic energy (e kin ) per particle for large number of atoms in the trap (N).The chemical potential is calculated quantally ͑QM͒, with the full Thomas-Fermi approach ͑TF͒ and with the asymptotic formula for large N (TF N→ϱ ).The kinetic energy is obtained quantally, with the exact TF approximation, with the asymptotic TF for large N and using the WKB quantization in the limit of large number of atoms.The frequency of the harmonic oscillator and the scattering lenght are the same as in Table I.Next, let us compare in Figs. 1 and 2 the densities ͑normalized to unity͒ in TF-approximation and calculated exactly for small ͑200͒ and large ͑200 000͒ particle numbers.As expected, the TF densities almost agree with the quantal ones for very large particle numbers.In view of the still quite reasonable expectation values shown in Table I for Nϭ200, the strong deviation of the TF density from the quantal result is somewhat a surprise.However, one always should remember that the TF solution for the densities is to be understood as a distribution ͓see Eq. ͑10͔͒, which for expectation values of ''slowly varying'' operators can still yield very reasonable values in spite of the fact that the detailed shape may only be a caricature of the exact one.
In Figs. 3 and 4 we show the self-consistent potentials VϭV ex ϩg corresponding to the densities of Figs. 1 and 2.Not astonishingly, V deviates from the harmonic oscillator only slightly for Nϭ200, both quantally and semiclassically.On the contrary for Nϭ200 000, the potential V deviates strongly from V ex being practically a constant equal to up to the classical turning point from where the harmonic oscillator takes over quite abruptly.Again, both the quantal and TF-solution are in close agreement.Figure 4 also teaches us why the TF approximation ͑10͒ to the quantal distribution function is very good for large N.The distribution function corresponds to a wave function with very large energy .In phase space it therefore is very much concentrated around the surface of the hypersphere with radius .
Let us now present the attractive case for the same atoms and external potential with, however, the scattering length aϭϪ1.0ϫ10Ϫ9 m.In Table III we again show chemical potential and kinetic, harmonic oscillator, and self-interaction energies per particle as a function of the particle number in TF and quantal calculation.For small particle numbers (N р1000) the agreement of TF with the quantal case is of FIG.
3. Self-consistent potential ͑in ប units͒ corresponding to a spherical trap containing 200 atoms as a function of the distance (a HO units͒ in the repulsive case calculated from the solution of the GPE ͑solid line͒ and using the TF approach described in the text ͑dashed line͒.similar quality as in the repulsive case.However, for N у1000 the agreement quickly deteriorates, indicating that the whole mean field approximation breaks down.Indeed even quantally the solution of the GPE, Eq. ͑1͒ becomes unstable for NϾ1500 for Cs atoms.
In Figs.5-8 we also show the densities ͑normalized to unity͒ and self-consistent potentials for the particle numbers Nϭ250 and Nϭ1500.We see that, whereas the case N ϭ250 is not dissimilar to the corresponding one with aϾ0, for Nϭ1500 the situation becomes quite unfavorable for the TF approximation.This is for instance manifest in looking at the graph for the densities.In the attractive case TF and quantal solutions diverge with increasing N whereas in the repulsive case they converge.
In Figs. 9 and 10 we plot the kinetic energy density per particle (/N) calculated quantally and in the TF approximation for Nϭ200 000 in the repulsive case and for Nϭ250 in the attractive case.In these figures, the quantal kinetic energy density is given by in order to compare with the TF one according to Ref. ͓7͔.
For large number of particles, when the density profile has a relatively flat region at the interior ͑see Fig. 2͒, the quantal kinetic energy density is peaked at the surface ͓see Eq. ͑40͔͒ whereas the TF one is rather a bulk term ͓see Eq. ͑31͔͒.
Inspite of the rather different form of the quantal and TF kinetic energy densities in this case, the corresponding integrals are in good agreement ͑see Table II͒.This fact points again to the distribution character of the TF-kinetic energy density.However, if the number of particles is small, the quantal and TF kinetic energy density profiles are quite similar.This is due to the fact that the particle density fall off abruptly from Rϭ0 ͑see Fig. 5͒ and consequently its derivatives contribute in all the range of R.
Other quantities where one directly needs the kinetic energy are, e.g., the collective monopole and quadrupole excitations of the condensate.Using the sum rule approach ͓16͔ they are given by: M ϭ 0 (5Ϫe kin /e HO ) 1/2 and D ϭͱ2 0 (1ϩe kin /e HO ) 1/2 .Table IV collects    equation and also from the TF approach.The quantal results obtained by solving the time-dependent Gross-Pitaevskii equation ͓17͔ are also given for comparison.From this table one can see that the quantal results are very well reproduced for any number of atoms in the trap by the TF approximation presented in this paper.Table III also collects the quantal and TF frequencies for the monopole and quadrupole excitations obtained in the sum rule approach for the attractive case.From these results it can be seen that the TF values reproduce well the quantal ones in the region where the semiclassical approximation can be confidently applied (NϽ1000).

IV. DISCUSSION AND OUTLOOK
In the preceding sections we have derived the Thomas-Fermi approximation, i.e., the ប→0 limit of the density matrix corresponding to the wave function of the GPE for the Bose condensate of atoms confined by magnetic traps.We have pointed out some misconceptions on this point that appeared in the past in the literature which for instance prevented the direct calculation of the kinetic energy in the large N limit.On the contrary with our Thomas-Fermi approach the evaluation of the kinetic energy causes no problem and the results are globally in quite satisfactory agreement with the quantal solution of the Gross-Pitaevskii equation.For particle numbers where the kinetic energy represents a sig-  nificant fraction of the total energy the (បϭ0) TF expression yields very satisfying results as compared with the corresponding quantal values.As it has been discussed previously, for large N values, nonanalytical quantal effects dominate the ͑small͒ kinetic energy and consequently this quantity cannot be well reproduced in a pure TF approximation in the large N regime.The semiclassical theory of condensed inhomogeneous Bose gases has also been considered in some detail in Ref. ͓8͔.There, however, the main emphasis was laid on the scattering terms out of the condensated ͑the Bogoliuvov approach͒.On the other hand no attempt was made to improve the kinetic energy in the pure ប→0 limit, i.e., in Thomas-Fermi approximation.However some estimates of the kinetic energy coming from gradients, i.e., ប corrections are given in the large N limit.On the contrary in the present work no quantal corrections to the lowest order Thomas-Fermi result have been worked out.Finally, we want to make some comments why for very large N the (ប ϭ0) TF values for the kinetic energy are very different ͑smaller͒ than the corresponding quantal values ͑see Table III͒.This simply means that the បϭ0 part of the kinetic energy goes to zero much faster for large N than for the full quantal solution.In other words, for very large N the kinetic energy is dominated by the quantal corrections.Of course, finally for N→ϱ, the latter ones also drop to zero.The above discussion shows again that the semiclassical approximations are a powerful tool but not devoid of subtleties and pitfalls.As a matter of fact also in this paper we, for simplicity, avoided developing the full complexity of the theory.One major simplification resides in the fact that we assume a spherical trap.This results in an isotropic momentum distribution f 0 ϰ␦(ϪH c ) where H c ϭp 2 /2mϩV is the classical Hamiltonian.Deforming the trap leads to a non-trivial modification of the theory, since squeezing the condensate wave function in one direction and relaxing it in the other entails in turn a deformation of the momentum distribution which is opposite to the spatial one, i.e., momenta are strongest in the squeezed direction and lowest in the long direction of the deformation ͓15͔.Our TF approach has also been useful for the evaluation of collective excitations of droplets of small or intermediate sizes.For this purpose, we have used the sum rule approach where the kinetic energy enters and for which we have used the TF values.In the deformed case the momentum becomes anisotropic.This has also been determined experimentally.The detailed determination of the anisotropy of the momenta ͑which may be position dependent͒ is theoretically not a completely trivial task in the general case and we will elaborate on this in future work.In the present case, however, there exists an evident first guess of the momentum deformation that results from a scaling argument of the harmonic oscillator coordinates.Assuming a prolate quadrupole deformation in the z direction we have to replace the classical Hamiltonian in Eq. ͑10͒ by

͑41͒
where the ratios z / 0 and Ќ / 0 are the frequency rela- tions in z and x,y with respect to the spherical case ( 0 ).From Eq. ͑41͒ one easily calculates the so-called aspect ratio in the TF-approximation a result that has been given previously ͓3͔.One other important consequence of the momentum deformation is that with Eq. ͑41͒ the moment of inertia of the condensate becomes equal to the irrotational flow value ͓16͔.On the contrary using Eq.͑10͒ with the isotropic momentum distribution the rigid momentum of inertia results.Consequently, the deformed case needs more detailed studies, which we reserve for future work.It is also evident that the present TF approach can be extended to finite temperature and to the Bogoliubov theory.
Another interesting subject of a more formal aspect is the evaluation of the ប-correction to the present lowest order theory.In principle this can easily be performed in posing in Eq. ͑3͒ ϭ 0 ϩប 2 2 and ϭ 0 ϩប 2 2 and properly sorting out different powers in ប.However, the proper elimination of divergencies and handling the normalization ͑19͒ and quantization ͑25͒ are slightly subtle problems.Investigations on the above mentioned directions are in progress.

(
NӍ10 8 ) the kinetic energy (បϭ0) TF values are still different ͑smaller͒ than the corresponding quantal results.This is due to the fact that for very large N the quantal kinetic energy is dominated by the quantal corrections.As was shown by Dalfovo, Pitaevskii, and Stringari ͓3͔, these corrections are nonanalytical ͑logarithmic͒ in ប.Consequently, they cannot be accounted for by a pure TF approach and a partial resummation of all orders in ប would be necessary to reproduce the quantal kinetic energy.Of course, when finally N→ϱ these quantal corrections drop to zero and one recovers the result of the standard TF theory where the kinetic energy vanishes.

FIG. 1 .
FIG. 1. Density ͑normalized to unity͒ of 200 atoms in a spherical trap ͑in a HO Ϫ3 units͒ as a function of the distance ͑in a HO units͒ in the repulsive case calculated from the solution of the GPE ͑solid line͒ and using the TF approach described in the text ͑dashed line͒.
FIG. 4. The same as Fig. 3 but with 200 000 atoms in the trap.

FIG. 5 .
FIG. 5.The same as Fig. 1 but with 250 atoms in the trap in the attractive case.

FIG. 6 .
FIG.6.The same as Fig.1but with 1500 atoms in the trap in the attractive case.

FIG. 8 .
FIG. 8.The same as Fig. 3 but with 1500 atoms in the trap in the attractive case.
͓8͔ by Timmermans et al.The essential point in Ref. ͓8͔ is the semiclassical treatment of scattering processes out of the condensate in the Bogoliubov approach.Though this is certainly a worthwhile generalization of the Gross-Pitaevskii approach to be considered semiclassically, a sensible comparison with our theory can only be performed once the Bogoliubov terms in Ref. ͓8͔ e kin (QM) e kin (TF) e kin (TF N→ϱ ) e kin (WKB N→ϱ )

TABLE III .
The same as in TableIbut with a scattering length aϭϪ1.0ϫ10Ϫ9 m.The frequencies corresponding to the monopole ( M ) and quadrupole ( D ) calculated in the sum rule approach are also given.

TABLE IV .
Monopole (Lϭ0)and quadrupole (Lϭ2) frequencies calculated from the sum rule estimate using the Gross-Pitaevskii and Thomas-Fermi results for the kinetic energy.The results obtained by solving the time-dependent Gross-Pitaevskii equation are also given ͑TDGP label͒.
FIG. 10.The same as Fig. 9 but with 250 atoms in the trap in the attractive case.