quantum-mechanical many-body problem with hard-sphere interaction

Progress of Theoretical Physics, Vol. 19, No. 6, June 1958

Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction

Ryuzo ABE

699

Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo

(Received January 27, 1958)

Theory of the surface charge pseudopotential for the hard-sphere potential is developed. It is shown that the formulas known to be exact are obtained by the present method. The excitation

energy spectrum of the Bose system is calculated, and is shown to have the nonmonotonic behaviour similar to the roton spectrum in liquid He4. The dispersion formula for the sound velocity is calculated and compared with that derived by Landau and Khalatonikov.

§ 1. Introduction

In the previous paper1> we have suggested that the " surface charge pseudopotential "

method for the hard-sphere potential is more suitable for the investigations of the excita

tions in the short-wave regions than the "point charge pseudopotential" method. The

purpose of this paper is to construct the surface charge pseudopotential in a systematic

way.

Before entering into the main subject, it may be helpful first to give a simple example

as an illustration of our method, the spherically symmetric wave function of a system of

two particles with hard-sphere interaction. The wave function in the center-of-mass coordinate system then satisfies the equations

(d2jdr2+k2) cft=O (r >a)

cft=O (r<a)

(1 ·1)

where a is the hard-sphere diameter, r= !xl, with x the relative position vector, and eft the wave function multiplied by r. We replace the above equations by

(d2jdr2 +k2) cft=Af3 (r-a) (1·2)

without any condition in the region r < a, and see how the solution of ( 1 · 2) reproduces

the vanishing of eft for r <a. Here A is an arbitrary constant which is not zero. To

settle the problem we impose the boundary conditions

eft= 0, for r = R and r = 0, ( R > a) . (1· 3)

A complete set of eigenfunctions in the region 0 r < R satisfying the condition

(1 · 3) are

(1·4)

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700 R. Abe

with eigenvalues ten= rrn / R. Expanding ¢, a solution of ( 1 · 2), in terms of the above

set, under the assumption that k is equal to none of ten, we obtain

R

¢(r) oo A J 2_i o (r1 -a) ¢n (r1) dr' ¢n (r)

n=l k2-tc 2 n 0

.--··1----·-- {cos-n_rc_ (r-a) n a 2-n2 R

with a= kR/ 7r. By using the equations

co

::i~ cos nO/ (a2-n2) = (n/2) cos a (rr -t)) /a sin an -1/2 a 2

, (0 (} 2rr) n=l

(nj2)cosa(n tJ)jasinan-1j2a2, ( 21r (} 0),

we have

¢ (r) = (ARjna sin an) sin {an (1- r / R)} sin (ana/ R), (r >a)

=- (ARjna sin an) sin {an (1-a/ R)} sin ( anrj R), (r a).

(1 ·Sa)

(1·5b)

From (1·5b) it follows that ¢(r) identically vanishes for r a once it vanishes at r=a. We have then

which are just the exact eigenvalues for the present problem. It is easily verified that

the same considerations can also be applied to the higher partial waves.

we condude from this argument that the wave function vanishes inside the hard

sphere if the additional term Aa (r-a), which corresponds to the surface charge on the sphere, is added to the Schroqinger wave equation and once the solution vanishes on the surface of the hard-sphere. In other words, the hard-sphere potential can be replaced effectively by the term Aa (r-a) in this case. On the other hand, in the point charge pseudopotential method the hard-sphere potential is replaced by the differential operator which includes the term a (r) ; in addition, the solution is extended to the region 0 < r <a and does not vanish inside the hard-sphere though it does on the surface of the hardsphere. These points are the essential differences between the two methods.

It should be pointed out here that (1 · 2) includes another type of solution by the

requirement ¢(a) =0 alone. This is the case when ana/R=nn, (n=1, 2, ... ). Then

¢ (r) vanishes for r? a but does not vanish for r <a. Of course, the solution of this

type should be excluded from the solutions of ( 1 · 2) . We shall return to this point later in the next section (see also the footnote in § 3) .

§ 2. Generalized two-body system

In the previous section we have demonstrated a simple example as an illustration of our method. In this section we proceed to a more general case in which the in

homogeneous term appears in the two-body Schrodinger equation and find an effective

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Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction 701

operator which is equivalent to the hard-sphere potential. Once such an operator is

found in this case, generalization to a many-body system is straightforward.

In accordance with the above statements, we consider the equation

(fll

with the boundary condition

(72 2

lJl' 0 for jx1 x2 j a.

Here 'h2 JC

2/2m corresponds to the energy eigenvalue per particle.

We assume here :

(a) [(xu x2) does not include the singularity J(r12-a) and

(b) f( Xu X 2) vanishes for I X1 - x2 1 a, with relevant conditions at infinity.

(2 ·1)

(2·2)

As we have done in § 1, we replace (2 ·1) and (2 · 2) by the equation including

the surface charge at I X 1 - x2 \ =a . Thus, we consider the equation

(fl/ (72 2 2J(r-a) b exp(iK·X) Yzm(O, ([J)Aij. lj.

(2. 3)

where K and X are the wave vector and the coordinate of the center of mass, respec

tively, Yzm the normalized spherical harmonic, r, fl, (/) the polar coordinates of the relative

position vector x ( = x1 - x2) , and )ct a constant to be determined later on. a stands

for K, l and m. In the following we denote the unit vector with z-direction as z and write Yzrn(H, cp) as Yzm(x·z).

Expanding lJI' and f in terms of exp (iK ·X) as

(2·4)

(2·5)

we have

(f72 +k2)cp(x) =f(x) bJ(r-a) Yzm(x·z))lj. (2·6) l.m

where k ts the magnitude of the wave vector for the relative motion defined by

(2·7)

and the suffix K is not written down explicitly.

In solving (2 · 6), we impose on cp (x) the periodic boundary condition in a cubical

box of volume V and expand cp ( x) in terms of exp (iq · x), where q is defined by

n3 ± integers.

Then the solution of (2 · 6) is given by

(/) (X) = J G (X' x') { f( x1) + ~ a ( r1 -a) Ylm ( x' . z) )a;} dx'' (2·8)

where G ( x, x 1) is the Green function defined by

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702

Using the equation

R. Abe

G(x, x') =-1-::E-1 - exp[iq · (x-x')]. v q k2-q2

co

exp(iq·x) = ::E (2l+1)i 1jz(qr)Pz(q·x) l=O

and replacing ::E by an integral, we have q

00

G(x, x') = (1/4n) ::E (21+1) Pz(x·x') Gz(r, r1)

l""'O

where

(2·9)

(2 ·10)

(2·11)

(2 ·12)

This integral is evaluated by means of the theory of residues (taking the principal values)

to give

G1 (r, r') =kjz(kr)nt(kr'), (r_:<=r')

=kn1 (kr) jz (kr'), (r > r').

Now, if we expand f(x) in the spherical harmonics,

f(x) = ~fzm (r) Ytm (x ·z), l,m

we have 00

tp (x) = ~ { J G1 (r, r') [ftm (r') Act() (r' -a)] r12 dr'} Yzm (x · z), 0

with the aid of the theorem,

l

P1 (x·x') (4nj2l+ 1) ~ Ytm(x·z) Yzm* (x' ·z). m=-l

From (2 ·15) the solution for a becomes

co

tp(x) = 2J{k f n1 (kr')fzm(r')r'2 dr'+kActa2 n1 (ka)}jz(kr) Yzm(x·z), l,m J

a

(2 ·13)

(2 ·14)

(2. 15)

(2 ·16)

if we use the assumption (b) mentioned previously. We see from (2 ·16) that tp(x) = 0 for r <a once it vanishes at r =a, that is, if Act is determined by

00

J n1 (kr')fzm(r1)r12 dr'+Acta2 n1 (ka) 0. (2 ·17) a

On the other hand, the solution outside the hard-sphere is given by, with the aid

of ( 2 · 13) , ( 2 · 15) and ( 2 · 17) ,

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r

+k J [ nz (kr) jz (kr') - jz (kr) nz (kr')] r'2fzm (r') dr'} Yzm (x · z), (2·18) a

which is shown to be the solution of (2 · 6) by a direct substitution. It is easily seen that the above solution does not identically vanish, since if (/) (X) 0 for r ~a, then f(x) 0 for r>a, [see (2·6)] or fzm(r)oc/J(r-a) [see (2·18)]. The latter case is forbidden by the assumption (a). Hence the appearance of the solution discussed at the end of § 1 is rather exceptional and, in general, such a solution is automatically excluded from the solutions of (2 · 6). (See also the footnote in § 3).

In order to have an effective Hamiltonian which is equivalent to the boundary condition · for a, it is necessary to eliminate the constants )r:t' s from the equation by using some operational procedure. This may be performed by a differential procedure in the pseudopotential method, but is not suitable here, for the wave function has the discontinuous derivative at r=a. We shall therefore choose another way in which )r:t's are eliminated by an integral procedure. It is not, however, convenient to use (2 ·17) or (2 · 18), since these equations do not take account of the periodic boundary condition which is to be imposed on the wave function. Thus we shall make use of the original form for the solution expressed in ( 2 · 8) ,

(j) (x) =V-1 ~ exp.(iq ·')x) I f(x') exp ( -iq · x') dx' + :8)r:t Yzm (x ·z) G1 (r, a) a2• q P-q· J l,m

(2·19)

Multiplying both sides by Yzm * ( x · z) and integrating over the solid angles, and by using

J Yzm * (x ·z) exp(iq · x) d.!2=4rril Ytm * (q ·z) jz (qr),

we have

47r ::8 it Ylk; * ( ~. z) jz (qa) v-1 r exp ( -iq. x') f(x') dx', (2. 20) a2 Gz (a, a) q -q J

from the condition that (/) ( x) should vanish at r =a . This equation together with ( 2 · 3) and (2 · 5) leads to

(2. 21)

as an equation which replaces the .boundary condition for r a. Here K12 is an integral operator defined by

K - - ~~ ( 2l + 1) it ( 'K X) ,~, Pz ( q . X) ( ) V 2 J ( K X' ') dX' d ' 12 - .L..J-~--~exp t • 2...J jz qa - exp -i · -iq ·x x. Ic,t a2Gz (a, a) q k2-q2

(2·22) We have so far implicitly assumed that k2 is equal to none of q2• Since we have

shown in § 1 that exact eigenvalues are obtained under this assumption in the twc-body

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704 R. Abe

problem, we may take it for granted that this still remains to hold in the many-body problem. A more complete discussion on this problem will be given in the following

paper2) from a general point of view.

§ 3. Many-body system (Fermi particles)

In this section we will try to find a many-body operator equivalent to the hardsphere potential, using the results obtained in § 2. For a while, we neglect quantum statistics ; this will be taken into account after the hard-sphere potential is replaced by

an equivalent operator. In this section we shall discuss the Fermi system and in the

next section the Bose system. In our theory, the equivalent operator is independent of quantum statistics, just as the pseudopotential obtained by Huang and Yang:l) is.

It is clear that the product vlJ! ( v : hard-sphere potential) is a delta function at

j xi- xJ J =a, which we shall write ~ ( riJ- a) tiJ lJ! with tiJ some operator to be determined later on. The Schrodinger equation then becomes

Writing this equation as

(f7/ f7l + 2JC2) qr

"C1 (f7 •. 2 + 1C2) 1TI' "C1 I) ( ) ,_ 11/' L.J • ':l:' L.J u riJ-a tiJ 'f:'. i i<j

~ (f7 n2 1C2) lJ! n::f1,2

(3 ·1)

(3 ·1)'

we consider the first and second terms on the right-hand side as corresponding to f(x1

,

x2) · in ( 2 · 21) ( X 3 , x4 , · · ·, x N are considered as parameters) . Here the prime on 2_j means that the pair (12) is excluded from the summation. Then it is easily seen that the function f(x1 , x 2) defined in this way satisfies the assumptions (a) and (b) mentioned at the beginning of § 2, and hence the results obtained in § 2 can be applied.

Substituting f ( Xn X 2) defined above in ( 2 · 21) and equating K1 2{ to t 12 W in ( 3 · 1) 1

(see also the following paper), we have

f12qr K12[ 2J (f7n2 JC

2) 2J'~(rkz a)tkz]lJ! n::f1;2 k<l

from the condition that the wave function should vanish for r12 <a. If we use ( 3. 1) 1

and drop W, * we have

Or, if we write

UJ2 = K12 (f7/ f7 / + 21C2)

the equation for the t-operator becomes

f12 = U12- K12 ~ (rlz-a) f12 ·

(3·2)

(3. 3)

(3 ·4)

* As a matter of fact any operator P12 defined by P121Jf=O may be added to t12 and thus t 12 is not uniquely defined. For our. purpose, however, it is sufficient to find one operator equivalent to the hardsphere potential.

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In this way, the effective operator which is mathematically equivalent to the boundary

conditions imposed on the wave function is found.* To carry out the further calcula

tions, it is convenient to make use of the second-quantization formalism ; we consider a

system of Fermi particles with spin 1/2, and introduce the field operator ¢(1) which

can be expanded in a Fourier series:

¢(1) = v- 112 :Sa(t exp(ika:. xl) a (sen (}1) (3·5) a:

where (} is the spin coordinate and a(t, aa: *, respectively, denote the annihilation and

creation operator for the single particle state of momentum k(t and spin sa:. The total

Hamiltonian can be written as

X (h2/2m) Jr¢*·P¢dr (f52/4m) J¢*(1)¢*(2)a(r12 -a)t12 ¢(2)¢(1)dr1 dr2,

(3. 6)

where dr means the integration over the coordinates together with the summation over

the spin coordinates. Introducing ( 3 · 5) into ( 3 · 6), we have

X= (f52/2m) 2~k/aa:*aa:+ (f52/4mV) 2Jaa:*a~*a,a.\a(sa:, s.\)a(sr~, s,)t(a/9, J.r), a: <t~i.A

(3·7)

where t(a/9, ..<r) is defined by

t(af9, Ar) = (1/V) J J exp ( ik(t. x1 ik,~. x2) a (rl2-a) t12 exp(ikA. x1 ik,. x2) dx1 dx2.

(3·8)

The matrix element t(a/1, Ar) should be determined by the linear integral equation

( 3 · 4) . Let us first calculate the matrix elements of the inhomogeneous part and the

kernel of the integral equation. If we define K ( af9, }. r) with t replaced by K in ( 3 · 8),

we have

K (a/9, l.r) = -4r.a (af9, J.r) ~ (2l + 1) t=o Gz (a, a)

where

Gz(a, a) =kjz(ka)nz(ka),

(3 ·9)

(3 ·10)

k<t:~=(k<t-k,l)/2, k). 1 =(k).-k,)/2 and k2 K

2 (kJ..+k,) 2/4. (3·11)

In (3·9) a(a/9, J.r) expresses the conservation of total momentum, i.e. r'J(a/9, l.r)

a(ka: k[>,-k).. k,).

* The solution of the eigenvalue problem. (3 ·1) with tij given by (3 · 4) m.ay include the wave function

of the following type: 1Jf=O for rij>a and 1/f-:;£:-o for rtj<a, since in this case f(x) =0 (r<a) and the

discussion in § 2 can equally be applied; therefore K12 does not exclude such a solution. This may be

excluded, however, as long as the perturbation procedure starting from the free particles is used, for the

energy eigenvalue corresponding to such a solution becomes co as a----)0 in order that 1Jf may stay normalized

and is not connected to the counterpart for a system of free particles.

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706 R. Abe

If we further define u(a(d, J.r) analogously to (3·8), we have from (3·3)

OJ

u(a(d, J.r) = -snJ(a(d, J.r) 2J (2!+ 1)Gi"1 (a, a)h(kll.fla)jz(kA1 a)Pz(k(1.r. ·k)-1 ). l=O

(3 ·12)

Fit;:tally, from (3 · 4) it follows that the integral equation for the t-operator becomes in the momentum representation,

t(a(d, J.r) =u(a(d, J.r)- (1/V) 2JK(a:(d, a-r)t(a-r, J.r) (3 ·13) 0,"1:

with the aid of the equation

J(r12 -a)t12 exp(ik).·X1 ik,·x2) (1/V) ~exp(ik0 ·X1 ik,·x2)t(o-r, Ar). Ci,'t

Before discussing the solution of ( 3 · 13), we wish to compare the u-matrix with the

corresponding matrix obtained by the point charge pseudopotential method. The latter

can be written as

(3 ·14)

where { } r=o means that the substitution r= 0 is carried out after the operation of a11.'s and the summation over a:, p, A, r are performed.4

) For the wave function which is

regular at riJ 0, we may put r=O in (3 ·14) and are led to

Up ( a:(d, A r) =- sna ( a:(d, A r) 2.J (2! + 1) tan al k~r'> k~i k-'2l-l pl (k(/.fl. k),). ( 3. 15) l

If the phase shifts are expanded in powers of k a and the lowest order term is retained,

(3 ·15) is reduced to

( (.) · ) ~ ( (.) ·• ) ,, { ( f ) 11} -2 2t kz kl p (k k ) up a:jJ, Ar =87Tau a:jJ, Ar L...J 2 -1 .. a 11.[1 Aj l (!.[!' ),j • (3 ·16) l

On the other hand, under the same approximation, that is, if G1 (a, a) IS expanded

m terms of k a and the lowest order term is retained, ( 3 · 12) is reduced to

OJ

u(a:fi, ).r) =BnaJ(a:(d, J.r) 2.J (2l+1) 2 jz(kCY.r'>a)jt(k).1 a)P1 (kCY.;>.·k). 1 ). (3·17) l=O

If jz(k(/.r'>a) and jz(k).1 a) in the above expression are expanded in terms of a, the lowest

order term turns out to be identical with u 11 ( a:(d, ). r) given by ( 3 · 16) . The essential

difference between (3 ·16) and (3 ·17) lies in the behaviours of the u-matrix for large

momentum transfers, that is, it is oscillatory in (3 ·17), while monotonic in (3 ·16). Such behaviours of u7) ( a:(d, ). r) indicate that the point charge pseudopotential method

cannot treat correctly the problems in which the large momentum transfers are important,

e. g. the roton spectrum in liquid He\ as we have discussed previously.1)

Returning now to the integral equation (3 ·13), we try to solve the equation by an

iteration procedure. The first iterated term is

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t1 (a{j, J.r)=-(1/V)~K(a{j, O"r)u((rr, J.r). (3·18) ·a,'t

By substituting ( 3 · 9) and ( 3 · 12) in this and replacing the summation over k 0 , by

integration, we have

If we formally carry out further iterations, t(a{3, J.r) becomes

t(a{j, J.r) =u(a{j, J.r) +u(a{j, J.r) +u(a{j, J.r)

which is obviously divergent. This indicates that there ts no solution of the integral

equation (3 ·13). In order to clarify the reason why such situations are present, we shall retain the

term of the order a in K(af3, ). r) and in u ( afj, ). r), and see how the iterated term

contributes to the energy. We have then

u(a{j, J.r) =Bnao(af3, J.r),

K(a{j, J.r) =4n:ao(af3, J.r) 1 (k2 -k'i.).

Hence t(a/3, J.r) becomes

t(a/9, .Ar)=Brcao(afi, J.r)-(321r2 a2/V)o(afi, .Ar)~1/(k2 -q2), (3·19) q

up to the order of a2• Now, let us calculate the perturbed energy of the order of a2

•

It consists of the second-order energy resulting from the first term in ( 3 · 19) and the

first-order one from the second term. They are, respectively, given by

(16 7r2 a21:

2/mV2) ""-'' ))(•vP, 1 Y)n,+n-,-(1 + -)/('k2 k'!) rJ .L...J u ""P 11 1 ,.. . n(j, - ntl , ), 1 - o;fl , (3·20)

(3·21)

where ~' in ( 3 · 20) means that k(j, :::1:- k1~ ~ k 1 :::1:- k;., no;' s are the occupation numbers and

+ or - denotes the spin direction. It is to be noted that both expressions are diver

gent. However, when added together, the result is no longer divergent and is given by

(3·22)

In deriving this equation, we have used

which can be easily proved by evaluating Cauchy's principal value of the integral.

The energy of the order of a2 given by ( 3 · 22) is just the same as obtained by

the pseudopotential method and is exact as long as the order of a2 is concerned. For

the ground state, after rather lengthy calculations, we obtain

(3·23)

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708 R. Abe

where p is the average density N /V ( N : particle number) and k F the Fermi momentum

defined by kF= (37r 2p) lf:l. The above expression is identical with that given by Lee and

Yang.5)

Now, it is easy to see how the iterated solutions of (3 ·13) contribute to the energy.

We have noticed above that the terms given by (3 · 20) and (3 · 21) are, respectively,

divergent. However, the sum of both terms leads to a correct result. In other words,

the iterated terms lead to subtractions which yield a correct finite result. Such situations

may hold to any higher orders of a, for we have so far made no approximations. In

general, all the iterated terms which give rise to the energy of a specified order should

be added before integration, and then the integration over the momentum space is performed.

It is interesting to note that we have encountered with the similar situation in the

pseudopotential method. There the operation a I or [see ( 3 . 14) J must not be carried

out before the summation over the momentum space is done. Contrary to the pseudo

potential method, many-body collision terms may be automatically included by the present

method, if one performs formally the perturbation calculation of the u-matrix and of the

iterated terms, with the remark given above in mind.

§ 4. Many-body system (Bose particles)

It is evident that the Hamiltonian for the Bose system is given by (3 · 7), with

a ( s<:J.' s;,J a ( s,~' s,) omitted and a<:/.' a<:/.* the annihilation and creation operators obeying

the commutation rules for the Bose statistics. The perturbational procedure discussed in

§ 3 for the Fermi system is, however, no longer valid for the Bose system, because such

a procedure leads to the divergence of energy owing to the contributions arising from

particle excitations with small momentum transfers. These points were fully discussed

by Brueckner and Sawada()) ; they have proposed another way in which the multiple ex

citation of one or more pairs to the state with opposite momenta and their interaction

with the unexcited particles are taken into account from the beginning, then the particle

Hamiltonian is transformed into the phonon Hamiltonian by a transformation similar to

that used by Bogolyubov. 7) The result is written, using the notations in the present

paper, as

( 4 ·1)

Cj)

K2 = _f!__ t ( 00,00) + -~~- r [ (fJ2- a:2 N 2

) 112

- PJ q2 dq' 2 47r2 p J

0

(4 ·2)

where ow is the excitation energy corresponding to a single phonon or rotan state, V2K

2/2m

the ground state energy per particle, and a:N and ;9 are defined by

aN= (p/2) {t(OO, q-q) t(OO, - qq)}

fl=q2 +p{t(Oq, Oq) +t(Oq, qO) -t(OO, 00)}.

( 4 ·3)

(4 ·4)

Before calculating ( 4 · 1) or ( 4 · 2) , we wish to make some remarks about the

contributions to the energy from phonon-phonon interactions. The lowest order term of

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Quantum-Mechanical Many-Body Problem with Hard-Sphere Interactton 709

these contributions was given by Bogolyubov and Zubarev/> and was shown to be ex

pressed as the multiple integrals over the momentum space, i. e., J J dq1 dq2 ( • • ·). If we

take account of multiple excitations of phonons, there should be similar terms such as

JJJ dq1 dq2 dq3 , JJJJ d%dq2 dq3 dq4 , etc.

Now we consider to solve the equation ( 3 · 13) by an iteration procedure. As we mentioned in the previous section, the iterated terms must not be integrated at once, but should remain in the original form of the integration over the momentum space. Substituting the iterated terms in ( 4 · 2) and expanding the integrand of the last term in ( 4 · 2) in powers of such terms, we have

K2=L[u (00, 00) -~1 - r K(oo, q-q) u(q-q, oo)dq+ r dqldq2+ ... J 2 (27r) 3 J . . J.

where u J indicates that t is replaced by u in ( 4 · 3) and ( 4 · 4). As a matter of fact, the contributions from phonon-phonon interactions should be added to the above equation. Those terms are written in a form of the multiple integrals as we have mentioned above, and are combined with the corresponding terms arising from the iteration; thus an exact energy is expanded as

which is the well-defined series, though extremely complicated one. The same consideration can be applied to the excitation energy spectrum. Under

the assumption that the phonon-phonon interaction may be neglected, we have

~1 ~JK(OO, q-q) u(q-q, OO)dq] (21r) 3

(4 ·5)

with t's replaced by u's in (4·1), (4·3) and (4·4). From (3·9), (3·12), (4·3) and ( 4 · 4) , it follows

and

ttN=-8rrpj0 (qa)jG0 (a, a),

(.) _ 2 + { "' (2t + 1) . 2 c I ) tJ-q p - L..J 16Tr-----}t qa 2 t<even) Gz (a, a)

1)0

1 8Tr----Go(a, a)

1)0

(4 ·6)

(4. 7)

Bp J G02 (a, a)

_I_J[cfd2-a2 N2)1f2_fi]q2dq. 4n2p

0

(4·8)

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710 R. Abe

It is noted that equations ( 4 · 6) and (4 · 7) are quite similar to those obtained by

Brueckner and Sawada/>) the only difference being that G1 (a, a)'s are given by (3 ·10)

in the present method while the ground state energy is proportional to G01 (a, a) in

Brueckner and Sawada's theory.

To see how the formula ( 4 · 8) leads to an exact result obtained by Lee and Y ang5)

and how the iterated term contributes to the energy, we shall confine ourselves only to

the S-wave scattering, approximating j0 (qa) by 1 ; then we have

Furthermore, if we expand G01 (a, a) in powers of ka and retain the lowest order term,

we have

r>IN-k 2 (.) __ q2+ko2, .._., -o, tJ (4 ·9)

where k0 2 1s given by

(4·10)

From (4·8) and (4·9), we obtain

{ 128 ( 3)1/'1} =4rrap 1 +J:sv7r~ pa , (4·11)

which is just the exact formula obtained by Lee and Yang") or by Lee, Huang and

Yang. 9)

In these calculations, the iterated term plays a role such that it leads to subtrac

tions which yield a finite result, in the same manner as in the Fermi system. This

makes the energy higher than that calculated without subtractions, since, neglecting the

iterated term, K2 is given by

which is smaller than 4rrpjG0 (a, a), for the second term is always negative. However,

when the iterated term is added, the negative contribution from the second is cancelled

by the added term and K2 becomes larger than 47r p jG0 (a, a). The same considera

tions can also be applied to the Fermi system. In general, one may attribute such a

behaviour of the iterated term to the non-hermitic property of the t-operator, since, if

it were possible to find an operator of the order of a, which is hermitian and equivalent

to the hard-sphere potential, then the second-order perturbation energy for the ground

state, E2 , should naturally be negative as usual, which is, however, contradictory to the

exact positive value given by (3 · 23). In order to save this, a positive contribution should

be added by the addition of the iterated term.

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Formula ( 4 · 11) forms an expansion of the energy in powers of (pa3)

112, appropriate

for the low density limit. On the other hand, at high densities such an expansion is no longer valid and another procedure should be taken.

As the density approaches the dose-packed density, it is expected that the energy becomes very large; this suggests that ka is assumed to be large compared with 1. Then the asymptotic expansions for the spherical Bessel or Neumann function may be applied. It is easily verified in this case that

Gi1 (a, a)=(-1)zG01 (a, a). ( 4 ·12)

By using (4·12), we have from (4·6) and (4·7),

_ _ 1 sinqa aN- -8r.pG0 ---,

a 2 _l sinqa p=q -8r.pGo ---, (4·13) qa qa

where use is made of the equation

~ (2l+1)jl(x) (1/2) +sin2xj4x. l(even)

From ( 4 · 1) and ( 4 · 13), the excitation energy spectrum becomes

, ( ) -- ~-- ~ q_ j ~;--~-·---~-_-G .. =1 ~in qa l'J q - q -167ro 0 -- ~---~,

2m ' 1 qa (4·14)

or, if we eliminate G01 by the relation OJ(q) =c0 q (at small q, c0 : sound velocity), we

have

(V (q) =q j c02 sinq!_ +(_!!!_)2

,

qa 2m (4·15)

which is of the form suggested in the previous paper.1> Also, this expression is formally

identical with that given by Brueckner and Sawada/;) The excitation spectrum given by ( 4 · 15) has a characteristic nonmonotonic behaviour similar to the roton spectrum in liquid He4

•

It is interesting here to calculate the dispersion formula for the sound velocity from (4·15). A simple computation yields

c ( q) =Co ( 1 - r I 2 . q2 + ... ) ( 4 ·16)

where r 1S given by

( 4 ·17)

Substituting a=2.2 X 10-S em and values appropriate for liquid He\ we have r=6.9 X 10-17 cm2

, which agrees fairly with the value 6.2 X 10-17 cm2 deduced by Landau and Khalatonikov.10

)

§ 5. Concluding remarks

A method which makes it possible to treat the hard-srhere rotential by the perturbational procedure has been developed. The essential point of the method lies in the

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712 R. Abe

transformation of the original potential function into an equivalent integral operator. Such a method is not only restricted to the hard-sphere potential, but can be generalized to arbitrary singular potentials. In the following paper2> we shall discuss the generalization of the method, with an application to the Lennard-Jones potential.

We must mention here some ambiguous feature of the t-operator defined in this paper. As shown in § 3, a ( r 12 - a) t12 is equivalent to the hard-sphere potential v12 •

However, the defining equation for t12 , (3 · 2), is valid for an arbitrary value of K2 not

necessarily proportional to the energy eigenvalue per particle. Hence k2 defined by ( 3 · 11) may be assumed to be arbitrary and yet the subsequent discussion is valid. As we have shown in (3 ·17), when the u-matrix is expanded in powers of k2

, the lowest order term is independent of k2 and free from such an arbitrariness. Though it is not yet clear how the arbitrariness of k2 plays a role in the determination of the energy eigenvalue, we hope that the results may be independent of the value of k2 if the perturbation calculation using the u-matrix is carried out up to infinite order.

Now the question is how to remove the ambiguity for k2 when the perturbation series is terminated in the finite terms, (this series is dependent on k2 owing to the dependence of the u-matrix on k2

) • We have not yet a definite answer to this question ; however, we consider that the following procedure will be useful. We assume that k2 is given by (3 · 11) with o2

K2/2m the energy eigenvalue per particle and consider that K

2

is an unknown quantity, then the perturbation series is some function of 1c2• This series

in turn is equal to K2 and we obtain an equation to determine K

2• The approximate

nature of this procedure is not clarified yet, but we expect that it is a good approximation for a system at low density.

In conclusion the author wishes to express his sincere thanks to Prof. N. Fukuda and Mr. M. Kawai for valuable discussions on this work.

References

1) R· Abe, Prog. Theor. Phys. 19 (1958), 1. 2) R. Abe, Prog. Theor. Phys. 19 (1958), 713. 3) K. Huaug and C. N. Yang, Phys. Rev. 105 (1957), 767. 4) K. Huang, C. N. Yang and ]. M. Luttinger, Phys. Rev. 105 (1957), 776. 5) T. D. Lee and C. N. Yang, Phys. Rev. 105 (1957), 1119. 6) K. A. Brueckner and K. Sawada, Phys. Rev. 106 (1957), 1117, 1128. 7) N. Bogolyubov, ]. Phys. U.S.S.R. 11 (1947), 23. 8) N. Bogolyubov and N. Zubarev, J. Exp. Theor. Phys. 28 (1955), 129. 9) T. D. Lee, K. Huang and C. N. Yang, Phys. Rev. 106 (1957), 1135.

10) L. D. Landau and I. M. Khalatonikov, J. Exp. Theor. Phys. 19 (1949), 637.

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quantum-mechanical many-body problem with hard-sphere interaction

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