Dilute Fermi gas: kinetic and interaction energies

0.32 MB·English

by  A. A. Shanenko

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 be-
tween 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-bodySchr¨odingerequation (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 men-
tioned 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 en-
ergies 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 ma-
trix 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 Schr¨odinger 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 Nozi´eres). This double ac-
count 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 →