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Dilute Fermi gas: kinetic and interaction energies A. A. Shanenko Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region,Russia 4 (Dated: January 13, 2004) 0 0 A dilute homogeneous 3D Fermi gas in the ground state is considered for the case of a repulsive 2 pairwiseinteraction. Thelow-density(dilution)expansionsforthekineticandinteractionenergiesof n thesysteminquestionarecalculateduptothethirdorderinthedilutionparameter. Similartothe a recentresultsforaBose gas, thecalculated quantitiesturnouttodependon apairwise interaction J through the two characteristic lengths: the former, a, is the well-known s-wave scattering length, 3 and the latter, b, is related to a by b = a−m(∂a/∂m), where m stands for the fermion mass. To 1 take control of the results, calculations are fulfilled in two independent ways. The first involves the Hellmann-Feynman theorem, taken in conjunction with a helpful variational theorem for the ] scattering length. This way is used to derive the kinetic and interaction energies from the familiar h low-density expansion of the total system energy first found by Huang and Yang. The second c way operates with the in-medium pair wave functions. It allows one to derive the quantities of e interest“from the scratch”, with no use of the total energy. An important result of the present m investigation is that the pairwise interaction of fermions makes an essential contribution to their - kineticenergy. Moreover, thereis a complicated and interesting interplay of thesequantities. t a t PACSnumbers: PACSnumber(s): 03.75.Ss,05.30.Fk,05.70.Ce s . t a m I. INTRODUCTION derexperimentalstudy. Inviewofthisfact, reconsidera- tionofthebasicaspectsofthetheoryofadiluteuniform - d Fermi gas in the ground state is of importance. Recent experiments with magnetically trapped alkali n o atoms significantly renewed interest in properties of In the present paper a dilute Fermi gas with repulsive c quantum gases. As it is known,the initial series of these pairwiseinteractionis under consideration. Why the sit- [ experiments concerned a Bose gas (87Rb [1], 23Na [2], uation of a repulsive Fermi gas is of interest whereas the and 7Li [3]) and resulted in extensive reconsiderations s wave scattering length is negative for 6Li [19] and, 1 − v and new investigations in the field of the Bose-Einstein most likely, for 40K [18] so that a trapped 6Li is con- 8 condensation. In so doing theoretical and experimental sidered as a good candidate for observation of BCS-like 1 observations were made that not only confirmed conclu- transition [19, 22, 23]? The point is that the experi- 2 sions made more than forty years ago but also provided ments can produce (and is now producing) the tempera- 1 a new horizon of the boson physics. In particular, one tures at which the BCS pairing does not occur yet. So, 0 shouldpointoutgoodagreementoftheresultsofsolving at these temperature a Fermi gas with attractive pair- 4 the Gross-Pitaevskii equation derived in 1960s [4] with wise interaction is close enough in properties (with the 0 / experimental data on the density profiles of a trapped corrections of O((T/Tf)2), where T/Tf 0.2) to a re- at Bosegas[5]. Theso-calledreleaseenergymeasuredinthe pulsive Fermi gas in the ground state. O≈f course, with m experiments with rubidium wasalso found to be in good one obvious alteration: the positive s wave scattering − - agreement with theoretical expectations based on the length should be replaced by a negative one in the final d time-dependent Gross-Pitaevskii equation [6]. Among expressions (see, for example, [19]). Thus, the experi- n new theoretical achievements the exact derivation of the ments with magnetically trapped atoms of 6Li and 40K o Gross-Pitaevskii energy functional [7] can be mentioned offer exciting possibility of exploring the both superfluid c : alongwiththeproofthataBosegaswithrepulsiveinter- and normal states of a Fermi gas. In addition, the most v actionis100%superfluidinthedilutelimit[8]. Astothe recentpublications[20,21]demonstratethatthereexists i X experimental innovations, observations of interference of interesting possibility of ruling the scattering length of two Bose condensates [9, 10] is a good example (for in- 6Liwhichvariesina wide rangeofvalues, fromnegative r a teresting theoretical details see the papers [11], [12] and topositiveones,whenamagneticfieldisimposedonthe reviews [13],[14]). system. Thefirstcommunicationsconcerningexperimentswith The particular problem to be investigated here con- trappedfermionicatomsappearedintheliteratureabout cerns the kinetic E and interaction E energies of kin int three years ago [15] when a temperature near 0.4T was a uniform dilute 3D Fermi gas in the ground state and f claimed to be reachedfor a trapped 6Li,where T is the with a repulsive interparticle potential. This problem f temperaturebelowwhichtheFermistatisticsisofimpor- is connected with a more general question related to all tance. Nowadaysthetemperaturescloseto0.2T [16]and the quantum gases. The question is if the pairwise in- f 0.5T [17]arereportedforthe6Li-vapor. Whereasatoms teractionofquantumparticlesmakescontributiontothe f offermionic40K wererecentlycooleddownto0.3T [18]. kinetic energyofa quantumgasornot? Itiswell-known f So,the regimeofthedegenerateFermigasisalreadyun- thatfora classicalimperfectgasthe pairwiseinteraction 2 does not make any contribution to the kinetic energy. withthe pairwiseinteractionV(r)=γΦ(r), whereγ >0 The usual expectations regarding the kinetic energy of is the coupling constant and Φ(r) > 0 stands for the dilute quantum gases comes from the pseudopotential interaction kernel (r = r). Below the particle spin is | | approach. According to these expectations the kinetic assumed to be s =1/2 [27]. The ground-state energy of energyofadilutequantumgasisnotpracticallyaffected thesysteminquestionE = 0Hˆ 0 obeysthewell-known h | | i by the pairwise interaction. It means that taken in the relation leading order of the expansion in the dilution parame- ter, the total system energy of a ground-state Bose gas δE = 0δHˆ 0 (2) h | | i coinsides with the interaction one if calculated with the pseudopotential (see Refs. [13, 25, 26]). For the Fermi calledtheHellmann-Feynmantheorem,δEandδHˆ being case the same approach dictates that the kinetic energy infinitesimal changes of E and Hˆ. An advantage of this does not include terms depending on the pairwise po- theorem is that it yields important relations connecting tential in the leading and next-to-leading orders of the thetotalground-stateenergyE withthekineticE and kin dilution expansion (see Ref. [24] and Eqs. (24) and (25) interaction E energies. These relations read int below). In other words, in what concerns relation between the kinetic and interaction energies, a quantum ∂E ¯h2 m = 0 2 0 =E , (3) gas is very similar to a classical one from the pseudopo- − ∂m − 2m∇i kin tential viewpoint. However, this result was proved to be D (cid:12) Xi (cid:12) E ∂E (cid:12) (cid:12) wrong. An adequate and thorough procedure of calcu- γ = 0 (cid:12) V(r r )(cid:12)0 =E . (4) i j int lating E and E of a cold dilute Bose gas has been ∂γ | − | recentlykdinevelopedintin Refs. [25, 26]. It proves that the D (cid:12)(cid:12)Xi>j (cid:12)(cid:12) E (cid:12) (cid:12) pairwise interaction in a Bose gas has a strong effect on Ifthe dependence ofthe ground-stateenergyonthe cou- the kinetic energy. Moreover, there are quite real situ- pling constant and particle mass were known explicitly, ations when the kinetic energy of a uniform dilute Bose one would readily be able to calculate E and E by kin int gas is essentially more than the interaction one! It is meansofEqs.(3)and(4). However,itisnotthecaseasa now necessary to clarify this situation in the Fermi case. rule,andthe dependence is usuallygivenonlyimplicitly. The more so, that the interaction and kinetic energies In the situation of the repulsive Fermi gas the depen- of imperfect trapped quantum gases are now under ex- dence of the ground-state energy on γ and m is indeed perimental study [20, 24]. Thus, the aim of the present known only implicitly. According to the familiar result publication is to generalize the procedure developed in of Huang and Yang [28] found with the pseudopoten- [25, 26] to the Fermi case. tial approach but then reproduced within the bound- The paper is organized as follows. The Section II arycollisionexpansionmethod[29]beyondanyeffective- presents the kinetic and interaction energy of a ground- interaction arguments, the energy per fermion ε = E/N state repulsive Fermi gas found with the Hellmann- reads Feynmantheoremonthebasisofanauxiliaryvariational relationgiveninRefs. [25,26]. The SectionIII is to con- ε= 3h¯2kF2 1+ 10k a+ 4 (11 2ln2)k2a2 , (5) sider the derived expressions in various regimes: from a 10m 9π F 21π2 − F (cid:20) (cid:21) weak coupling to a strong one. This is needed to discuss the failure of the pseudopotential approach in operating whichisaccuratetothetermsoforderk2a2. InEq.(5)a F with Ekin and Eint. Derivation of Ekin and Eint in Sec- stands for the s−wavescatteringlength, kF is the Fermi tion2is simple but ratherformalso thatsomequestions wavenumber given by can remain. This is why Sections IV and V give a more physicallysoundwayofcalculatingthekineticandinter- kF =(3π2n)1/3, (6) actionenergies. Thiswayinvokesamethoddevelopedin where n = N/V, and the thermodynamic limit N the papers [25, 26] and dealing with the pair wave func- → , V , n = N/V const is implied. Inserting tions, which allows one to go in more detail concerning ∞ → ∞ → Eq. (5) in Eqs. (3) and (4), one can arrive at the microscopic features of dilute quantum gases. ε = kin 3h¯2k2 ∂a 10 8 II. HELLMANN-FEYNMAN THEOREM =ε F k a+ (11 2ln2)k2a2 , (7) − 10a ∂m 9π F 21π2 − F (cid:20) (cid:21) 3h¯2k2 ∂a 10 8 Let us consider the system of N identical fermions ε = F γ k a+ (11 2ln2)k2a2 , (8) placed in a box with the volume V and ruled by the int 10ma ∂γ 9π F 21π2 − F (cid:20) (cid:21) following Hamiltonian: whereε =E /N andε =E /N. Hence,toderive kin kin int int thekineticandinteractionenergiesfromEq.(5)withthe ¯h2 Hˆ = 2+ V(r r ) (1) help of the Hellmann-Feynman theorem, we should have − 2m∇i | i− j| an idea concerning the derivatives of a with respect to i i>j X X 3 the particle mass m and coupling constant γ. As E = nothing moreto calculatethe kinetic andinteractionen- E +E , then from Eqs. (7) and (8) it follows that ergies of the uniform repulsive Fermi gas in the ground kin int state. Equations (7) and (8) taken in conjunction with m∂a/∂m=γ∂a/∂γ. (9) Eq. (15), result in the following expressions: This property of the derivatives becomes clear if we re- 3h¯2k2 10 ε = F 1+ k b mind that in the 3D case the s wave scattering length kin F 10m 9π is given by − (cid:20) 4 b + (11 2ln2) 2 1 k2a2 , (17) mγ 21π2 − a − F a= d3rϕ(r)Φ(r), (10) (cid:18) (cid:19) (cid:21) 4π¯h2 3h¯2k2 b Z ε = F 1 int 10m − a whereϕ(r)obeysthethetwo-bodySchrödingerequation (cid:18) (cid:19) in the center-off-mass system: 10 8 k a+ (11 2ln2)k2a2 , (18) × 9π F 21π2 − F (h¯2/mγ) 2ϕ(r)+Φ(r)ϕ(r) =0. (11) (cid:20) (cid:21) − ∇ whosesum is, ofcourse,equalto Eq.(5). We againhave The pair wavefunction ϕ(r) representsthe zero-momen- series expansions in k a but with coefficients depending F tum scattering state, and ϕ(r) 1 when r . The on the ratio b/a. → → ∞ scattering part of the pair wave function given by the definition ϕ(r) = 1+ψ(r) is specified by the following asymptotic behavior: III. FROM WEAK TO STRONG COUPLING ψ(r) a/r (r ). (12) →− →∞ TogoinmoredetailconcerningEqs.(17)and(18),let usconsidertheminvariousregimes. Wespeakaboutthe Note once more that the pairwise potential involved in week coupling when the interaction kernel Φ(r) is inte- Eqs. (10) and (11) is V(r) = γΦ(r) but not Φ(r) which grable and the coupling constant γ 1. The integrable is the repulsive interaction kernel. As it is seen from ≪ kernel with γ 1 and a singular pairwise interaction Eqs. (10) and (11), the scattering length depends on the ≫ like the hard-sphere potential are related to the strong- particlemassandcouplingconstantthroughtheproduct coupling regime. The expansionparameter k a involved mγ. Hence, to use Eqs. (7) and (8) we should know the F in the expressions mentioned above corresponds to the derivative of a with respect to mγ. dilutionlimitk 0. Inthissituationoneisabletoop- This derivative can be found with very useful varia- F → eratewithEq.(5)inthe bothweak-andstrong-coupling tionaltheoremprovedin the papers[25, 26]. After small cases. However, for the weak coupling k a is small even algebra the result of this theorem is rewritten in the fol- F beyond the dilute regime due to a γ 1. This is why lowing form: ∝ ≪ Eq. (5) can be used and rearranged in such a way that δ(mγ) to derive the weak-coupling expansion for ε. δa= d3rϕ2(r)Φ(r), (13) 4π¯h2 In the weak-coupling regime the scattering length a is Z given by the Born series: where, remind, ϕ(r) is a realquantity. In view of crucial a=a +a +... (19) importance of this theorem, let us make an explaining 0 1 remark concerning the proof. The key point here is to with represent Eq. (10) as mγ mγ2 d3k Φ2(k) a = Φ(k =0), a = , (20) a= mγ d3rϕ2(r)Φ(r)+ 1 d3r ψ(r)2, (14) 0 4π¯h2 1 −4π¯h2 Z (2π)3 2Tk 4π¯h2 4π |∇ | Z Z where T = h¯2k2/(2m), and Φ(k) is the Fourier trans- k which is realized with the help of Eqs. (11), (12) and form of the interaction kernel ( for more detail see (ψ ψ)= ψ ψ+ψ 2ψ. So, from Eq. (13) one gets Ref. [30]). Inserting Eq. (19) in Eq. (5), one gets the ∇ ∇ ∇ ∇ ∇ following expression: m∂a/∂m=γ∂a/∂γ =a b, (15) − 3h¯2k2 10 where the additional characteristic length b>0 is of the ε= F 1+ kFa0 10m 9π " form 10 4 b= 41π d3r|∇ψ(r)|2. (16) +(cid:18)9πkFa1+ 21π2(11−2ln2)kF2a20(cid:19)#, (21) Z Emphasize that b can not be represented as a function where terms of order γ3 are ignored. Due to Eq. (20) ofa inprinciple,andthe ratiob/a depends onaparticu- thedependenceofEq.(21)ontheparticlemassandcou- lar shape of a pairwise potential involved. Now we need plingconstantisknownexplicitly. Hence,onecanreadily 4 employ the Hellmann-Feynman theorem that, taken to- usuallyfulfilledviaaregularizationproceduresimulating gether with Eq. (21), yields themomentumdependenceofthet matrix. Inthepseu- − dopotential scheme of Huang and Yang this corresponds ε =3h¯2kF2 1 to use of the effective interaction (4π¯h2a/m)δ(r)(∂/∂r)r kin 10m " − ratherthan(4π¯h2a/m)δ(r). Fromthisonecanlearnthat to generalize Eqs. (21), (22) and (23) to the situation of 10k a + 4 (11 2ln2)k2a2 , (22) a finite coupling constant, one should replace a0 by a −(cid:18)9π F 1 21π2 − F 0(cid:19)# and remove all the terms depending on a1 in the mentioned equations. This yields Eq. (5) and the following pseudopotential predictions for the kinetic and interac- 3h¯2k2 10 ε = F k a tion energies: int F 0 10m 9π " 3h¯2k2 4 + 20k a + 8 (11 2ln2)k2a2 . (23) ε(kpins) = 10mF 1− 21π2(11−2ln2)kF2a2 , (24) (cid:18)9π F 1 21π2 − F 0(cid:19)# 3h¯2k2 10 (cid:20) 8 (cid:21) ε(ps) = F k a+ (11 2ln2)k2a2 . (25) So,thederivedresultssuggestthatthepairwiseinterac- int 10m 9π F 21π2 − F (cid:20) (cid:21) tion influences the both kinetic and interaction energies of a Fermi gas. In the weak-coupling regime the major Note that these results can be derivedin another way as part of the γ-dependent contribution to Eq. (21) is re- well. For example, the first term in Eq. (25) can readily lated to E , this part being proportional to γ. While be reproduced with the pseudopotential in the Hartree- int thetermsoforderγ2 appearinbothE andE . This Fockapproximation(seeRef.[28]andthenextsectionof kin int conclusion meets usual expectations according to which the present paper). From Eqs. (24) and (25) one could the contribution to the mean energy coming from the conclude that the second term in Eq. (5) is related to pairwisepotentialismostlytheinteractionenergyfordi- the interaction energy, and, hence, the contribution to lute quantum gases (see, for example, Refs. [13, 19] and the mean energy of a dilute cold Fermi gas coming from the discussion in Introduction of the paper [25]). On the pairwise potential is mainly the interaction energy. the contrary, beyond the weak-coupling regime the situ- However, now we know that actually it is not the case. ation with E and E turned out to be rather curious So, one should be careful with the pseudopotential pro- kin int and differs significantly from that of the weak-coupling cedure which has serious limitations in spite of the cor- case. However, before any detail let us discuss the pseu- rect result for the mean energy. Here it is worth noting dopotential predictions for E and E being the basis thatthe pseudopotentialscheme preservessome features kin int of usual speculations involving the kinetic and interac- of the weak-coupling regime even being applied in the tionenergiesofquantumgasesbeyondtheweak-coupling strong-couplingcase. Thisconcernstherelationbetween regime. thekineticandinteractionenergyforbothadiluteFermi At present the customary way of operating with the ground-stategasandaBoseone[25,26]. Thesameprob- thermodynamicsofadilutecoldFermigaswithrepulsive lem appears when the pseudopotential is used to calcu- pairwise potential is based on the effective-interaction latethetwo-particleGreenfunctioninaBosegas,which procedure: one is able to use either the t matrix for- manifests itself in abnormal short-range boson correla- mulation like in the Galitskii original pape−r [31] or the tions [25]. Similar troubles can also be expected for the pseudopotential scheme applied in the classical work of two-fermion Green function. Huang andYang [28]. Inthe dilution limit thet matrix Now let us consider Eqs. (17) and (18) beyond the isreducedtot=4π¯h2a/m,whichyieldsthemom−entum- weak-couplingregime,theratiob/abeingofspecialinter- independent result 4π¯h2a/m for the Fourier transform est. We start with the simplified situation of penetrate- of the effective interaction. This is why one is able not able spheres that are specified by the interaction kernel to make essential difference between these two effective- Φ if r r , interaction formulations both referred to as the pseu- Φ(r)= ≤ 0 (26) dopotential approach here. The key point of this ap- (cid:26)0 if r >r0. proach is that to go beyond the weak-coupling regime, Inserting Eq. (26) in Eq. (11), one can find one should replace the Fourier transform of the pair- wiseinteractionΦ(k)bythequantity4π¯h2a/minallthe 2Asinh(αr)/r if r r , expressions related to the weak-coupling approximation. ϕ(r)= ≤ 0 (27) In so doing, some divergent integrals appear due to ig- (cid:26) 1−a/r if r>r0, norance of the momentum dependence of the t matrix. Indeed, substituting t = 4π¯h2a/m for Φ(k) in −Eq. (14), whereα2 =mγΦ/¯h2 (Φ>0)andAisaconstant. Equa- onegetsadivergentquantitya d3k/2T thatmakes tion (27) taken together with the usual boundary condi- 1 k contribution to the total energy∝of the system. To de- tions at r=r0 leads to R rive the classical result of Huang and Yang, the diver- genttermproportionaltoa1 shouldberemoved,whichis a=r0 1−tanh(αr0)/(αr0) (28) h i 5 30 and 1 b=r 1 3tanh(αr )/(αr ) csch(αr ) , (29) 0 0 0 0 −2 − 20 (cid:20) (cid:16) (cid:17)(cid:21) where csch(x)=1/cosh2(x). One canreadily checkthat in the weak-coupling regime, when αr γ1/2 0, 0 10 ∝ → Eqs. (28) and (29) are reduced to 1 2 a≃ 3α2r03 ∝γ, b≃ 15α4r05 ∝γ2 (30) b/a 0 and, hence, b a. This means that the next-to-leading ≪ term in the expansion in kFa given by Eq. (5) is mostly -10 theinteractionenergy,asitwasmentionedabove. Onthe contrary,in the strong-coupling regime, when αr , 0 →∞ Eqs. (28) and (29) give -20 a r , b r . (31) 0 0 → → Hence, b/a 1, and the ground-state energy of a dilute -30 0 2 4 6 8 10 Fermi gas w→ith the hard-sphere interaction is exactly ki- x netic! Note that the same conclusionis valid for a dilute Bose gas of the hard spheres [25, 26, 32]. FIG. 1: The ratio b/a versus x = rc2/(2r02) for the pairwise Another,amorerealisticexampleconcernsasituation interaction kernel(32), r =(mγC/¯h2)1/4. c whentheinteractionkernelcombinesashort-rangerepul- sive sectorwith a long-rangeattractiveone. Here we are especiallyinterestedinanegativescatteringlength. Itis with D = r Γ(3/4)/[23/2Γ(5/4)]. Note that to derive 0 usually considered (see, e.g., [33]) that for alkali atoms Eq. (34), the useful formula one can employ the following approximation: J (x)J (x)+J (x)J (x)= 2sin(πν)/(πx) ν+1 −ν ν −(ν+1) + if r r , − 0 Φ(r)= ∞ ≤ (32) C/r6 if r >r . should be applied. Equation (34), taken in conjunction (cid:26)− 0 with Eq. (15), yields The scatteringlength for the pairinteractionkernel(32) is of the form (see Ref [34]) b/a=3/4+1/ π√2J (x)J (x) . (35) 1/4 −1/4 a/r =Γ(3/4)J (x)/ 2Γ(5/4)J (x) , (33) c −1/4 1/4 As it is seen from Eq. ((cid:2)35), in the limit x (cid:3) 0 we get → where x = r2/(2r2), r = (m(cid:2)γC/¯h2)1/4, whe(cid:3)reas J (x) the hard-sphere result b/a = 1 (see Fig. 1). The quan- c 0 c ν tity b (remind that b is always positive!) is finite at and Γ(z) denote the Bessel function and the Euler gamma-function. It is known that Jν(x) ≃ xν/[2νΓ(1+ x= x0(i), while b →+∞ for x→x∞(i). In the latter situ- ν)] for x 0. Therefore, Eq. (33) reduces to a = r ation b goes to infinity in such a way that b/a + in this lim→it. In other words, when the attractive sec0- though a for x x(i), too. Hence|, |x→(i) an∞d | | → ∞ → ∞ 0 tor is “switched off”, we arrive at the hard-sphere re- x(i) are both singular points of b/a. Let us stress that ∞ sult discussed in the previous paragraph of the present the zeros of the scattering length in the case considered section. For x > 0 the scattering length (33) is a de- have nothing to do with the weak-coupling regime for creasing function of x with the complicated pattern of which, remind, b/a 1. Operating with the kernel behaviour specified by the infinite set of singular points | | ≪ (32) we are not able to reach the weak-coupling regime x(∞1), x(∞2), x(∞3),... . These points are the zeros of at all because in this kernel is not bound from above. { } J (x) so that a when x x(i) 0 and Nowletusconsiderthesituationofanegativescattering 1/4 ∞ → −∞ → − a + when x x(i) +0. In addition, there is also length being of special interest in the experimental con- ∞ → ∞ → text. The scattering length givenby Eq. (33) is negative the infinite sequence of the zeros ofthe scatteringlength {x(01), x(02), x(03),...} being the zeros of J−1/4(x). Note fporrovaindyedofthtahtesxe0(ii)nt<erxva<ls xth∞(ie).raAtisoitb/saeehnafsroammFaxigi.m(u1m), (i) (i) (i+1) that x <x <x . Keeping in mind this informa- 0 ∞ 0 value[b/a](i) ,anditdecreaseswhileiincreases. Inpar- tion and Eq. (33), we can explore the ratio b/a for the max pair interaction kernel (32). Equation (33) leads to ticular, [b/a](m1a)x 5, whereas [b/a](m2a)x 12 and ≈ − ≈ − [b/a](3) 18 (see Fig. (1)). Hence, Eq. (35) turned max γ∂a/∂γ =D√x J−1/4(x)/ 2J1/4(x) out to m≈ake−it possible to get some information about h −(cid:0)√2/ πJ12(cid:1)/4(x) . (34) bu/easoefvexn(winitshpoituetosfptehceifyfainctgtthhaetrtahnigseraonfgtheeisreinlepvarinntcvipalle- (cid:0) (cid:1)i 6 known). Indeed, according to the mentioned above, the IV. INTERACTION ENERGY VIA THE PAIR ratiob/a does notexceed[b/a](1) 5 if the scattering WAVE FUNCTIONS max ≈− lengthisnegative. Thissuggeststhatthecontributionof thepairwisepotentialtothekineticenergyismuchlarger The derivation of the kinetic and interaction energies thantheabsolutevalueofthecorrespondingcontribution of a dilute ground-state Fermi gas given in Section II is to the mean energy for a normal-state dilute Fermi gas mathematically adequate. However, from the physical with a negative scattering length at temperatures close point of view it has an obvious disadvantage. Namely, enoughtozero! Theinteractionenergyisnegativeinthis the microscopic information remains hidden in Eqs. (5), caseandalsomuchlarger,iftakeninabsolutevalue,than (17) and (18) due to its implicit usage in Section II. To the sum of the a dependent terms in the Huang-Yang eliminate this shortcoming, εint and εkin are considered − result. Note that for alkali atoms [34] one can expect below witha physically soundapproachbasedonthe in- thatx 10,whichmeansthatb/a >20(seeFig.1). In medium pair wave functions (PWF). view of∼the recent results [21] on|a|tr∼apped Fermi gas, it Forthesakeofconvenience,letusbeginwiththeinter- isalsoofinteresttoconsiderbehaviourofb/ainvicinities action energy. It is well-known that all the microscopic of the special points x(i) and x(i). Varying the magnetic information concerning the N particle system is con- ∞ 0 − field acting on the system of 6Li atoms, the authors of tained in the N particle density matrix. In the case of − − interest the N matrix is defined by the paper [21] were interested in the regime of the Fesh- − bach resonance, for which a , and in the situation → ∞ ̺ (x′,x′,...,x′ ;x ,x ,...,x )= when a 0, as well. The both variants, as it follows N 1 2 N 1 2 N from our→consideration, are characterized by b/a 1. =Ψ∗(x1,x2,...,xN)Ψ(x′1,x′2,...,x′N), (38) | | ≫ Note that passage to the limit a in Eqs. (17) and → ∞ where Ψ(x ,x ,...,x ) is the ground-state normalized (18) is not correctbecause it violates the expansioncon- 1 2 N wave function, x = r,σ stands for the space coor- dition k a 1. On the contrary, one can set a = 0 in F { } | |≪ dinates r and the spin z projection σ = 1/2. It is these equation, which leads to the exact result, beyond − ± also known that actually we does not need to know the the perturbation theory, N matrix in detail. In particular, to investigate the to- − tal system energy together with the kinetic and interac- 3h¯2k2 ¯h2k3b tion ones, we can deal with the 2 matrix defined by ε a 0 = F + F , (36) − kin → 10m 3πm ε (cid:0)a 0(cid:1)= ¯h2kF3b. (37) ̺2(x′1,x′2;x1,x2)= dx3...dxNΨ∗(x1,x2,x3,...,xN) int → − 3πm VZ (cid:0) (cid:1) Ψ(x′,x′,x ,...,x ), (39) × 1 2 3 N Equations (36) and (37) correspond to an unusual and where in general extreme situation that, nevertheless, is experimentally attainable now (see Ref. [20, 21]). The total energy of the system is here equal (or practically equal) to that ...dx= ...d3r. of an ideal Fermi gas, while the interaction and kinetic Z σ Z V XV energies taken separately have nothing to do with those ofagasofnoninteractingfermions. Themostinteresting Let us introduce the eigenfuctions of the 2 matrix − particular case concerns the regime kFb 1, where the ξν(x1,x2) given by ≫ first term in Eq. (36) is negligible as compared to the second one. In this case |εint|≈εkin ∝n. dx1dx2 ̺2(x′1,x′2;x1,x2)ξν(x1,x2)= Thus, the examples listed above show that in phys- VZ ically relevant situations the correct results for the ki- =w ξ (x′,x′), (40) ν ν 1 2 netic andinteractionenergiesof adilute Fermigasgiven by Eqs. (17) and (18) differ significantly (more than by wherew standsfortheν stateeigenvalue. Theseeigen- ν − order of magnitude!) from the pseudopotential predic- fuctions are usually called in-medium PWF [35]. The tions (24) and (25). As it is seen, in a Fermi gas the 2-matrix can be expressed in terms of its eigenfunctions pairwise interaction has a profound effect on the kinetic and eigenvalues as follows: energycontrarytoaclassicalimperfectgas. Andthiswell meets the conclusion on an interacting Bose gas derived ̺ (x ,x ;x′,x′)= w ξ∗(x ,x )ξ (x′,x′), (41) 2 1 2 1 2 ν ν 1 2 ν 1 2 in Refs. [25, 26]. A physically sound way of explaining ν X this feature of quantum gases is to invoke the formalism where it is implied that ofthein-mediumpairwavefunctions. Thisiswhybelow, in Sections (IV) and (V), the interaction and kinetic energies of a Fermi gas are investigated through the prism dx1dx2 ξν∗(x1,x2)ξν′(x1,x2)=δνν′. of this formalism. Z V 7 From Eq. (39) it follows that In Eqs. (47) and (48) 0 if σ =0, 1 if σ 0, dx1dx2 ̺2(x1,x2;x1,x2)=1, (42) ∆(σ)= 6 Θ(σ)= ≥ 1 if σ =0, 0 if σ <0. VZ (cid:26) (cid:26) Now, from Eq. (47) it follows that and, hence, χ (σ ,σ )= χ (σ ,σ ). (49) wν =1, (43) 0,0 1 2 − 0,0 2 1 Xν Then, the Fermi statistics dictates which allows one to interprete the eigenvalue w as the ν ϕ (r)=ϕ ( r)=ϕ (r), (50) probability of observing a particle pair in the ν state. q,Q,0 q,Q,0 −q,Q,0 − − Now let us remind that the total momentum of and for r the wave function ϕ (r) obeys the the system of interest, the total system spin and its → ∞ q,Q,0 asymptotic regime z projection are conserved quantities [36]. This means − that they commute with the N particle density matrix because the latter is permut−able with the system ϕq,Q,0(r)→√2cos(qr). (51) Hamiltonian. As the total pair momentum h¯Qˆ, the to- In turn, for triplet states Eq. (48) yields tal pair spin Sˆ and its Z component Sˆ commute with Z − the total system momentum, total system spin and its χ (σ ,σ )=χ (σ ,σ ), (52) Z projection, correspondingly, they commute with the 1,mS 1 2 1,mS 2 1 N− matrix,too. Ifso,thenone canderivethatQˆ, Sˆ and which leads to Sˆ−are permutable with the 2 matrix. This is why we Z − can choose the eigenfunctions of the 2 matrix in such a ϕ (r)= ϕ ( r)= ϕ (r). (53) way that [35] ν = λ,Q,S,m , wher−e m is an eigen- q,Q,1 − q,Q,1 − − −q,Q,1 S S { } value of SZ and λ stands for other quantum numbers. Now, another boundary regime Hence, in the homogeneous situation one arrives at (see Refs. [35] and [37]) ϕ (r) √2sin(qr). (54) q,Q,1 → ξ (x ,x )=ϑ (r,σ ,σ ) exp(iQR)/√V, (44) is fulfilled when r . ν 1 2 ν 1 2 →∞ Working in the thermodynamic limit N ,V where r = r1 r2 and R = (r1 + r2)/2. As the in- ,N/V = n = const, it is more convenie→nt∞to lea→ve mediumbound−pairstatesliketheBCS-pairsarebeyond ∞the 2 matrix in favour of the so-called pair correlation the scope of the present publication, here we deal only functi−on withthescatteringstates. Inotherwords,onlythesector of the “dissociated” pair states is taken into considera- F (x ,x ;x′,x′)= ψˆ†(x )ψˆ†(x )ψˆ(x′)ψˆ(x′) , (55) 2 1 2 1 2 1 2 2 1 tion. Hence, ν = q,Q,S,m , where q stands for the S relative wave vecto{r. This is w}hy it is convenient to set where Aˆ standsfor t(cid:10)he statisticalaverageofth(cid:11)e opera- by definition tor Aˆ,hanid ψˆ†(x), ψˆ(x) denote the field Fermi operators. The pair correlation function differs from the 2 matrix ϑ (r,σ ,σ )=ϕ (r,σ ,σ )/√V. (45) − ν 1 2 ν 1 2 by the normalization factor (see Ref. [35]), The pair interaction of interest does not depend on F (x ,x ;x′,x′)=N(N 1)̺ (x′,x′;x ,x ), (56) the spin variables, which means that ϕ (r,σ ,σ ) is ex- 2 1 2 1 2 − 2 1 2 1 2 ν 1 2 pressed as so that F remains finite while ̺ approaches zero in 2 2 the thermodynamic limit. Indeed, when V , N ϕν(r,σ1,σ2)=ϕq,Q,S(r)χS,mS(σ1,σ2), (46) , N/V = n const, Eqs. (41) and (56→), ∞taken →in ∞ → conjunction with Eqs. (44) and (45), yield where for the singlet states (S =0) one gets F (x ,x ;x′,x′)= χ (σ ,σ )=∆(σ +σ )sign(σ )/√2, (47) 2 1 2 1 2 0,0 1 2 1 2 1 d3qd3Q = ρ (q,Q)ϕ∗ (r)ϕ (r′) while for the triplet wave functions (S =1) (2π)6 S,mS q,Q,S q,Q,S SX,mSZ χ1,mS(σ1,σ2)= ×χ∗S,mS(σ1,σ2)χS,mS(σ1′,σ2′)exp{iQ(R′−R)}, (57) Θ( σ1)Θ( s2) if mS = 1, where the momentum-distribution function − − − = ∆(σ +σ )/√2 if m = 0, (48)  1 2 S Θ(σ1)Θ(σ2) if mS = 1. ρS,mS(q,Q)= V,lNim→∞ N(N −1)wq,Q,S,mS (58) n o  8 is finite because w 1/V2 (this follows from where q,Q,S,mS ∼ Eq. (43) when V ). For ρ (q,Q) one gets the → ∞ S,mS d3qd3Q relation η = ρ (q,Q) (65) S,mS (2π)6 S,mS d3qd3Q Z (2π)6 ρS,mS(q,Q)=n2 (59) and SX,mSZ ϕ (r)= lim ϕ (r). (66) S q,Q,S resulting from Eqs. (43) and (58). q,Q→0 All the necessary formulae are now discussed and dis- FromEqs.(53)and(66)itfollowsthatϕ (r)=0. This played,andonecanturntocalculationsoftheinteraction S=1 result,takenin conjunctionwiththe low-momentumap- energy. Using the well-known expression proximation of Eq. (64), makes it possible to conclude 1 that Eq. (61) reduces for n 0 to E = dx dx V(r r )F (x ,x ;x ,x ) (60) → int 1 2 1 2 2 1 2 1 2 2 | − | η Z ε 0,0 d3rV(r)ϕ (r)2. (67) int 0 ≃ 2n | | and keeping in mind Eq. (57), one gets the following im- Z portant relation: The triplet states do not make any contribution to the interaction energy in the approximation (64), and this 1 ε = d3r V(r) completely meets the usual expectations. int 2n Z Now,toemployEq.(67),oneshouldhaveanideacon- d3qd3Q ρ (q,Q) ϕ (r)2, (61) cerning ϕ0(r) and η0,0. As to the limiting wave function × (2π)6 S,mS | q,Q,S | ϕ (r), it can be determined by means of the following SX,mSZ si0mple and custom arguments. In the dilution limit the provided the equality pair wave function ϕq,Q,S(r) approaches the solution of the ordinary two-body Schrödinger equation χ∗ (σ ,σ )χ (σ ,σ )=1 S,mS 1 2 S,mS 1 2 (h¯2/m) 2ϕ (r)+V(r)ϕ (r)= q,Q,S q,Q,S σX1,σ2 − ∇ =(h¯2q2/m)ϕ (r) (68) q,Q,S is taken into account. Equation (61) directly connects the interaction energy per fermion with PWF and, thus, with the boundary conditions given by Eqs. (50) and with the scattering waves defined by (53). Hence, in the limit n 0 the quantity ϕ (r) has 0 → to obey the equation ψ (r)=ϕ (r) √2cos(qr) (62) q,Q,0 q,Q,0 − (h¯2/m) 2ϕ (r)+V(r)ϕ (r)=0, (69) 0 0 − ∇ and where ϕ (r) √2 when r . Comparing Eq. (11) 0 ψq,Q,1(r)=ϕq,Q,1(r) √2sin(qr). (63) with Eq. (69)→, for n 0 one→fin∞ds − → The scattering waves are immediately related to the ϕ (r)=√2ϕ(r). (70) 0 pairwise-potential contribution to the spatial particle correlations. Setting ψ (r) = ψ (r) = 0, or, in Tocompletecalculationofthe interactionenergy,itonly q,Q,0 q,Q,1 other words, ignoring that contribution and taking no- remainstofindη . Onecanexpectthatwhennumbers 0,0 tice only of the correlations due to the statistics, one offermionswithpositiveandnegativespinz projections − arrives at the Hartree-Fock scheme. arethesame,themagnitudeofρ (q,Q)appearstobe S,mS So far we did not invoke any approximation when independent of the spin variables. In this case Eqs. (59) operating with the 2 matrix and pair correlation func- and (65) give − tion [38]. However,takenin the regime of a dilute Fermi gas, Eq. (61) can significantly be simplified. Indeed, the ηS,mS =n2/4. (71) lowerdensities,thelowermomentaaretypicalofthesys- Note that Eq. (71) can readily be found in a more rigor- tem. This means that the pair momentum distibution ous way concerning the relation ρ (q,Q) is getting more localized in a small vicinity S,mS of the point q = q = 0 when n 0. Consequently, the 1 low-momentum approximation c→an be applied according V2 d3r1d3r2F2(x1,x2;x1,x2)=nσ1nσ2, (72) to which for n 0 we get Z → where n = ψˆ†(x )ψˆ(x ) . This relation results from d3qd3Q σ h 1 1 i ρ (q,Q) ϕ (r)2 the definition of the pair correlation function (55). To (2π)6 S,mS | q,Q,S | ≃ Z derive Eq. (71) from Eq. (72), one should employ the ϕ (r)2 η , (64) latter in conjunction with Eqs. (47), (48) and (57) and, ≃| S | S,mS 9 then, take account of n = ψˆ†(x )ψˆ(x ) =n/2. Let us placing V(r) by V(ps)(r) = (4π¯h2a/m)δ(r) in Eq. (67) σ 1 1 h i stressthatEq.(71)isnotgeneral. Forexample,whenall and setting ψ(r) = 0 (ϕ(r) = 1), for n 0 one de- → theconsideredfermionshavethespinz projectionequal rives ε(ps) (π¯h2an/m). It is just the leading term in to +1/2, one gets η1,1 =n2 and η0,0 =−η1,−1 =η1,0 =0. Eq. (25in)t. ≃This supports the conclusion that the pseu- As it is seen, in this case the interaction energy result- dopotential is not able to produce correctresults for the ing from Eq. (67) is exactly amount to zero: one should kinetic and interactionenergies of dilute quantum gases. go beyond the approximation defined by Eq. (64) to get Hereletusmakesomeremarksonthe momentumdis- an idea about εint of such a weekly interacting system. tributionofthe“dissociated”pairsρS,mS(q,Q). Thecal- Here it is worth remarking that this week interaction is culationalprocedureleadingto Eq.(67)doesnotinvolve an obstacle that can prevent experimentalists from ob- adetailedknowledgeofthisdistribution. However,itcan serving possible BCS-like pairing of fermions due to an becompletelyrefined. InRef.[37]itwassuggestedtode- extremely low temperature of the BCS-transition. To rive ρ (q,Q) via the correlation-weakening principle S,mS strengthen the interaction effects, it was, in particular, (CWP).AccordingtoCWPthepaircorrelationfunction suggested [19] to complicate the experimental scheme in obeys the following relation: suchawaythatfermionswithvariousspinz projections − would be trapped. In this case the low-momentum ap- F (x ,x ;x′,x′) F (x ;x′)F (x ;x′) (73) 2 1 2 1 2 → 1 1 1 1 2 2 proximation yields a finite result for ε . It is deductive int to go in more detail concerning this situation because when hereanotherchoiceoftheeigenfunctionsofthe2 matrix turnedouttobeconvenientratherthanthatofE−q.(46). |r1−r2|→∞, |r1−r′1|=const, |r2−r′2|=const. The details are in Appendix. In Eq. (73) we set F (x ;x′) = ψˆ†(x )ψˆ(x′) . So, the At last, inserting Eqs. (70) and (71) in Eq. (67) and 1 1 1 h 1 1 i pairmomentumdistributionρ (q,Q),whichappears making use of Eqs. (14) and (16), in the dilution limit S,mS in the expansion of F in the set of its eigenfunctions, one can derive ε π¯h2n(a b)/m, which is nothing 2 int ≃ − can be expressed in terms of the single-particle momen- else but the leading term in Eq. (18). Note that to de- tum distribution n (k)= a†(k)a (k) , that comes into rive the next-to-leading terms in the expression for the σ h σ σ i the plane-wave expansion for F . In the case of inter- interactionenergyviaPWF,oneshouldconstructamore 1 est, when the both distribution functions turn out to be elaboratedmodelsimilartothatofRefs.[25,26]concern- independent of spin variables, this leads to ing a dilute Bose gas. The model like that has to take intoaccountin-mediumcorrectionstoPWFtogobeyond ρ (q,Q)=n Q/2+q n Q/2 q , (74) theapproximationofEq.(64). Thoughthisinvestigation S,mS | | | − | is beyond the scope of the present publication, there are where, by definition,(cid:0)n(k) = n(cid:1)(k(cid:0)). Conclud(cid:1)ing let us σ someimportantremarksonthein-mediumcorrectionsto set, for the sake of demonstration, n(k) = 1 for k k , F the eigenfunctions of the 2 matrix in the next section. n(k) = 0 for k > k and return to Eq. (64). Ins≤erting − F HereitisofinteresttocheckifEq.(67)yieldsEq.(25) Eq. (74) in the right-hand side of Eq. (64) and utilizing when replacing V(r) by the pseudopotential V(ps)(r). this single-particle momentum distribution of an ideal The simplest way of escaping divergences while oper- Fermigas,we arriveatthe left-handsideofEq.(64)due ating with the pseudopotential is to adopt V(ps)(r) = to k 0 when n 0. This example is a good illus- F (4π¯h2a/m)δ(r) in conjunction with the Hartree-Fock tration→of the idea o→f the low-momentum approximation scheme. For example, exactly this way was used in introduced by Eq. (64). the classical paper by Pitaevskii when deriving the well- Thus, in Section IV it is demonstrated how to calcu- known Gross-Pitaevskii equation for the order parame- latetheinteractionenergyofadiluteFermigasfromthe ter of the Bose-Einstein condensation in a dilute Bose first principles, beyond the formula by Huang and Yang gas [4]. A more elaborated variant, which goes beyond taken in conjunction with the Hellmann-Feynman the- the Hartree-Fock framework, requires a more sophisti- orem. Though the results of this section make Eq. (18) cated choice of the pseudopotential which, for the par- physicallysoundandsupporttheconclusionaboutstrong ticular case ofthe hard-sphereinteraction, is of the form influenceofthepairwiseinteractiononthesystemkinetic V(ps)(r) = (4π¯h2a/m)δ(r)(∂/∂r)r [28]. The aim of this energy,thenatureofthisinfluenceisnothighlightedyet. variant is to calculate not only the total system energy The detailed discussion of this nature is given in Sec- but some additional important characteristics (for in- tion V. stance, the pair correlation function) which can not be properly considered in the former way. Complicating V. KINETIC ENERGY VIA THE PAIR WAVE the pseudopotential construction allows one to escape a FUNCTIONS double account of some scattering channels (this fact is known since the Thesis by Noziéres). This double account appears due to the fact that the particle scatter- Some hints as to how to proceed with the problem of ing makes contribution to the pseudopotential. In our the influnce of the pairwise particle interaction on the case, of course, the simplest choice is enough. Now, re- kinetic energy can be found in the Bogoliubov model of 10 a weakly interacting Bose gas and in the BCS-approach. thefunctional (k),weshouldtrytocalculatethekinetic L AsitshowninRefs.[25,26],thereexistssomeimportant energy, starting durectly from n(k) of Eq. (77). This relation which mediates between the pairwise boson in- equation results in teractionandsingle-bosonmomentumdistributionn (k) B in the Bogoliubov model. For the ground-state case this 1+ 1 4 (k) /2 if k , F n(k)= − L ≤K (78) relation is the form ((cid:2)1 p1 4 (k)(cid:3)/2 if k > F, − − L K n (k) 1+n (k) =n2ψ2(k), (75) B B 0 B where ¯h s(cid:2)tandps for the Fe(cid:3)rmi momentum. In the F K where ψ (k) is the(cid:2) Fourier tr(cid:3)ansform of the scattering presentpaper a weakly nonidealgasoffermions is under B part of the bosonic PWF corresponding to q = Q = 0, investigation [39], which means that the single-fermion andn standsforthedensityofcondensedbosons. When momentum distribution approaches the ideal-gas Fermi 0 the pairwise boson interaction is “switched off”, there is distribution in the dilution limit: (k) 0 and F L → K → no scattering. So, bosonic PWF are the symmetrized kF forn 0. Then,Eq.(78)canbe rewrittenforn 0 → → plane waves and ψ (k) = 0. In this case Eq. (75) has as B the only physical solution n (k) = 0, that corresponds B n(k) 1 ℓ(k) Θ(k k)+ℓ(k)Θ(k k ), (79) to an ideal Bose gas with the zero condensate depletion F F ≃ − − − and the zero kinetic energy. On the contrary,“switching where the (cid:2)dilution e(cid:3)xpansions (k) = ℓ(k)(1+...) and on” the pairwise interaction leads to ψB(k)=0, and we L 6 = k (1+...) are implied. Taken together with the arrive at the regime of a nonzero condensate depletion, KF F familiar formula when n (k) = 0 and the kinetic energy is not equal to B 6 zero, as well. d3k A similar situationoccurs in the BCS-model. There is Ekin =V (2π)3 Tknσ(k), (80) again some corner-stone relation mediating between the σ Z X pairwise interaction and the single-particle momentum Eq. (79) leads for n 0 to distribution nBCS(k). It can be expressed[37] as → n (k) 1 n (k) =n2ψ2 (k), (76) 3h¯2k2 2 d3k BCS − BCS s BCS εkin = 10mF + n (2π)3Tkℓ(k)+.... (81) where ψ (k) is(cid:2)the Fourier (cid:3)transform of the internal Z BCS wave function of a condensedbound pair of fermions, ns Note that the characteristic length b given by (16) can is the density of this pairs.“Switching off” the pairwise be rewritten as attractionleadstodisappearnceoftheboundpairstates: ψBCS(k)=0. In this case there are two branches of the b= m d3k T ψ2(k), solution of Eq. (75): nBCS = 1 and nBCS = 0. Below 2π¯h2 Z (2π)3 k the Fermi momentum the first branch is advantageous fromthe thermodynamicpoint ofview, while the second where ψ(k) is the Fourier transform of the scattering one is of relevance above. So, one gets the regime of an waveψ(r)(seeEq.(12)). Keepingthisinmindandcom- ideal Fermi gas with the familiar kinetic energy, often paring Eqs. (17) with (81), we can find for n 0 that → called the Fermi energy. When “switching on” the at- ℓ(k)=(n2/4)ψ2(k). (82) traction, some significant corrections to the momentum distribution of an ideal Fermi gas arise. This corrections Hence, the quantuty (k) is indeed a functional of are dependent of the mutual attraction of fermions and L ψ (r) that reduces to the right-hand side of Eq. (82) make a significant contribution to the kinetic energy ad- p,q,S in the limit n 0. So, our expectations about the re- ditional to the Fermi energy. → lation mediating between the pairwise interaction and Now, keeping in mind the examples listed above, one single-particle momentum distribution in a dilute Fermi can suppose that the relation connecting PWF (strictly gasturnouttobeadequate. LetusremarkthatEq.(79) speaking, the scattering waves and bound waves) with is a good approximation only when calculating the dilu- the single-particle momentum distribution is some gen- tion expansion for the kinetic energy (strictly speaking, eral feature of quantum many-body systems. If so, ex- the leading and next-to-leading terms). However, to go actly this relation has to be responsible for the influ- in more detail concerning the single-fermion momentum ence of the pairwise interaction on the kinetic energy of distribution,oneshouldbebasedonEq.(78)ratherthan the quantum gases. For a ground-state dilute Fermi gas on Eq. (79). Indeed, Eq. (11) can be rewritten as withnopairingeffectstherelationofinterestcanbecon- structed in the form 1 ψ(k)= d3rϕ(r)V(r)exp( ikr). n(k)[1−n(k)]=L(k), (77) −2Tk Z − where (k)standsforafunctionalofthein-mediumscat- FromEq.(10)itfollowsthat d3rϕ(r)V(r)=4π¯h2a/m. teringLwaves ψ (r). To go in more detail concerning Therefore, ψ(k) 1/k2 when k 0. Taken in the q,Q,S ∝ − R →