Wu Tai Tsun

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CERN-TH/96-250Bose-Einstein Condensation in an External Potential IITai Tsun WuGordon McKay Laboratory, Harvard UniversityCambridge, Massachusetts 02138-2901, U.S.A.andTheoretical Physics Division, CERNCH-1211 Geneva 23, SwitzerlandAbstractThe theory of the preceding Paper I is applied to the Bose-Einstein condensationin a trap for a dilute gas with short-range pairwise repulsive interaction. Sincethe trap is of macroscopic dimensions, the resulting external potential is oftenslowly varying. In these cases, the partial dierential equation for the condensatewave function and the integro-dierential equation for the pair excitation aresolved approximately.CERN-TH/96-250September 1996Work supported in part by the United States Department of Energy under Grant No. DE-FG02-84ER40158.I. INTRODUCTIONIn the beautiful pioneering experimental observation of Bose-Einstein condensation [1,2] indilute atomic gases [3{5], the atoms are spin-polarized and conned by the magnetic trap. Fortheoretical understanding, the spin of the bosons is not of primary concern, and hence the eectof the magnetic trap can be taken into account by the introduction of an external potential.On the theoretical side, the systematic eort begun forty years ago to study the properties ofa dilute Bose gas with short-range repulsion [6{15] has been generalized to include an externalpotential in the preceding paper, to be referred to as Paper I [16]. It is the purpose of thepresent paper to apply the results of Paper I to a trap.Since the bosons are kept inside the trap, strictly speaking thermodynamics does not apply.However, since the number of atoms in the condensate ranges from the thousands to the hundredthousands or even millions [3{5], it is not unreasonable to use approximations such as (I.2.19)and (I.3.5).II. ATOMIC TRAPOne immediate consequence of the trapping potential is that the volume Ω no longer has awell-dened meaning. It is indeed possible to assign a somewhat arbitrary value to Ω, and thenal results will be independent of this arbitrarily assigned value. A better procedure is to useN , the total number of particles, as the normalization instead of Ω. In terms of the condensatewave function of Paper I, dene~(r) = (N=Ω)1=2(r); (2.1)so that (I.2.10) becomesN−1Zdr j~(r)j2 = 1: (2.2)Similar to the  of (I.2.18), dene~ = N−1Zdr j~(r)j4; (2.3)which is equal to Ω=N .In terms of this ~(r), the basic equations (I.2.22), (I.3.23), and (I.4.6) with (I.4.7) for ~,K0, and ’ are, respectively,[−r2 + Ve(r) + 8aj~(r)j2 − 4a~ − E]~(r) = 0; (2.4)−r2K0(r; r0)−r02K0(r; r0) + 8a~(r)2(r− r0)+ f−2 − 16a~ − 2e + Ve(r) + Ve(r0) + 16a[j~(r)j2 + j~(r0)j2]gK0(r; r0)+ 8aZdr00 ~(r00)2K0(r; r00)K0(r0; r00)= 8aN−1f~(r)~(r0)[j~(r)j2 + j~(r0)j2 − ~]+ ~(r)Zdr00K0(r0; r00)j~(r00)j2 ~(r00) + ~(r0)Zdr00K0(r; r00)j~(r00)j2 ~(r00)g (2.5)1and[−r2 −  − 8a~ − e + Ve(r) + 16aj~(r)j2]’(r) + 8aZdr0K0(r; r0)~(r)2’(r0)= ’(r) + ~~(r); (2.6)with~ = N−1Zdr ~(r)[−r2 + Ve(r) + 16aj~(r)j2]’(r): (2.7)In (2.2){(2.7), the volume Ω does not appear anywhere.In the next three sections, (2.4){(2.7) will be studied under the assumption that the externalpotential Ve(r) is suciently slowly varying. This case is of interest because in the experiments[3{5] the magnetic traps are necessarily of macroscopic dimensions.III. CONDENSATE IN A TRAPWhen the condensate is in the lowest state, its wave function ~(r) can be chosen to benon-negative. Thus, (2.4) is[−r2 + Ve(r) + 8a~(r)2 − 4a~ − E]~(r) = 0: (3.1)When Ve(r) is slowly varying, a rst approximation to (3.1) is obtained by neglecting the rstterm −r2 ~(r), i.e.,[Ve(r) + 8a~(r)2 − 4a~ − E]~(r) = 0: (3.2)This means that~(r) = f[4a~ + E − Ve(r)]=(8a)g1=2 (3.3a)in some region, say R0, and~(r) = 0 (3.3b)outside of R0. It follows from (3.3a) thatVe(r)  4a~ + E (3.4)in R0. Furthermore, since ~(r) is necessarily continuous,Ve(r) = 4a~ + E (3.5)on the boundary of R0.Multiplying (3.2) by ~(r) and integrating over r giveE = e + 4a~: (3.6)2Given N , the value of ~ is determined by (2.2) and (2.3). With (3.6), the approximate solution(3.3) is~(r) =(f~ + (8a)−1[e − Ve(r)]g1=2 inside R00 outside R0:(3.7)As mentioned above, in writing down (3.2), the term−r2 ~(r) has been neglected. However,if the approximate solution (3.3), or equivalently (3.7), is used, this neglected term −r2 ~(r) isin general not even bounded on the boundary of R0. An improved solution is thus needed nearthis boundary. Let x be the rectangular coordinate normal to the boundary, chosen such thatthe positive x-axis points toward the interior of R0. The geometry is shown schematically inFig. 1. Instead of neglecting −r2 ~(r), replace it by the corresponding second derivative withrespect to x. Expand the potential Ve(r) near x = 0 on the boundary of R0 to getVe(x; y; z) = Ve(0; y; z)− (y; z)x+ o(x); (3.8)where (y; z) is assumed to be positive. By (3.5) and suppressing the variables y and z, (3.8)isVe(x)  4a~ + E − x: (3.9)The substitution of (3.9) into (3.1) gives the ordinary dierential equation−d2dx2+ 8a~(x)2 − x~(x) = 0: (3.10)This dierential equation can be put in a canonical form by the scalingx = −1=3;and~(x) = (8a)−1=21=3 (): (3.11)The equation for  is then 00()−  ()3 +  () = 0; (3.12)with the boundary conditionslim!−1 () = 0 (3.13)andlim!1−1=2 () = 1: (3.14)Equation (3.12) with the boundary conditions (3.13) and (3.14) denes a unique universalfunction  (). It is plotted in Fig. 2 and describes through (3.11) the behavior of the condensatewave function near the boundary of R0.3IV. PAIR PRODUCTION IN A TRAPThe next step is to study the integro-dierential equation (2.5) when the external potentialVe(r) is slowly varying. In this case, it is convenient to use the \center-of-mass" coordinate bydeningP (R; r) = K0(r1; r2); (4.1)whereR = 12(r1 + r2); r = r1 − r2:In terms of these coordinates, (2.5) is−12r2RP (R; r)− 2r2P (R; r) + 8a~(R)2(r)+ f−2 − 16a~ − 2e + Ve(R +12r) + Ve(R−12r)+ 16a[j~(R + 12r)j2 + j~(R− 12r)j2]gP (R; r)+ 8aZdr0 ~(R + r0)2P (R + 14r + 12r0; 12r− r0)P (R− 14r + 12r0;−12r− r0)= 8aN−1~(R + 12r)~(R− 12r)[j~(R + 12r)j2 + j~(R− 12r)j2 − ~]+ ~(R + 12r)Zdr0 P (R− 14r + 12r0;−12r− r0)j~(R + r0)j2 ~(R + r0)+ ~(R− 12r)Zdr0 P (R + 14r + 12r0; 12r− r0)j~(R + r0)j2 ~(R + r0); (4.2)where r2 means r2r.Motivated by (3.2), simplify (4.2) by the following approximations:(i) neglect the rst term −12r2RP (R; r);(ii) replace Ve(R+12r)+Ve(R− 12 r) by 2Ve(R), and similarly replace j~(R+ 12r)j2+j~(R− 12r)j2by 2j~(R)j2;(iii) in the same spirit, simplify the last term on the left-hand side to8aZdr0 ~(R)2P (R; 12r − r0)P (R;−12r − r0);and, nally,(iv) neglect the entire right-hand side because of the factor N−1.With (i){(iv), (4.2) becomes−r2P (R; r) + 4a~(R)2(r)+ [− − 8a~ − e + Ve(R) + 16aj~(R)j2]P (R; r)+ 4a~(R)2Zdr0 P (R; r− r0)P (R; r0) = 0: (4.3)In this approximation, R can be considered to be a parameter and the ideas of Lee, Huang,and Yang [9] can be applied.4Since there is translational invariance in the variable r, dene the Fourier transformP (R;k) =Zdr eikrP (R; r): (4.4)In terms of this P , (4.3) is a quadratic equation4a~(R)2 P (R;k)2 + [k2 + k0(R)2] P (R;k) + 4a~(R)2 = 0; (4.5)wherek0(R)2 = − − 8a~ − e + Ve(R) + 16aj~(R)j2: (4.6)ThereforeP (R;k) = [8a~(R)2]−1h−k2 − k0(R)2 + f[k2 + k0(R)2]2 − (8a)2j~(R)j4g1=2i: (4.7)So far, no property of ~ has been used except that it is slowly varying. Let us now use theapproximate solution (3.7). With this approximation, it follows from (4.5) thatK0(r; r0) = 0 (4.8)if 12(r+ r0) is outside of the region R0. If it is inside, then, with  negligible because ~ is slowlyvarying, (4.6) reduces simply tok0(R)2 = 8a~(R)2; (4.9)and the substitution into (4.7) givesP (R;k) = −k0(R)−2fk2 + k0(R)2 − k[k2 + 2k0(R)2]1=2g; (4.10)very similar to the result of Lee, Huang, and Yang [9].It is of some interest to carry out the inverse Fourier transform to get P (R; r) explicitly.The result isP (R; r) = −2(4a)3=2 ~(R)3−1 Im[S04(i)− S00(i)]; (4.11)where = (16a)1=2 ~(R)jrj; (4.12)and S04 and S00 are Lommel’s functions [17].5V. CONCLUSIONS AND DISCUSSIONSeventy years after the theoretical prediction of Bose-Einstein condensation [1,2], it wasobserved in several beautiful experiments for dilute atomic gases [3{5]. In these experiments,it is necessary to introduce an external trap in order to keep the atoms together. The presenceof such traps is the most important dierence between the theory and the experiments. In thispaper, the theory of Bose-Einstein condensation is generalized so that such external potentialscan be included systematically.In the case treated here of pairwise short-range repulsive interactions, the condensate wavefunction satises a non-linear Schro¨dinger equation while the pair excitation from the conden-sate is described by an integro-dierential equation. When the external potential is slowlyvarying, as is often the case because of the macroscopic nature of the trap, both the non-linearSchro¨dinger equation and the integro-dierential equation are solved approximately in termsof elementary functions.It should perhaps be emphasized that, since Bose-Einstein condensation is a quantum phe-nomenon, the present treatment is also entirely quantum in nature, without the introductionof any classical concepts.It is clear from this paper and the preceding Paper I [16] that there are many open problems.Since the present treatment is limited to zero temperature, the most important next step isto remove this limitation so that the properties of Bose-Einstein condensation at non-zerotemperatures can also be studied in the presence of an external potential. An equivalent wayof stating this generalization is to deal with a mixture of superfluids and normal fluids. In theabsence of the external potential, this case has been solved by Lee and Yang [10].Even at zero temperature, there are a number of physically interesting problems that remainto be studied. As an example of the many such problems, it can be asked how the pairexcitation of Sec. IV should modify the condensate wave function. In other words, the non-linearSchro¨dinger equation gives the zeroth-order approximation to the condensate wave function,and the problem is to nd the rst-order approximation, which is expected to be an integro-dierential equation. Some of these problems will be treated in future publications.ACKNOWLEDGMENTSI am greatly indebted to Professor Chen Ning Yang for numerous discussions, includingthose concerning his recent papers with Chou and Yu [18]. I would also like to thank ProfessorHarry Lehmann for helpful discussions, Dr. Haimo Zobernig for his help in solving eq. (3.12)to get the curve of Fig. 2, and the Theory Division at CERN for their kind hospitality.6REFERENCES[1] S. N. Bose, Z. Phys. 26, 178 (1924).[2] A. Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. 1924, 261 (1924) and 1925, 3 (1925).[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science269, 198 (1995).[4] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687(1995).[5] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn,and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).[6] K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957).[7] K. Huang, C. N. Yang, and J. M. Luttinger, Phys. Rev. 105, 776 (1957).[8] T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957).[9] T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).[10] T. D. Lee and C. N. Yang, Phys. Rev. 112, 1419 (1958).[11] T. T. Wu, Phys. Rev. 115, 1390 (1959).[12] N. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959).[13] K. Sawada, Phys. Rev. 116, 1344 (1959).[14] C. N. Yang, Physica 26, S49 (1960).[15] T. T. Wu, J. Math. Phys. 2, 105 (1961).[16] T. T. Wu, preceding paper. The equations in this paper will be referred to with the desig-nation I; for example, (I.2.19) means eq. (2.19) of this reference.[17] Bateman Manuscript Project, Higher Transcendental Functions (edited by A. Erdelyi),McGraw-Hill Book Company (1953), Vol. II.[18] T. T. Chou, C. N. Yang, and L. H. Yu, Phys. Rev. A 53, 4257 (1996), and to be published(1996).7FIGURESFIG. 1. Geometry for the condensate wave function.FIG. 2. The universal function () that describes the behavior of the condensate wave functionnear the boundary of the region R0.8xx = 0R0FIG. 1. Geometry for the condensate wave function.2.2521.751.51.2510.750.50.250– 4 – 2 0 2 4ητ(η)FIG. 2. The universal function () that describes the behavior of the condensate wave functionnear the boundary of the region R0.

Bose-Einstein condensation in an external potential, 2

CERN-TH/96-249Bose-Einstein Condensation in an External Potential ITai Tsun WuGordon McKay Laboratory, Harvard UniversityCambridge, Massachusetts 02138-2901, U.S.A.andTheoretical Physics Division, CERNCH-1211 Geneva 23, SwitzerlandAbstractThe theory of Bose-Einstein condensation for a dilute gas with short-range pair-wise repulsive interaction is developed when an external potential is present.At low temperatures when the Bose-Einstein condensation is nearly complete,a partial dierential equation is obtained for the condensate wave function andan integro-dierential equation for the pair excitation.CERN-TH/96-249September 1996Work supported in part by the United States Department of Energy under Grant No. DE-FG02-84ER40158.I. INTRODUCTIONA most interesting development in physics last year was the rst experimental observationof Bose-Einstein condensation [1,2] in dilute atomic gases [3{5]. From the point of view ofstatistical mechanics in general and Bose-Einstein condensation in particular, a novel feature ofthese experiments is the presence of a trap which supplies an external potential that keeps theatomic gas together. It is the purpose of this and the following papers to study Bose-Einsteincondensation in the presence of such an external potential.In the fties and early sixties, there was a systematic eort to study theoretically theproperties of a dilute Bose gas with short-range repulsion [6{15]. In most of these papers,periodic boundary conditions are assumed and play an important role. The case of the rigidwall where the wave function vanishes is briefly mentioned by Lee, Huang, and Yang [9] andstudied further in Ref. [15]. Since such a rigid wall is described by an innite potential, thiscase can be considered to be an example of an external potential. Accordingly, the method ofRef. [15] is to be generalized to treat the present case of the Bose-Einstein condensation in anexternal potential. This Paper I and the following Paper II are devoted to the study of thesimplest situation where the Bose-Einstein condensation is nearly 100%.II. THE CONDENSATEWhen there is no external potential and the periodic boundary condition applies, the con-densate is the zero-momentum state. In other words, the single-particle zero-momentum statehas a macroscopic occupation number. Clearly, the rst question to be answered is: In thepresence of an external particle, what is the condensate wave function?Consider a system of N pairwise interacting particles in an external potential Ve. TheHamiltonian is (h = 2m = 1)NXi=1p2i +Xi<jV0(rij) +NXi=1Ve(ri); (2.1)whererij = jri − rjj; (2.2)and V0 is a short-range repulsive potential with scattering length a > 0. For example, V0may be the hard-sphere potential which is zero for rij > a and innite for rij < a. When themethod of pseudopotential of Huang and Yang [6] is applied, the Hamiltonian (2.1) is replacedapproximately byH 0 = T + V 0; (2.3)withT =Xi[p2i + Ve(ri)] (2.4)1andV 0 = 4aXi6=j(ri − rj)@@rijrij: (2.5)For the present purpose, it is sucient to useH = T + V; (2.6)whereV = 4aXi6=j(ri − rj): (2.7)The only diculty involved in using H instead of H 0 is the appearance of a familiar type ofdivergence, which may be removed using the procedure of Refs. [9] and [11].It is convenient to rewrite this H in the language of quantized elds:T =Zdr [jr (r)j2 + Ve(r)j (r)j2] (2.8)andV = 4aZdr (r)2 (r)2; (2.9)where  (r) satises the usual commutation rules for a boson eld.Let (r) be the wave function of the condensate, normalized byΩ−1Zdr j(r)j2 = 1; (2.10)where Ω is the volume of the box. The creation and annihilation operators for this condensatearea0 = Ω−1=2Zdr (r) (r); a0 = Ω−1=2Zdr (r) (r): (2.11)They, of course, satisfy[a0; a0] = 1: (2.12)With the parts of  (r) and  (r) corresponding to this  singled out, dene  1(r) and  1(r)by (r) = Ω−1=2(r)a0 +  1(r);  (r) = Ω−1=2(r)a0 +  1(r): (2.13)Since this state  is macroscopically occupied, for low densities these  1 and  1 may beconsidered to be a small perturbation [9,15]. Moreover, at low temperatures very near zero,the relationN =Zdr (r) (r) = a0a0 +Zdr 1(r) 1(r) (2.14)2may be approximated byN  a0a0: (2.15)This means that the Bose-Einstein condensation is nearly 100%.To rst order in  (r) and  (r), the T and V are approximated byT1 = N( + e) + Ω−1=2a0Zdr [−r2(r) + Ve(r)(r)] 1(r)+ Ω−1=2a0Zdr [−r2(r) + Ve(r)(r)] 1(r) (2.16)andV1 =4aNΩN + 2Ω−1=2a0Zdr j(r)j2(r) 1(r)+ 2Ω−1=2a0Zdr j(r)j2(r) 1(r); (2.17)where = Ω−1Zdr jr(r)j2;  = Ω−1Zdr j(r)j4; e = Ω−1ZdrVe(r)j(r)j2: (2.18)Note that eq. (2.17) here is identical to eq. (2.14) of Ref. [15], but T1 is modied by the presenceof the external potential Ve(r).Since the Hamiltonian T1 + V1 is linear in  (r) and  (r), it is a simple matter to take aSchro¨dinger state vector of the formΨ = (N !)−1=2aN0 jvaci; (2.19)where jvaci is dened by (r)jvaci = 0: (2.20)The Schro¨dinger equation for Ψ,(T1 + V1)Ψ = NEΨ; (2.21)then leads to a non-linear Schro¨dinger equation for (r):−r2 + Ve(r) +8aNΩj(r)j2 −4aNΩ −E(r) = 0: (2.22)This is the desired equation for the condensate wave function in the presence of the externalpotential Ve(r).Note that eq. (2.22) diers from the non-linear Schro¨dinger equation (2.21) of Ref. [15] onlyin the addition of the term Ve(r) due to the external potential. This Ve(r) does not lead to anycomplication in the derivation of (2.22); for a more detailed discussion, see Appendix A of Ref.[15].There are two basic lengths in eq. (2.22): (i) the length over which Ve(r) varies; and (ii) thelength scale determined by (4aN=Ω)j(r)j2. It is the interplay between these two lengths thatis responsible for some of the interesting properties of this non-linear Schro¨dinger equation.The simplest example of this interplay is the case of the rigid boundary [9,15]. Some furthercases are studied in Sec. III.3III. PAIR PRODUCTIONIn the absence of an external potential, pair production of particles with opposite momentaplays an important role [9,10,15]. In Appendix A here, the procedure of dealing with pairproduction as discussed by Lee, Huang, and Yang in Appendix I of Ref. [9] is recast in a formmore suitable for generalization to the present case.Equations (2.16) and (2.17) give approximations to the T and V of (2.8) and (2.9) to therst order in  (r) and  (r). In order to study pair production, what is needed is the second-order approximation to T and V . Here (2.15) for N is no longer accurate enough, and (2.14)needs to be used. Thus,T = N( + e) + Ω−1=2a0Zdr [−r2(r) + Ve(r)(r)] 1(r)+ Ω−1=2a0Zdr [−r2(r) + Ve(r)(r)] 1(r)+Zdr [jr 1(r)j2 + Ve(r) 1(r) 1(r)− ( + e) 1(r) 1(r)] (3.1)andV2 =4aNΩN + 2Ω−1=2a0Zdr j(r)j2(r) 1(r)+ 2Ω−1=2a0Zdr j(r)j2(r) 1(r)+Zdr f[−2 + 4j(r)j2] 1(r) 1(r)+N−1a20(r)2 1(r)2 +N−1a20 (r)2 1(r)2g: (3.2)The rst terms in (3.1) and (3.2) are constants, while the sum of those terms involving afactor of Ω−1=2 also leads to a constant by virtue of the non-linear Schro¨dinger equation (2.22)for (r). We therefore concentrate on the terms that are quadratic in  1(r) and  1(r):H2 =Zdrjr 1(r)j2 +− − e −8aNΩ + Ve(r) 1(r) 1(r)+4aNΩ[4j(r)j2 1(r) 1(r) +N−1a20(r)2 1(r)2 +N−1a20 (r)2 1(r)2]: (3.3)This is a quadratic form in  1(r) and  1(r).Following the procedure in Appendix A, deneH 02 = e−PH2eP ; (3.4)where [15]P = (2N)−1Zdr dr0  1(r) 1(r0)K0(r; r0)a20: (3.5)Without loss of generality, choose K0 to satisfy4K0(r; r0) = K0(r0; r) (3.6)and Zdr (r)K0(r; r0) = 0: (3.7)Since it follows from (2.13) that[ 1(r);  1(r0)] = 3(r− r0)− Ω−1(r)(r0); (3.8)it is straightforward to verify that[ 1(r); eP ] = N−1Zdr0K0(r; r0) 1(r0)ePa20; (3.9)[ 1(r) 1(r); eP ] = N−1Zdr0K0(r; r0) 1(r) 1(r0)ePa20; (3.10)[jr 1(r)j2; eP ] = N−1Zdr0rrK0(r; r0)  rr 1(r) 1(r0)ePa20; (3.11)and[ 1(r)2; eP ] = N−1ePK0(r; r) + 2Zdr0K0(r; r0) 1(r0) 1(r0)+N−1Zdr0 dr00K0(r; r0)K0(r; r00) 1(r0) 1(r00)a20a20: (3.12)Using these commutation relations, the H 02 of (3.4) is given byH 02 = H2 + e−P [H; eP ] = H 002 +H02c; (3.13)whereH 002 =Zdrjr 1(r)j2 +− −8aNΩ − e + Ve(r) 1(r) 1(r)+4aNΩ[4j(r)j2 1(r) 1(r) +N−1(r)2a20  1(r)2]+ (r)24aNΩK0(r; r) + 2Zdr0K0(r; r0) 1(r0) 1(r); (3.14)and H 02c contains all the terms that are quadratic in  1(r) and hence no  1(r):H 02c = N−1Zdr4aNΩ(r)2 1(r)2 −Zdr0 [r2rK0(r; r0)] 1(r) 1(r0)+− −8aNΩ − e + Ve(r) Zdr0K0(r; r0) 1(r) 1(r0)+4aNΩ4j(r)j2Zdr0K0(r; r0) 1(r) 1(r0)+ (r)2Zdr0 dr00K0(r; r0)K0(r; r00) 1(r0) 1(r00)a20: (3.15)Similar to (3.5){(3.7), dene L(r; r0) by5H 02c = (2N)−1Zdr dr0 L(r; r0) 1(r) 1(r0)a20; (3.16)L(r; r0) = L(r0; r); (3.17)andZdr (r0)L(r; r0) = 0: (3.18)As a generalization of (A4) in Appendix A, the pair-production distribution K0(r; r0) is deter-mined by the conditionL(r; r0) = 0: (3.19)It only remains to rewrite (3.19) in the form of an integro-dierential equation for K0(r; r0):L(r; r0)= −r2K0(r; r0)−r0 2K0(r; r0) +8aNΩ(r)2(r− r0)+−2 −16aNΩ − 2e + Ve(r) + Ve(r0) +16aNΩ[j(r)j2 + j(r0)j2]K0(r; r0)+8aNΩZdr00(r00)2K0(r; r00)K0(r0; r00)− (r)(r0)− (r0)(r)= 0; (3.20)where (r) is to be determined by (3.18).This determination is carried out by multiplying by (r0) and then integrating over r0. Anintegration by parts givesZdr0K0(r; r0)−r0 2 + Ve(r0) +16aNΩj(r0)j2(r0) +8aNΩj(r)j2(r)= (r)Ω + (r)Zdr0(r0)(r0): (3.21)Equation (2.22) may be used to give(r) =8aNΩ2j(r)j2(r) +Zdr0K0(r; r0)j(r0)j2(r)− 12(r): (3.22)For completeness, we write down explicitly the integro-dierential equation for K0(r; r0):−r2K0(r; r0)−r0 2K0(r; r0) +8aNΩ(r)2(r− r0)+−2 −16aNΩ − 2e + Ve(r) + Ve(r0) +16aNΩ[j(r)j2 + j(r0)j2]K0(r; r0)+8aNΩZdr00(r00)2K0(r; r00)K0(r0; r00)=8aNΩ2(r)(r0)[j(r)j2 + j(r0)j2 − ]+ (r)Zdr00K0(r0; r00)j(r00)j2(r00) + (r0)Zdr00K0(r; r00)j(r00)j2(r00): (3.23)6IV. LOW-LYING EXCITATIONSWhen (3.23) is satised, H 02c = 0 and henceH 02 = H002 : (4.1)>From (3.14), H 002 contains three types of terms:i) a constant4aNΩZdr (r)2K0(r; r);ii) a pair interaction that creates two condensate particles4aΩa20Zdr (r)2 1(r)2; andiii) the rest of the terms.Terms of the types i) and ii) have no eect on the energies of the low-lying excitations. Itis therefore useful to consider terms of type iii) rst:H 0002 =Zdrjr 1(r)j2 +− −8aNΩ − e + Ve(r) +16aNΩj(r)j2 1(r) 1(r)+8aNΩ(r)2Zdr0K0(r; r0) 1(r0) 1(r): (4.2)Since this H 0002 is linear in  1(r) and also linear in  1(r), the excitations can be described by acreation operator of the forma =Zdr’(r) 1(r); (4.3)where ’(r) is a c-number wave function for the excitation. Similar to (3.7) and (3.18), it isconvenient to choose ’(r) so that Zdr (r)’(r) = 0: (4.4)This wave function ’(r) is determined by[H 0003 ; a] = a; (4.5)where  is the energy of the excitation. This is the generalization of the phonon spectrum. Itfollows from (4.5) that ’(r) satises the linear integro-dierential equation−r2 −  −8aNΩ − e + Ve(r) +16aNΩj(r)j2’(r)+8aNΩZdr0K0(r; r0)(r)2’(r0)= ’(r) + (r); (4.6)where  is to be determined so that (4.4) is satised = Ω−1Zdr (r)−r2 + Ve(r) +16aNΩj(r)j2’(r): (4.7)Note that (4.6) describes a low-lying excitation due to the addition of a boson to the system.7V. CONCLUSIONUsing the method of the pseudo-potential as applied to many-body problems [6,7], the eectsof an external potential have been incorporated in the theory of Bose-Einstein condensationfor a dilute gas with short-range pairwise repulsive interaction. The condensate wave function(r) is found rst by solving the non-linear Schro¨dinger equation (2.22); the pair production asdescribed by K0(r; r0) of (3.5) is then determined by the non-linear integro-dierential equation(3.23); and nally, the wave function ’(r) for low-lying excitations is governed by (4.7) here.As is often the case for a dilute boson gas, the expansion parameter is typically (Na3=Ω)1=2.In terms of this expansion parameter, (2.22) and (3.5) give essentially the leading order andthe next-to-leading order description of the system. Extension to higher orders, where thelogarithm of this expansion parameter will appear, has not been accomplished.This present work has been motivated by the beautiful experiments on Bose-Einstein con-densation in dilute atomic gases [3{5]. In the following paper, the results obtained here will beapplied to the case where the external potential is a trap to keep the atoms together.ACKNOWLEDGMENTSI am greatly indebted to Professor Chen Ning Yang for numerous discussions, including thoseconcerning his recent papers with Chou and Yu [16]. I would also like to thank Professor HarryLehmann for helpful discussions, and the Theory Division at CERN for their kind hospitality.8APPENDIX AConsider the HamiltonianH = 12(aa+ bb) + y(ab + ab); (A1)where (a; a) and (b; b) are a pair of independent creation and annihilation boson operators.Let us calculateH 0 = eabHe−ab; (A2)where the constant  remains to be chosen. This H 0 is of course not hermitian. The result isH 0 = (12− y)(aa+ bb) + yab− y + (y −  + 2y)ab: (A3)Let  be chosen such that the coecient of ab vanishes:y − + 2y = 0: (A4)This gives =12y[1− (1− 4y2)1=2]; (A5)which is (A12) of Lee, Huang, and Yang [9]. With this , H 0 isH 0 = −12+ 12(1− 4y2)1=2(aa+ bb+ 1) + yab: (A6)Since the last term yab has no eect on the energy spectrum, H 0 leads immediately to theground-state energy per particle and the phonons.9REFERENCES[1] S. N. Bose, Z. Phys. 26, 178 (1924).[2] A. Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. 1924, 261 (1924) and 1925, 3 (1925).[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science269, 198 (1995).[4] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687(1995).[5] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn,and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).[6] K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957).[7] K. Huang, C. N. Yang, and J. M. Luttinger, Phys. Rev. 105, 776 (1957).[8] T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957).[9] T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).[10] T. D. Lee and C. N. Yang, Phys. Rev. 112, 1419 (1958).[11] T. T. Wu, Phys. Rev. 115, 1390 (1959).[12] N. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959).[13] K. Sawada, Phys. Rev. 116, 1344 (1959).[14] C. N. Yang, Physica 26, S49 (1960).[15] T. T. Wu, J. Math. Phys. 2, 105 (1961).[16] T. T. Chou, C. N. Yang, and L. H. Yu, Phys. Rev. A 53, 4257 (1996), and to be published(1996).10

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