Seth T. Rittenhouse et al- Hyperspherical description of the degenerate Fermi gas: s-wave...

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Hyperspherical description of the degenerate Fermi gas: s-wave interactions

Seth T. Rittenhouse,1

M. J. Cavagnero,2

Javier von Stecher,1

and Chris H. Greene1

1Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440, USA

2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA

Received 12 July 2006; published 28 November 2006

We present a unique theoretical description of the physics of the spherically trapped N-atom degenerate

Fermi gas at zero temperature based on an ordinary Schrdinger equation with a microscopic, two-bodyinteraction potential. With a careful choice of coordinates and a variational wave function, the many-body

Schrdinger equation can be accurately described by a linear, one-dimensional effective Schrdinger equation

in a single collective coordinate, the rms radius of the gas. Comparisons of the energy, rms radius, and peak

density of the ground-state energy are made to those predicted by Hartree-Fock HF theory. Also the lowestradial excitation frequency the breathing mode frequency agrees with a sum rule calculation, but deviatesfrom a HF prediction.

DOI: 10.1103/PhysRevA.74.053624 PACS numbers: 03.75.Ss, 31.15.Ja

I. INTRODUCTION

The realization of a degenerate Fermi gas DFG in a

dilute gas of fermionic atoms has triggered widespread inter-est in the nature of these systems. This achievement com-

bined with the use of a Feshbach resonance allows for a

quantum laboratory in which many quantum phenomena can

be explored over a wide range of interaction strengths. This

leads to a large array of complex behaviors including the

discovery of highly correlated BCS-like pairing for effec-

tively attractive interactions 15. While these pairing phe-nomena are extremely interesting, we believe that the phys-

ics of a pure degenerate Fermi gas, in which the majority of

atoms successively fill single-particle states, is worthy of fur-

ther study even in the absence of pairing. This topic is ad-

dressed in the present paper, which follows a less complete

archived study of the topic 6.The starting point for this study is that of a hypersphericaltreatment of the problem in which the gas is described by a

set of 3N1 angular coordinates on the surface of a

3N-dimensional hypersphere of radius R. Here N is the num-

ber of atoms in the system. This formulation is inspired by a

similar study of the Bose-Einstein condensate BEC 79.Furthermore, these same coordinates have been applied to

finite nuclei 10. The formulation is a rigorous variationaltreatment of a many-body Hamiltonian, aside from the limi-

tations of the assumption of pairwise, zero-range interac-

tions. In this paper, we consider only filled energy shells of

atoms. This is done for analytic and calculational simplicity,

but the treatment could apply to any number of atoms in

open shells with modest extensions that are discussed briefly.

The main goal of this study is to describe the motion of

the gas in a single collective coordinate R, which describes

the overall extent of the gas. The benefit of this strategy is

that the behavior of the gas is reduced to a single one-

dimensional 1D linearSchrdinger equation with an effec-tive hyperradial potential. The use of a real potential then

lends itself to the intuitive understanding of normal

Schrdinger quantum mechanics. This method also allows

for the calculation of physical quantities such as the energy

and rms radius of the ground state; these observables agree

quantitatively with those computed using Hartree-Fock

methods. The method also yields a visceral understanding of

a low-energy collective oscillation of the gasi.e., a breath-

ing mode.Beyond the intuitive benefits of reducing the problem to

an effective one-dimensional Schrdinger equation, another

motivation for developing this hyperspherical viewpoint is

that it has proven effective in other contexts for describing

processes involving fragmentation or collisions in few-body

and many-body systems 11,12. Such processes would bechallenging to formulate using field theory or the random

phase approximation RPA or configuration interactionviewpoints, but they emerge naturally and intuitively, once

the techniques for computing the hyperspherical potential

curves for such systems are adequately developed. To this

end we view it as a first essential step, in the development of

a more comprehensive theory, to calculate the ground-stateand low-lying excited-state properties within this framework.

Then we can ascertain whether the hyperspherical formula-

tion is capable of reproducing the key results of other, more

conventional descriptions, which start instead from a mean-

field theory perspective.

The paper is organized as follows: Section II develops the

formulation that yields a hyperradial 1D effective Hamil-

tonian. In Sec. III we apply this formalism to a zero-range

s-wave interaction and find the effective Hamiltonian in both

the finite-N and large-N limits. In Secs. III A and III B we

examine the nature of the resulting effective potential for a

wide range of interaction strengths and give comparisons

with other known methods, mainly the Hartree-Fock HFmethod. Section III C is a brief discussion of the simplifica-

tions that can be made in the limit where N. Finally in

Sec. IV we summarize the results and discuss future avenues

of study.

II. FORMULATION

The formalism is similar to that of Ref. 7, but we willreiterate it for clarity and to make this article self-contained.

Consider a collection of N identical fermionic atoms of mass

m in a spherically symmetric trap with oscillator frequency

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, distributed equally between two internal spin substates.

The governing Hamiltonian is

H= 2

2mi=1

N

i2 +

1

2m2

i=1

N

ri2 +

ij

Uintrij , 2.1

where Uintr is an arbitrary two-body interaction potentialand rij = ri rj. We ignore interaction terms involving three or

more bodies. In general, the Schrdinger equation that comesfrom this Hamiltonian is very difficult to solve. Our goal is

to simplify the system by describing its behavior in terms of

a single collective coordinate. To achieve this aim we trans-

form this Hamiltonian into a set of collective hyperspherical

coordinates; the hyperradius R of this set is given by the

root-mean-square distance of the atoms from the center of

the trap:

R 1N

i=1

N

ri2

1/2

. 2.2

So far, we only have one coordinate for the system, but we

need to account for all 3N spatial coordinates and also thespin degrees of freedom. This leaves 3N1 angular coordi-

nates left to define. We have 2N of the angles as the

independent-particle spherical polar coordinate angles

1 ,1 ,2 ,2 , . . . ,N,N. The remaining N1 hyperanglesare chosen by the convention used in Ref. 10, which de-scribes correlated motions in the radial distances of the at-

oms from the trap center,

tan i =

j=1

i

rj2

ri+1,

i = 1,2,3, .. . ,N 1 . 2.3

Alternatively we may write this as

rn = NR cos n1 j=n

N1

sin j ,

0 j

2, j = 1,2, . .. ,N 1 , 2.4

where we define cos 0 1 and j=NN1 sin j 1. For the pur-

poses of this study, the set of 3N1 hyperangles will be

referred to collectively as . The particular definition of thehyperangles will not play a significant role in the actual for-

malism, but we give the definitions here for completeness.

After carrying out this coordinate transformation on the

sum of Laplacians, the kinetic energy becomes 13

2

2M 1

R3N1

RR3N1

R 2

R2 . 2.5

Here M=Nm and is the grand angular momentum

operator which is similar to the conventional angular mo-

mentum operator and is defined by

2 =

ij

ij2 ,ij = xi

xj xj

xi2.6

for all Cartesian components xi of the 3N-dimensional space.

The isotropic spherical oscillator potential becomes simply

1

2m2

i=1

N

ri

2 =1

2M2R2. 2.7

The sum of Eqs. 2.5 and 2.7 gives a time-independentSchrdinger equation HR , =ER , of the form

0 = 22M

2R2

+3N 13N 3

2R2

2

R2 + 1

2M2R2

+ ij

Uintri rj ER3N1/2R,, 2.8where R , has been multiplied by R3N1/2 to removefirst-derivative terms in the hyperradius. These mathematical

transformations have not yet accomplished much. We startedwith a 3N-dimensional Schrdinger equation to solve, and

we still have a 3N-dimensional Schrdinger equation. To

simplify, we assume that is approximately separable into

an eigenfunction of the operator 2, a hyperspherical har-

monic, multiplied by an unknown hyperradial function. Hy-

perspherical harmonics HH; see Ref. 13 for more detailsare generally expressed as products of Jacobi polynomials

for any number of dimensions. Their eigenvalue equation is

2 = + 3N 1, 2.9

where =0,1,2,... and omitted below for brevity stands

for the 3N2 other quantum numbers that are needed todistinguish between the usually quite large degeneracies fora given . The separability ansatz implies that

R, = FR . 2.10

Here we assume that is the HH that corresponds tothe lowest value of the hyperangular momentum that is al-

lowed for the given symmetry of the problemi.e., that is

antisymmetric with respect to interchange of indistinguish-

able fermions. This choice of trial wave function indicates

that we expect the overall energetics of the gas to be de-

scribed by its size; as such, fixing the hyperangular behavior

is equivalent to fixing the configuration of the atoms in the

gas. In nuclear physics this is known as the K harmonicmethod. Alternatively, this may be viewed as choosing a trial

wave function whose hyperradial behavior will be variation-

ally optimized, but whose hyperangular behavior is that of a

noninteracting-trap-dominated gas of fermions.

To utilize this as a trial wave function, we evaluate the

expectation value H, where the integration is takenover all hyperangles at a fixed hyperradius. This approach

gives a new effective linear 1D Schrdinger equation

Hef fR3N1/2FR =ER3N1/2FR in terms of an effective

Hamiltonian Hef f given by

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2

2M d2

dR2

KK+ 1

R2 + 1

2M2R2 +

ij

Uintrij .

2.11

Here K=+ 3N 1/2.We now must force our trial wave function to obey the

antisymmetry condition of fermionic atoms. To antisymme-

trize the total wave function FR

note that Eq. 2.2indicates that R is completely symmetric under all particlecoordinate exchange; thus the antisymmetrization of the

wave function must only affect . Finding a completelyantisymmetric for any given is generally quite dif-ficult and is often done using recursive techniques like coef-

ficient of fractional parentage expansions see Ref.10,14,15 for more details or using a basis of Slater deter-minants of independent-particle wave functions 1619. Weuse a simplified version of the second method combined with

the following theorem, proved in Ref. 15 and developed inAppendix B.

Theorem 1. The ground state of any noninteracting set of

Nparticles in an isotropic oscillator is an eigenfunction of2 with minimal eigenvalue + 3N 2 where is given

by the total number of oscillator quanta in the noninteracting

system:

=ENI

3N

2, 2.12

where ENI is the total ground-state energy of the noninteract-

ing N-body system.

With theorem 1 in hand we may find in terms ofthe independent-particle coordinates. The noninteracting

ground state is given by a Slater determinant of single-

particle solutions:

NIr1,r2, . . . ,rN,1,2, . . . ,N

= P

1pPi=1

N

Rniiriyimiimsi . 2.13

Here i is the spin coordinate for the ith atom and Rnii

ri isthe radial solution to the independent-particle harmonic os-

cillator for the ith particle given by rRnr =Nn expr2/ 2l2r/l+1Ln

+1/2r/l2 where Lnr is an associated

Laguerre polynomial with l =/m. yimi is an ordinary3D spherical harmonic with i as the spatial solid angle for

the ith particle, and msi is a spin ket that will allow for twospin species of atoms,

and

. The sum in Eq.

2.13

runs

over all possible permutations P of the N spatial and spin

coordinates in the product wave function. We now apply

theorem 1 which directly leads to a NI that is separable into

a hyperangular piece and a hyperradial piece:

NIr1,r2, . . . ,rN,1,2, . . . ,N

= GR,1,2, . . . ,N. 2.14

Here GR is the nodeless hyperradial wave function describ-ing the ground state ofN noninteracting atoms in an isotropic

oscillator trap. A derivation of GR is given in Appendix A:

R3N1/2GR = A exp R2/2L2R

L+3N/21/2;

2.15

here, A is a normalization constant and L=/M= l/N.Combining Eq. 2.14 with Eq. 2.15 now gives us :

,1,2, . . . ,N =

P

1pPi=1

N

Rniiriyimiimsi

GR,

2.16

where we must make the variable substitutions in the nu-

merator using Eqs. 2.2 and 2.3. In the following, for brev-ity, the spin coordinates 1 , . . . ,N will be suppressedi.e., = ,1 ,2 , . . . ,N. Interestingly, this mustbe a function only of the hyperangles, and thus all of the

hyperradial dependence must cancel out on the right-hand

side of Eq. 2.16. We are now ready to start calculating theinteraction matrix element

U

int

.

III. ZERO-RANGE s-WAVE INTERACTION

Here we specify Uintr as a zero-range two-body interac-tion with an interaction strength given by a constant param-

eter g:

Uintr = g3r. 3.1

For a small two-body, s-wave scattering length a we know

that g =42a

m20. For stronger interactionsi.e.,

kfa1this approximation no longer holds and g must berenormalized.

Now we must calculate the effective interaction matrix

element given by

Uef fR = gij

3rij . 3.2

Degenerate ground states cause some complications for

this formulation which we avoid by restricting ourselves to

the nondegenerate ground states that correspond to filled en-

ergy shells of the oscillatori.e., magic numbers of par-

ticles. With moderate extensions the degeneracies can be

taken into account by creating an interaction matrix, but the

magic number restriction should still give a good description

of the general behavior of systems with large numbers of

atoms. The total number of atoms and the hyperangular mo-

mentum quantum number are most conveniently expressed

in terms of the number n of single-particle orbital energies

filled:

N=nn + 1n + 2

3, 3.3a

=n 1nn + 1n + 2

4, 3.3b

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kf = 2m

n + 12 , 3.3c

where kf is the peak noninteracting Fermi wave number. In

the limit where N1, we write and kf in terms of the total

number of particles, N,

3N4/3

4, 3.4a

kf2m

3N1/3 . 3.4b

Next we combine Eq. 3.4a with from Eq. 2.16 andcalculate the interaction matrix element.

Since is antisymmetric under particle exchange,we may do a coordinate transposition in the sum rir2 and

rjr1. Each transposition pulls out a negative sign from ,

and we are left with

Uef fR = gij

3r21 = g

NN 1

2

3r21.

Appendix C details the calculation of the matrix element

3r21. The result is

Uef fR = gCN

N3/2R3, 3.5

where CN is a constant that is dependent only on the number

of atoms in the system. While CN has some complex behav-

ior for smaller numbers of particles, we have seen that for

N100, CN quickly converges to

CN23

32N7/2

35

3 1 + 0.049N2/3 0.277

N4/3+

= 0.02408N7/21 + 0.049N2/3

0.277

N4/3+ , 3.6

where the higher-order terms in 1/N were found by fitting a

curve to numerically calculated data. Values of CN/N7/2 are

shown in Table I for several filled shells. Figure 1 shows

both the calculated values of CN and the values from the fit

in Eq. 3.6 versus 1/N. In both the table and the plot we canclearly see the convergence to the large N value of CN 0.024 08N7/2.

We now may write the effective one-dimensional, hyper-

radial Schrdinger equation Hef fR3N1/2FR

=ER

3N1/2

FR, where Hef f is given by

Hef f =

2

2M

d2

dR2+ Vef fR, 3.7

where Vef fR is an effective hyperradial potential given by

Vef fR =

2

2M

KK+ 1

R2+

1

2M2R2 + g

CN

N3/2R3. 3.8

We reinforce the idea that this effective Hamiltonian de-

scribes the fully correlated motion of all of the atoms in the

trap albeit within the aforementioned approximations. As ex-

pected, for N=1, with C1 =0, Eq. 3.8 reduces to a singleparticle in a trap with K as the angular momentum quantumnumber . Note that the form of Vef f is very similar to theeffective potential found for bosons by the authors of Ref.

7. What may be surprising is the extra term of 3N 1/ 2contained in K. The kinetic energy term in Vef f is controlledby the hyperangular momentum, which in turn reflects thetotal nodal structure of the N-fermion wave function. Thisadded piece of hyperangular momentum summarizes the en-ergy cost of confining N fermions in the trap. This repulsivebarrier stabilizes the gas against collapse for attractiveinteractionsi.e., g0.

A final transformation simplifies the radial Schrdinger

equationnamely, setting E=ENIE and R =R2NIR,where

ENI = + 3N2 , 3.9a

R2NI = l2

N+

3

2 3.9b

are the noninteracting expectation values of the energy and

squared hyperradius in the ground state. In the following any

TABLE I. N, , and CN/N7/2 for several filled shells. We can see

that CN/N7/2 quickly converges to the Thomas-Fermi limiting value

of 322/3/353 0.02408 to several digits.

n N CN/N7/2

1 2 01

82 0.0127

2 8 6 0.0637

3 20 30 0.02514 40 90 0.0244

5 70 210 0.02435

15 1360 14280 0.0241

30 9920 215760 0.02409

100 343400 25497450 0.02408

0 0.0005 0.001 0.0015 0.002 0.0025

1/N

0.999

1

1.001

1.002

1.003

1.004

1.005

1.006

CN

/(0.

0241N7/2

)

FIG. 1. Values of CN divided by the large-N limit CN0.02408N7/2 vs 1/N are shown. The circles are the calculated

value while the curve is the fit stated in Eq. 3.6.


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hyperradius with a prime, R, denotes the hyperradius in

units of R2NI. Under this transformation, with the use ofEqs. 3.4a and 3.4b, and in the limit where N, thehyperradial Schrdinger equation becomes

12m*

d2

dR2+

Vef fR

ENI ER3N1/2FR = 0 ,

3.10

where m* = + 3N/ 22. The effective potential takes on thesimple form

Vef fR

ENI

1

2R2+

1

2R2 +

kf

R3, 3.11

where =1024/28353 and = g/l2. We note that the

only parameter that remains in this potential is the dimen-

sionless quantity kf and that in oscillator units = g.In the limit where N the effective mass m* becomes

m*1

163N8/3 . 3.12

For large numbers of particles, the second derivative terms in

Eq. 3.10 becomes negligible, a fact used later in Sec. III C.The behavior of Vef fR versus R is illustrated in Fig. 2

for various values of kf. For kf= 0 solid curve, the non-interacting limit, the curve is exact and the ground-state so-

lution is given by Eq. 2.15. For nonzero values of g, Vef facquires an attractive kf0 or repulsive kf0 1 /R

3

contribution as indicated by the dot-dashed and dashed lines,

respectively. For kf0 the DFG is metastable in a region

which has a repulsive barrier which it may tunnel through

and emerges in the region of small R where the interaction

term is dominant. It should be noted, though, that small Rmeans the overall size of the gas is small. Thus the region of

collapse corresponds to a very high density in the gas. In this

region several of the assumptions made can fall apart, most

notably the assumption dealing with the validity of the two-

body, zero-range potential 21. For kf0 the positive1 /R3 serves to strengthen the repulsive barrier and pushes

the gas further out.

A. Repulsive interactions g0

For positive values of the interaction parameter g we ex-

pect the predicted energy for this K harmonic method to

deviate from experimental values, since the trial wave func-

tion does not allow any fermions to combine into molecular

pairs as has been seen in experiments 1,2,4,5. Our method

can only describe the normal degenerate Fermi gas. Thestrong repulsive barrier for repulsive interactions shown in

Fig. 2 arises as the gas pushes against itself which increases

the energy and rms radius of the ground state. Figures 3 and

4 compare the ground-state energy and average radius

squared, respectively, of 240 trapped atoms, plotted as a

function of kf with a Hartree-Fock HF calculation. Theinset in Fig. 3 shows that the K harmonic energies are

slightly above the HF energies; since both methods are varia-

tional upper bounds, we can conclude that the Hartree-Fock

solution is a slightly better representation of the true solution

to the full Schrdinger equation with -function interactions.

An added benefit of the K harmonic method is that we

now have an intuitively simple way to understand the energy

of the lowest radial excitation of the gasi.e., the breathing

mode frequency. Figure 2 shows that as kf increases, therepulsion increases the curvature at the local minimum,

whereby stronger repulsion causes the breathing-mode fre-

quency to increase. Figure 5 compares the breathing-mode

frequency calculated using the K harmonic method to the

sum rule prediction 22 based on HF orbitals and also thelowest eight radial excitation frequencies predicted by the

Hartree-Fock method. As anticipated, the K harmonic

method and the sum rule method agree that the breathing-

mode frequency will increase with added repulsion. Interest-

0 0.5 1 1.5 2 2.5 3

R R2 NI

0

1

2

3

4

5

6

Veff

R

ENI

FIG. 2. The dimensionless rescaled effective potential Vef f/ENIas a function of the rescaled hyperradius for kf= 0 solid line,kf= 15 dashed line, and kf= 5 dot-dashed line.

0 5 10 15

kf

1

1.05

1.1

1.15

E/ENI

0 5 10 15

kf

0

110-4

210-4

310-4

410-4

(EK-EHF

)/ENI

FIG. 3. The ground-state energy in units of the noninteracting

energy vs kf for 240 atoms calculated using the K harmonic

method curve and using the Hartree-Fock method circles. Inset:the difference in the ground-state energies predicted by the K har-

monic EK and Hartree-Fock EHF methods. Clearly the K har-

monic energies are slightly higher than the Hartree-Fock energies.


053624-5


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ingly both the K harmonic and sum rule methods disagreequalitatively with all eight of the lowest HF excitations. This

difference is attributed to the fact that the Hartree-Fock

method on its own can only describe single-particle excita-

tions while both the sum rule and the K harmonic methods

describe collective excitations in which the entire gas oscil-

lates coherently.

As another test of the K harmonic method we calculate

the peak density of the gas. To do this we first define the

density

r = j=1

N

d3rji=1

N

3ri r

2. 3.13

It can be seen that integration over r using this definition

gives rd3r=N. We recall that our separable approxima-

tion takes the form = FR. Use of Eq. 3.13 and

the antisymmetry of gives

0 = N dRR3N1FR2 d3rN2.3.14

The hyperangular integration is carried out in Appendix D.

The result is

0

NI0= dRR

3N1FR2

R3, 3.15

where NI0 is the noninteracting peak density and

=l3+3N/2

N3/2

(+3N1/2). For the nth filled energy shell the noninter-

acting peak density is given by

NI0 kf

3

62=

1

622m

n + 1/23/2 .Note that the peak density is not given by R3N1FR2

evaluated at R =0: this describes the probability of all of the

particles being at the center at once.

Figure 6 compares the K harmonic and HF peak densities.

Clearly the density does decrease from the noninteracting

value. The two methods are in good qualitative agreement,

but the Hartree-Fock method seems to predict a slightly

lower density, presumably a manifestation of the slightly in-

ferior K harmonic wave function FR.

B. Attractive interactions g0

In this section we examine the behavior of the gas under

the influence of attractive s-wave interactions kf0. Forattractive interactions the gas resides in a metastable region

and can tunnel through the barrier shown in Fig. 2. Figure 7

shows the behavior of Vef f for several values of kf. Thelocation of the local minimum gets pulled down with stron-

ger attraction as the gas pulls in on itself and deeper into the

0 5 10 15k

f

1

1.05

1.1

1.15

1.2

/NI

FIG. 4. The ground-state average squared radius of the gas at-

oms in units of the noninteracting rms squared radius is plotted vs

kf. The calculations considered 240 atoms in both the K harmonic

method curve and Hartree-Fock method squares.

0 2 4 6 8 10

kf

1.8

1.9

2

0

/

FIG. 5. The lowest breathing-mode excitation 0 in units ofthe trap frequency is plotted vs kf for the K harmonic method

solid curve and for the sum rule circles. Also shown as dashedcurves are the lowest eight radial excitation frequencies predicted in

the Hartree-Fock approximation.

0 2 4 6 8 10 12 14

kf

0.6

0.65

0.7

0.75

peakdens

ity(oscillatorunits)

FIG. 6. The peak density in units of /m3/2 vs kfpredictedby the K harmonic solid line and HF circles methods. Both setsof calculations were done for a filled shell of 240 atoms.


053624-6


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center of the trap. Further, as the strength of the interaction

increases, the height of the barrier decreases. In fact beyond

a critical interaction strength c the interaction becomes so

strong that it always dominates over the repulsive kinetic

term. At this critical point the local extrema disappear en-

tirely and the gas is free to fall into the inner collapseregion. The value of c can be calculated approximately by

finding the point where Vef f loses its local minimum and

becomes entirely attractive. This is not exact as the gas will

have some small zero-point energy that will allow it to tunnel

through or spill over the barrier before the minimum entirely

disappears. This critical interaction strength is given by

kfc = 1893

256

1

51/4 15.31.

Just before the minimum disappears, its location is given

by Rmin = 51/4, with an energy of Vef fRmin =5ENI/ 3

0.75ENI. This means that if the gas is mechanically stablefor all values of the two-body scattering lengthi.e., ain this approximation, there must be a renormaliza-

tion cutoff in the strength of the function such that kf15.31 for all a. With this in mind, we begin to examine

the behavior of the DFG for the allowed values of kf.Figures 8 and 9 show a comparison of the ground-state

energy and rms radius of the gas versus kfdown to the kfcas calculated in the K harmonic and Hartree-Fock methods.

Again the Hartree-Fock method does just slightly better in

energy, which we interpret as the Hartree-Fock method giv-

ing a slightly better representation of the actual ground-state

wave function. The energy difference becomes largest as the

interaction strength approaches the critical value. This in-

crease is due to the fact that the Hartree-Fock method pre-

dicts that collapse occurs slightly earlier with kfc 14.1.As the interaction strength increases, the energy and rms

radius of the gas decrease. As kfapproaches kfc the overall

size of the gas decreases sharply and from Eq. 3.15 weexpect to see a very sharp rise in the peak density. This

behavior is apparent in Fig. 10, which displays the peak den-

sity versus kf for both the K harmonic and Hartree-Fock

methods.

While the local minimum present in Vef f only supports

metastable states, it is still informative to examine the behav-

ior of the energy spectrum versus kf, beginning with the

breathing-mode frequency. As the interaction strength be-

comes more negative, Fig. 7 shows that the curvature about

the local minimum in Vef f decreases. This softening of the

hyperradial potential leads to a decrease in the breathing-

mode frequency in the outer well. Figure 11 shows the

breathing mode versus kf predicted by the K harmoniccurve method and also using the sum rule with Hartree-Fock orbitals circles. Also shown in Fig. 11 are the lowesteight Hartree-Fock excitation frequencies for a filled shell of

240 atoms. Again, the K harmonic method agrees quite well

with the sum rule, while both differ qualitatively from the

HF prediction. The sharp decrease in the breathing modefrequency that occurs as kfkfc is a result of the excitedmode falling over the barrier into the collapse region as the

barrier is pulled down by the interaction.

0.5 1 1.5 2

R R2 NI

0

1

2

3

4

5

VeffR

ENI

FIG. 7. The dimensionless rescaled potential curve Vef f/ENI is

shown as a function of the rescaled hyperradius R for several val-

ues of kf. kf= kfc dashed line and from top to bottom kf=5.3, 8.3, 11.3, 19.3 all solid lines

-14 -12 -10 -8 -6 -4 -2 0

kf

0.8

0.85

0.9

0.95

1

E/ENI - 12 - 1 0 - 8 -6 -4 -2 0

kf

0

510-4

110-3

210-3

210-3

(EK-EHF

)/ENI

FIG. 8. The ground-state energy in units of the noninteractingenergy vs kf for 240 atoms calculated using the K harmoniccurve and Hartree-Fockcircles methods. Inset: the difference inthe ground-state energies predicted by the K harmonic EK andHartree-Fock EHF methods. Clearly the K harmonic prediction isslightly higher.

-15 -10 -5 0

kf

0.5

0.6

0.7

0.8

0.9

1

/NI

FIG. 9. The average squared radius of the Fermi gas ground

state in units of the noninteracting value is plotted vs kf. The

calculations are for 240 atoms in both the K harmonic method

curve and Hartree-Fock method squares.


053624-7


8/14

Figure 12 displays some energy levels in the metastable

region as functions of kfnear kfc. Because of the singularnature of the 1/R3 behavior in the inner region, we have

added an inner repulsive 1/R12 barrier to truncate the infi-

nitely many nodes of the wave function in the inner region.

The behavior of the wave function is not correct within this

region anyway because recombination would become domi-

nant, and in any case the zero-range interaction is suspect

beyond kfa 1 and it must be renormalized. Figure 12shows three distinct types of energy level. Levels that are

contained in the local minimum shown in blue are decreas-ing, but not as quickly as the others; levels that are in the

collapse region shown in red have a very steep slope asthey are drawn further into toward R =0; and energy levels

that are above the barrier in Vef f shown in purple havewave functions in both the collapse region and the localminimum. As kf decreases, the higher-energy levels fall

over the barrier into the collapse region earlier, until finally just before kfc is reached the lowest metastable level fallsbelow the ground state. This corresponds to the breathing-

mode behavior seen in Fig. 11. Of course all of this applies

only if there is no further hyperradial dependence of g. If the

interaction becomes density dependent, as is the case in

some types of renormalization, a change in the hyperradius

will change the density and thus the interaction coupling

parameteri.e., ggR.

C. Large-N limit

In Secs. III A and III B the ground-state energy and ex-

pectation values discussed were found by solving the hyper-

radial effective Schrdinger equation 3.7 for a finite num-ber of particles. Here we discuss the behavior of the

N-fermion system in the limit where N is large. To do this we

exploit the fact that the 2/R2 term in the effective Hamil-

tonian, the hyperradial kinetic energy, becomes negligible.

In this limit we see that the total energy of the system is

merely given by E= VeffRmin, as is the case in dimensionalperturbation theory 23. To find the ground-state energy wemust merely find the minimum local minimum for g0value of Vef f. Accordingly, we find the roots of

dVef f

dR=0. Us-

ing Eq. 3.11 we simplify this to

kf=1

3

Rmin Rmin4 1, 3.16

where Rmin is the hyperradial value that minimizes Vef f. The

solutions to Eq. 3.16 are illustrated graphically in Fig. 13;for any given kf we need only look for the value of Rmin

that gives that value. The exact solution of Eq. 3.16 cannotbe determined analytically for all values of kf, but Fig. 13

shows that for kf0 there is always only one positive, real

Rmin that satisfies Eq. 3.16. This corresponds to the globalminimum discussed for repulsive interactions. For kf0

things are a bit more complicated. Figure 13 shows that for

kfckf0 there are two solutions to Eq. 3.16. The inner

-14 -12 -10 -8 -6 -4 -2 0

kf

0.8

1

1.2

1.4

1.6

1.8

2

peakdensity(oscillatorunits)

FIG. 10. The peak density in units of /m3/2 vs kf pre-dicted by the K harmonic solid line and HF circles methods.Both calculations were carried out for a filled shell of 240 atoms.

-15 -10 -5 0

kf

1.5

2

2.5

0

/

FIG. 11. The frequency of the lowest-energy radial transition in

units of the trap frequency vs kf predicted by the K harmonic

method solid line and by the sum rule method circles. Alsoshown are the lowest eight radial transitions predicted by the

Hartree-Fock method.

1217 16 15 14 13

kf

0.7

0.75

0.8

0.85

0.9

0.95

E

ENI

FIG. 12. Color online A portion of the energy spectrum vs kfclose to the critical point kf= kfc. Levels in the metastable region

blue decrease slowly while levels in the collapse region red de-crease very quickly. Energy levels above the barrier in Vef f

s purplereside in both the collapse region and the metastable region.


053624-8


9/14

solution is a local maximum and corresponds to the peak of

the barrier seen in Fig. 7; the outer solution corresponds to

the local minimum where the DFG resides. The local mini-

mum is the state that we are concerned with here as this will

give the energy and hyperradial expectation values of the

metastable Fermi gas. The value of kf where these two

branches merge is the place where the local maximummerges with the local minimum: namely, the critical value

kfc. Any value of kf less than kfc has no solution to Eq.3.16 and thus we cannot say that there is a region of sta-bility. For kf=0, the noninteracting limit, we see that there

are two solutions Rmin =0 and Rmin =1. The solution Rmin = 0

must be discounted as there is a singularity in Vef f at R = 0.

Thus in the noninteracting limit Rmin 1 and Vef fRminENI, as expected.

Substitution of Eq. 3.16 into Eq. 3.11 gives the energyof the ground state, in the large-N limit, as a function of the

size of the gas:

Vef fRminENI

=1 + 5Rmin4

6Rmin2

.

These solutions to Eq. 3.16 immediately give the ground-state energy of the gas versus kf. Figure 14 shows the per-centage difference of the ground-state energy found by this

minimization procedure and that of 240 particles found by

diagonalizing Hef f in Eq. 3.7. We see in Fig. 14 that forkf10 the energy difference is less than 0.5%. As kfgets larger the difference increases, but in the range shown

the difference is always less than 1.5%.

Another result from 3.10 is the fact that in the large-N

limit the commutator Hef f,R0. Thus for any operatorthat is solely a function of the hyperradius OR the ground-state expectation value in the large-N limit is given by the

operator evaluated at Rmin : i.e., OR = ORmin . This tells

us that the large-N-limit wave function is given by

R3N1/2GR2 =R Rmin. We can perturb this slightlyand say that the ground-state hyperradial wave function can

be approximated by a very narrow Gaussian centered at Rmin.

To find the width of this Gaussian we approximate Vef f about

Rmin as a harmonic oscillator with mass m* and frequency

0. We may find 0 by comparing the oscillator potentialwith the second-order Taylor series about Rmin in Vef fi.e.,

0 = 1

m*1

ENI2Vef f

R2 R=Rmin . 3.17The effective hyperradial oscillator length of R3N1/2GRis then given by l0 =/m*0.

The breathing-mode frequency is now simply found to be

0. The frequency in Eq. 3.17 is in units of the noninter-acting energy, so to get back to conventional units we mul-

tiply by ENI/. From Eq. 3.10 we know that m*

= mENINR2NI/

2. Noting that NR2NI= l2ENI/ this leads

to

0B

=

1

ENI

2Vef f

R2 R=Rmin 3.18

for 0B, the breathing-mode frequency in units of the trap

frequency. Using Eq. 3.11 and substituting into Eq. 3.16to evaluate at the minimum gives that

0B = 5 1

Rmin4

.

We note that this is now dependent only on the value of kf;

i.e., for a fixed kf the predicted breathing mode is indepen-

0.2 0.4 0.6 0.8 1

R'min

20

10

0

10

20

30

kf

FIG. 13. Plot of the interaction strength in the dimensionless

combination kf vs the rescaled locations of the potential curveextrema, Rmin =Rmin/R2NI, solutions of Eq. 3.16. Examinationof this plot tells us the behavior of Rmin for all allowed values of kf

including the existence of the critical point kfc located at the mini-

mum of the plot where the maximum and minimum coincide.

-15 -10 -5 0 5 10 15

kf

-1.5

-1

-0.5

0

Energy

difference(%)

FIG. 14. The percentage difference between the energy found by

minimizing Vef f and the energy found by explicitly solving the hy-

perradial Schrdinger equation for 240 atoms.


053624-9


10/14

dent of the number of atoms in the system in the large-N

limit.This prediction can be compared with that predicted by

the sum rule using the formula found by the authors of Ref.

22:

0B = 4 + 3

2

Eint

Eho. 3.19

Here Eint and Eho are the expectation values of the interaction

potential and the oscillator potential in the ground state, re-

spectively. If we insert the expectation values predicted here

by the K harmonic method, we find

0B

=

4 + 3

kf

Rmin5 .

Substituting Eq. 3.16 for kf gives

0B = 5 1

Rmin4

,

which agrees exactly with the frequency predicted in Eq.3.19. Figure 15 shows the breathing mode frequency pre-dicted by Eq. 3.19 versus kf. We see the same behavior inthis plot as was seen in Fig. 11 where the breathing-mode

frequency dives to zero as kfkfc.

IV. SUMMARY AND PROSPECTS

We have demonstrated an alternative approach to describ-

ing the physics of a trapped degenerate Fermi gas from the

point of view of an ordinary linear Schrdinger equation

with two-body, microscopic interactions. The use of a hyper-

spherical variational trial wave function whose hyperangular

behavior is frozen to be that of the K harmonic yields a

one-dimensional effective potential 3.8 in a collective co-ordinate, the hyperradius R. This approach yields an intuitive

understanding of the energy and size of the DFG in terms of

familiar Schrdinger quantum mechanics. Perhaps surpris-

ingly, this approximation gives good results for the ground-

state energy, rms radius, and peak number density of the gas.

These are all in quantitative agreement with the HF method,

while the breathing-mode frequency compares well with that

found using the sum rule method, but gives a qualitative

improvement over the lowest calculated HF radial excitation

frequencies.

The work presented here has been limited to the case of

filled energy shellsi.e., magic numbers of atomsand itshould be readily generalizable, with minor extensions, to

open shells as well. The present results are limited to a

spherically symmetric trap. Generalization to an cylindrically

symmetric cigar-shaped trap should be possible with a judi-

cious choice of coordinates and will be presented elsewhere.

A more comprehensive description of this system would in-

clude the use of higher-order hyperspherical harmonics to get

several adiabatic potential curves. The next-order term would

involve expansions of all Slater determinants for single-

particle excitations 1619 and is beyond the scope of thepresent study. Since the -function Hamiltonian is too singu-

lar to have well-behaved solutions, in the absence of renor-

malization, we defer this discussion to future research.For strongly attractive interactions, this picture predicts an

instability in the DFG in a manner qualitatively similar to the

physics of an explosive collapse of a BEC or Bosenova

7. For the gas to remain stable across the BEC-BCS cross-over regime the strength of interaction potential must be

bounded from below, kf15.31. Preliminary results from

another study 24 indicate that renormalization of the singu-lar -function interaction accomplishes precisely that and ap-

parently prevents collapse. The full interrelation between this

picture and that of pairing in the BEC-BCS crossover region

is beyond the scope of this study and is a subject that will be

relegated to future publications.

ACKNOWLEDGMENTS

This work was supported by funding from the National

Science Foundation. We thank N. Barnea for introducing us

to Refs. 18,19. Discussions with L. Radzihovsky and V.Gurarie have also been informative.

APPENDIX A: HYPERRADIAL SOLUTION TO THE

NONINTERACTING OSCILLATOR

We wish to derive Eq. 2.15, the nodeless hyperradialsolution to the Schrdinger equation for N noninteracting

particles in an isotropic oscillator. The Schrdinger equationfor this system is given by

i=1

N

22m

i2 +

1

2m2ri

2 E = 0 , A1where r1 , r2 , . . . , rN are the Cartesian coordinates for each

atom from the trap center.

To begin we examine the radial Schrdinger equation for

a single particle in an isotropic trap:

-10 0 10

kf

0

0.5

1

1.5

2

B

0

FIG. 15. The breathing mode in units of the oscillator fre-quency in the large-N limit vs kf. Note that as kfkfc thefrequency drops to zero as the local minimum disappears.


053624-10


11/14

22m

d2dr2

+ 1

r2 + 1

2m2r2 Enrfnr = 0 .

A2

The solution to this is well known and is given by

rfnr = An exp r2/2l r

l+1Ln+1/2 r

2

l2 , A3

with energy En=2n ++ 3 / 2 where l =/m and n isthe number of radial nodes in the wave function.

Transformation of Eq. A1 into hyperspherical coordi-nates using Eqs. 2.2, 2.3, 2.5, and 2.7 yields aSchrdinger equation that separates into hyperradial and hy-

perangular pieces. The hyperangular solution is a hyper-

spherical harmonic that diagonalizes 2. The resulting hy-

perradial Schrdinger equation is given by

22M

d2dR2

KK+ 1

R2 + 1

2M2R2 ER3N1/2FR = 0 ,

A4

where K=+ 3N 1 /2. Comparing this with Eq. A2 wesee that if we make the substitutions K, n, mM,

rR, and rfnrR3N1/2FKR, the single-particle ra-

dial Schrdinger equation becomes the N-particle hyperra-

dial Schrdinger equation. With these replacements the solu-

tion is evidently

R3N1/2GKR = AK exp R2/2LR

LK+1LK+1/2R

2

L2 ,

A5a

where L= l/N and is the number of hyperradial nodes inthe N-body system. For =0 this is the same hyperradial

wave function written in Eq. 2.15. The total energy is givenby

EK = 2+ K+ 32 = 2+ + 3N

2 . A5b

Now that we have a hyperspherical solution we can compare

it with the solution to Eq. A1 written in terms ofindependent-particle coordinates r1 , r2 , . . . ,rN. This equa-tion is clearly separable in each coordinate ri, and its solution

is a product of N single-particle wave functions

= i=1

N

fnii

riyimii,

where fniir is given by Eq. A3, i are the spherical polar

angular coordinates of the ith particle, and ym is a nor-mal 3D spherical harmonic. The energy for this independent-

particle solution is given by

E= 3N2

+ i=1

N

2ni + i . A6Now that we have seen that Eq. A1 is separable in bothhyperspherical and independent-particle coordinates, we may

compare Eqs. A5b and A6 to find that

2+ = i=1

N

2ni + i. A7

We should note that this does not mean that the independent-

particle solution and the hyperspherical solution are the

same, only that the hyperspherical solution FR mustbe a linear combination of independent-particle solutions of

the same energy.

APPENDIX B: PROOF OF THEOREM 1

Here we will use the results from Appendix A to prove

theorem 1. We proceed by assuming that there exists a fully

antisymmetric, ground-state, hyperspherical solution to the

Schrdinger equation for N noninteracting fermions in an

isotropic trap with energy given by Eq. A5b with 0. Ifwe can show that this leads to a contradiction, then we have,

from Eq. A7, that there is only one for all of the ground-state configurations. From Appendix A we know that this

hyperspherical solution must be a linear combination of an-

tisymmetric, degenerate, ground-state solutions in

independent-particle coordinates:

FKR,1,2, . . . ,N

=

Dr1,r2, . . . ,rN,1,2, . . . ,N,

where each D is of the form of Eq. 2.13, is used todistinguish between degenerate states of the same energy,

and FKR is given by Eq. A5a. Again, for brevity, wesuppress the spin coordinates 1 ,2 , . . . ,N in . We nownote that the hyperradius R is completely symmetric under

all transpositions of particle coordinates; thus all of the trans-

position symmetry must be contained in the function .From we construct a new completely antisymme-trized hyperspherical solution F0KR which has en-ergy

E= + 3N2 .

Use of Eqs. A5b and A7 gives

=ENI

3N

2 2, B1

where ENI=3N/2+i=1N 2ni +i is the ground-state en-

ergy as defined by any of the functions Drii=1N . Thus our

new function F0KR has energy

E= ENI 2,

which would lie below the ground-state energy, a contradic-

tion unless = 0.

In the above analysis we assumed a degenerate set of

solutions at the ground-state energy. For nondegenerate so-

lutions the proof becomes trivial, as any nondegenerate so-

lution must be the same no matter what coordinate system it

is expressed in. From this the rest of the proof follows in the

same way, and the final, unique for this system is given by

Eq. B1,


053624-11


12/14

=ENI

3N

2. B2

APPENDIX C: CALCULATION OF THE s-WAVE

INTERACTION MATRIX ELEMENT

To calculate CN it is useful to start with a more general

interaction. We assume that the interaction term in the totalN-body Hamiltonian is such that at a fixed hyperradius

Uintrij is separable into a hyperradial function times a hy-perangular integrali.e.,

Ur12 vRv . C1

From properties of the function and Eq. 2.4 it is easy tosee that Uintr = g

3r fits this criterion. While v mighthave some very complex form, it will be seen shortly that

only the form of VR and Uintrij will matter:

Uef fR = ij

Uintrij .We may again use the antisymmetry of to exchange rir2 and rjr1 and arrive at

Uef fR =NN 1

2Uintr21 .

From Eq. C1 we can write that

Uef fR = VRNN 1/2, C2

where =v is the analog of gCN/N3/2 in Eq.

3.5. To find we may substitute in the definition from Eq. 2.16, multiplying by R3N1GR2 on both sides,

integrating over R, and using Eq. C1 to replace VRv

,which gives a simple equation that may be solved for :

= , C3a

= R3N1GR2VRdR , C3b

=NN 1

2

j=1

N

d3rjD*rii=1

N Uintr21Drii=1N ,

C3c

where Dr

ii=1N

is the Slater determinant defined in Eq.2.13 and GR is defined as in Eq. 2.15. We have also

used the fact that R3N1dRd= j=1

N

d3rj 13. can now be

found as a ratio of two integrals.

The integral on the left-hand side LHS of Eq. C3acan be calculated directly. The integral on the right-handside of Eq. C3a is now a diagonal, determinantal matrixelement. may be drastically simplified by using the or-thogonality of the single-particle basis functions for detailssee Ref. 25, Sec. 6-1:

=1

2

i,j=1

N

d3r1d3r2i*r1j*r2Uintr21ir1jr2 msi

msji

*r2j*r1Uintr21ir1jr2 , C4

where ir is the ith single-particle spatial wave functionthat appears in the product in Eq. 2.13i.e.,

rir = Nnii exp r2/2l2r/li+1Lni

i+1/2r/l2yimi.

C5

With this result we can now specify to the s-wave inter-

action. Following Eq. C1 we may identifyUintr21g

3r21 and V21RR1 /R3. With this and Eq.

2.15, the integral in Eq. C3b is found to be

=

+ 3N 12

L3 + 3N

2

. C6

Evaluating Eq. C4 begins by integrating the function overr2. This is simple, and we can clearly see that the two terms

in the integral in Eq. C4 have a common factor ofir1

2jr12. Factoring this out gives

=g

2

i,j=1

N

1 msimsj d3r1ir12jr12 . C7This sum runs over all spatial and spin quantum numbers in

a filled energy shell; we may break this up into two factors,

a spin sum and a space sum:

=g

2

msi,msj=1/2

1/2

1 msimsj d3r1

r12

r12

= g d3r1

r12

r12 , C8

where the Greek letters and stand for the set of spatialquantum numbers n , , m. The spin sum is trivial and isgiven by msi,msj=1/2

1/2 1 msimsj =2. We may now calculate

CN for any given filled energy shell using Eq. C8 and plug-ging into the relationship

CN = N

3/2

g

. C9

This has been done for the first 100 filled shells, the results

of which are summarized by Fig. 1. CN for a selection of

shells is given in Table I.

To extract CN in the limit as N we may examine the

expression for in Eq. C8. We may note that the sumr =r

2 is the definition of the density of a spin po-

larized degenerate Fermi gas of noninteracting particles in an

isotropic oscillator. In the limit where N the trap energy

of the noninteracting gas dominates the total energy. This


053624-12


13/14

means that the Thomas-Fermi approximation becomes exact

in this limit. Thus we may write that

r =1

622m

2 3/21 m2r2

23/2 , C10a

Nms

= d3rr . C10bThe chemical potential may be found by the condition

given in Eq. C10b where Nms

is the number of particles

with the same spin projection ms. The system we are consid-

ering is an equal-spin mixture so that N=N=N/2. Thus we

find that =3N1/3. Plugging this into Eq. C8 gives

= g r2d3r= g23

256

315

N3/2

3l3. C11

We insert this into Eq. C9 with Eq. C6 to find that

CN =

2

3

256

3153N3/2

+ 3N2

+ 3N 12 . C12

Using Eq. 3.4a for the large-N behavior of , in the limitN,

CN23

32

353N7/2 , C13

which is the quoted large-N behavior in Eq. 3.6.It should be said that the same formalism that is presented

in this paper can be applied to this system in center-of-mass

coordinates. This is done by first choosing an appropriate set

of Jacobi coordinates: xi =i/i + 1j=1i rj/i ri+1 for i= 1 , . . . ,N1 with the center-of-mass vector defined as xcm=j=1

N rj/N. Hyperangular coordinates are now used to de-scribe the 3N3 degrees of freedom in the Jacobi coordi-

nates where R2 =j=1N1xj

2/N=j=1N rj

2 xcm2 /N. The 3N4 hy-

perangles needed are now defined with respect to the lengths

of the Jacobi vectors in the same way as in Eqs. 2.3 and2.4. Under this coordinate transformation we find theHamiltonian to be given by

H= HCM + HR,,

where HCM is the Hamiltonian for the center-of-mass coor-

dinate xcm and HR, is an operator entirely defined by hyper-

spherical coordinatesi.e.,

HCM = 2

2mcm

2 +1

2m2xcm

2 ,

HR, = 2

2M 1

R3N4

RR3N4

R

2

R2 + 1

2M2R2

+ ij

Uintrij.

With this Hamiltonian we make the ansatz that for H

=E, =xcmGR, where is a wave function

describing the center-of-mass motion, GR is a nodeless hy-perradial function, and is the lowest hypersphericalharmonic in the 3N 4 angular coordinates. We are then

looking for the matrix element HR, where the inte-gral is taken over all hyperangles at fixed R. Theorem 1 still

applies with the added idea that is given by the lowest

s-wave state of the center of mass in an oscillator. From here

the analysis presented in this section for trap centered coor-

dinates still holds with the added change that the dimension

of the hyperradial integral in Eq. C3b is three dimensionssmaller. This leads to a factor in the interaction matrix ele-

ment given by

CN

+ 3N 12

2

+ 3N2 + 3N 2

2

CN.

For smaller N this factor changes the interaction consider-

ably, but for larger N it quickly goes to 1. Thus, in the large-

N limit, besides extracting the center-of-mass sloshingmodes, there is very little difference from the trap center

coordinate systems in this Jacobi coordinate formalism.

APPENDIX D: CALCULATING 0

We start from Eq. 3.14:

0 = N dRR3N1FR2 d3rN2.If we can find d3rN

2, then we have the solu-

tion. From Eq. 2.4 and properties of the function we may

say that

N d3rN2 = R3

.

We multiply this on both sides by the noninteracting hyper-

radial function R3N1GR2 and integrate over R:

N 3rN2GR2R3N1dRd

= GR2

R3R3N1dR . D1

Inserting GR from Eq. 2.15 into the right-hand side thisgives

GR2

N3/2R3R3N1dR =

3N 1/2

L3 3N/2. D2a

From the definition of in Eq. 2.16 we see that the left-hand side of Eq. D1 is a determinantal matrix element of asingle-particle operator integrated over all independent-

particle coordinates:


053624-13


14/14

N 3rN2GR2R3N1dRd

= Nj=1

N

d3rjDrii=1N 23rN, D2b

where Drii=1N is the Slater determinant wave function de-

fined in Eq. 2.13. Referring to the definition of the density in Eq. 3.13 we see that this is merely the peak density ofthe non-interacting system NI0. Thus

= NI0l3 3N/2

N3/2 3N 1/2.

The peak density is then given by

0 = dRR3N1FR2

R3,

which is what we were seeking to show.

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