IL NUO¥O ('IMENTI) VoL. XX. X. 3 1 o Maggio 1961

Structure of a Quantized Vortex in Boron Systems (').

E. P. GRoss (**)

( ' E R N - (;e~eva

(ricevuto il 9 Gennaio 1961)

Summary. For a system of weakly repelling bosons, a theory of tile elementary line vortex excitations is developed. The vortex state is characterised by the presence of a finite fraction of the particles in a single particle state of integer angular momentum. The radial depend- ence of the highly occupied state follows from a self-consistent field equation. The radial function and the associated particle density are essentially constant everywhere except inside a core, where they drop to zero. The core size is the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the veloc- i ty has the radial dependence of a classical vortex. In this Hartree approximation the vorticity is zero everywhere except on the vortex line. VVhen the description of the state is refined to include the zero point oscillations of the phonon field, the vorticity is spread out over the core. These results confirm in all essentials the intuit ive arguments of ONSAOI~R and FEYNMAN. The phonons moving perpendicular to the vortex line are coherent excitations of equal and opposite angular momentum rela- tive to the substratum of moving particles that constitute the vortex. The vortex motion resolves the degeneracy of the Bogoljubov phonons with respe('t to the azimuthal quantum number.

1 . - I n t r o d u c t i o n .

The idea t h a t l iqu id he l i um permit~ macroscopic vo r t ex type mot ions , as

does an o r d i n a r y l iquid , has p l ayed a key role in sugges t ing a n d i n t e r p r e t i n g

a large n u m b e r of recen t exper iments . The expe r imen t s of VI~E.~ (x) p rovide

(*) Work supported by the Office of Scientific Research, U.~. Air Force and by the National Science Fundation.

(*') Permanent adress, Brandeis University, Waltham, 5fass. (1) H. E. HALL: Adv. i'~t Phys., 9, 89 (1960); K. R. ATKINS: Liquid Helium (Cam-

bridge, 1959); ~V. F. VINEX: Physica Suppl., 24, 13 (1958).

S T R U C T U I { E (}F A t2UANT1Z]£D VI~tCI'E'< IN t~I}SItN SYSTEMS 4 5 5

convincing evidence for the existence of free vor tex lines with a circulation

quant ized in units of h/m. Superposed on a field of vor tex motions is the gen- eral phonon field. The idea of O~SA(~EU (3) and FEV33[AN (3), t ha t (.ireu-

la t ion is quantized, meets the objection tha t if vor tex excitat ions of a rb i t ra ry

circulat ion and energy were permi t ted , there would be no superfluidity. On the other hand if there were only phonon- type excitations, i t follows f rom

Landau ' s well known a rgument based on Galilean invariance, tha t the critical flow veloci ty would be much higher than is observed exper imental ly . With

the assumpt ion of quantized vor tex lines and rings a qual i ta t ive explanat ion

of the low critical velocity becomes possible. The same is true of the behaviour

of ro ta t ing helium and of many other phenomena.

The description of a vor tex quan tum mechanical ly has a number of puzzling

aspects . Arguments , for and against the existence of vor t ic i ty have been made. ~None have been precise enough to be universal ly convincing. Against the existence of vor t ic i ty cue ('an argue as follows. In the wave function

~V(xx ... x ) = R exp[i(2/l i)] , S plays the role of a veloci ty potent ia l when the Schr5dinger equat ion is wri t ten in hydrodynamic form. The existence of a

potent ia l seems to imply tha t there can be no vort ic i ty . This has been one object ion to a t t em p t s to relate macroscopic cont inuum quan tum hydrodynam-

ics wi th vor tex motions to the propert ies of liquid helium. But it has not been shown, with reasonable definitions of the veloci ty and vort ic i ty , t ha t the vor t ic i ty is everywhere zero. For example, if we define the densi ty as

the velocity,

v (x ) -

• 3"

j i=1

(l'* b (x - x,) 7-- + = - 6 ( x - x~) T d ~ ,

one would have to show tha t the vor t ic i ty , curl v(x), is zero. At best one can make it plausible tha t if S is slowly varying, and R not very different

f rom the ground state in certain spatial regions, the vor t ic i ty is zero. There is an essential distinction between the existence of a potent ia l function S ( x l , ..., x~) in 3N dimensional configuration space and potent ia l flow of average values of operators in ordinary space.

On the other hand, there is the more convincing reasoning of O~SAGEn und F E ¥ ~ A ~ , which indicates tha t there m a y be vor t ic i ty in concentra ted regions. We will t ry to paraphrase their argument• One m a y s ta r t by con-

(3) L. 0NSAC, ER: Nuppl. iVuovo Gimento, 6, 2, 249 (1949), (a) R. P. FEYN.~IAN: Prog. Low Temp. Phys. . Vol. I (Amsterdam, 1957), ch. 2;

Physica Nuppl., 24, 18 (1958).

456 E.P. ~Re)ss

sidering the flow in a mul t ip ly connected region, for example~ between cox-

centr ie cylinders. Corresponding to a given state, say the ground state, there N

should be other s tates in which S differs by # ~ hv~, where /~ is an integer. 1 = 1

The new mot ions are those in which each part ic le has # units of angular mo-

m e n t u m , i.e. the sys tem has a to ta l angular m o m e n t u m N~#. Wi thou t vor-

t ic i ty anywhere, there is nevertheless a circulation. When the inner radius, ~o, of the region is made small, if i t is assumed tha t the real pa r t of the wave

funct ion is still essentially the ground s ta te function, the expecta t ion value

of the az imutha l veloci ty should be propor t ional to 1/9 , and the vor t ic i ty still essentially zero. B u t the s i tuat ion becomes unclear as ~ shrinks to a tomic dimensions. An ideal classical line vor tex has a vor t ic i ty zero everywhere,

except on the singular vor t ex line; and has a character is t ic 1/~ value of the

velocity. Quan tum mechanical ly, the 1/~ behaviour of the veloci ty can per-

sist as ~--> 0 only if the real pa r t of the wave funct ion drops essentially to

zero on the vor tex line. Otherwise the kinetic energy would become infinite.

Bu t there mus t be l imitat ions on the definition of the posit ion of the singular

line implied by the uncer ta in ty principle. So, one expects a core in which the

densi ty will become small bu t not necessarily zero, and in which the veloci ty will be finite. The vor t ic i ty should be spread out over the core and should

drop cont inuously and rapidly to zero, as one moves out. Tha t these things happen has not been shown in a detailed theoretical t r ea tment .

S tar t ing f rom the assumpt ion t ha t there exist such macroscopic exci tat ions with a core of a tomic dimensions and with a quantized circulation, but other- wise behaving like a classical vor tex , FE~-~rA~ ~ (~) in te rpre ted a large number

of phenomena occurring in helium. This point of view was extended and applied with great success by HALL and ¥INF~:~ (~) and others. I n spite of the

fundamenta l impor tance of these physical ideas, bo th pract ical ly and con- ceptually, ve ry li t t le work has been done to relate the ideas to basic quan tum

mechanics. Ins t ead a t t en t ion has been concentra ted recent ly on how the phonon- type excitat ions, which are relat ively well unders tood as a result of

the work of LA~NDA~r, ]~0GOLJUBOV and FEY~NMAN, follow f rom the m a n y - b o d y t Iami l ton ian . The purpose of the present paper is to construct a theory of the s t ructure of the simple line vor tex , and of the superposed phonon- type exci tat ions when a vor tex is present . We shall do this for the case of weakly repelling bosons (or for a dilute gas of hard spheres) in the quantized field description of the many-body problem. I n this l imit a sys temat ic theory is constructed which follows closely, and is essentially an applicat ion of the work of references (4~) and (~). I t is close in spirit to Bogol jubov 's fundamenta l

(4) E. P. GRoss: _1~. Phys., a) 4, 57 (1958); b) 9, 292 (1960).

STRUCTUI{~E (}1' A ~ U A N T I Z E D V~}ICI'EX I x, lI~}5~}h SYSTEMS 4 5 7

paper (5). The results agree in all essentials with the ideas of FEYN]~IAN and O . N S A G E R .

The main features are a l ready apparen t in the Har t r ee approximat ion. T h a t is to say, the possibil i ty of describing a vor tex mot ion in a Bose fluid

is contained in this approximat ion . Each particle is in the same single par-

ticle s ta te with angular m o m e n t u m ~ t about the z-axis. The single particle

s ta te is a solution [ ' (~)cxp[i#9] of a self-consistent field (or semiclassical self- interact ing field; ref. (4,)) equation. I t has the remarkable p roper ty tha t

the densi ty I]"(~)I ~ is substant ia l ly constant at distances greater than a

length a. ~ e a r the vor tex core the densi ty tends to zero like (~/a) ~.". In the semiclassical approx imat ion the expecta t ion value of the az imuthal veloci ty is s t r ic t ly propor t ional to 1/o, just as for a classical e lementary -cortex. The

core energy per uni t length of vor tex line is finite. The radius of the core is

given ve ry s imply as the de Broglie wavelength a = h /~ /2ME, where E is the

mean interact ion energy per part icle nfV(x)dSx. For ~ gas of dilute hard

spheres, described b y a pseudopotent ial , this is .15 v~(~ /~ /n~s ) , where ~ is the sphere radius and n is the density. These results are obtained in Section 2

f rom an exact solution of the semiclassical field equation. The connection of such a solution with a YIartree wave function is discussed in ref. (da).

I n Section 3 we consider the small oscillations of the semiclassical field abou t the exact solution, i.e. the phonon spec t rum in the presence of a vortex. The phonons are described as coherent excitat ions of pairs of particles of equal

~nd opposi te angulur m o m e n t u m rela t ive to the Har t r ee s ta te of angular mo- m e n t u m Nh/~. Far f rom the cortex core, where the fluid is hardly turning at all, the phonons go over to the usual exci tat ions of Bogol jubov (5). The vor tex

mot ion removes the degeneracy of the Bogol jubov spect rum with respect to the az imutha l quan tum number .

In Section 4 the classical considerations are put in a fully quan tum mechan-

ical form. One result is the lowering of the self consistent field estinaate of the energy of a vor tex line because of the shift in zero point energy of the

phonons provided by the normal mode analysis. In addition, consideration of the fo rm of the wave function of a line vor tex shows tha t the zero point mot ions of the oscillations smear out the densi ty pa t t e rn of the Har t r ee field~ yielding ~ smull finite value a t the vor tex line. At the saute t ime the expec- ta t ion value of the veloci ty drops to a finite value (in fact to zero in our approximat ion) , a t the vor tex line. There is now a definite finite vor t ie i ty differing f rom zero mainly in the core.

The sys temat ic quan t um theory has been obta ined by first s tudying in some detai l the associated semi classical field theory. :Notably, we use special solutions of the semi classical theory to suggest an appropr ia te single part icle

(5) ~. ~. ~BOGOLJUBOV: Journ. o] Phys. USSR. 9, 292 (196~).

4 5 8 E . P . GROSS

basis, in t e rms of which the quantized field is expanded. We use the small oscillation analysis of the classical field to suggest a suitable quasi-particle

t ransformat ion . This pa t t e rn of analysis (.an be extended to s tudy more gen- eral types of vor tex excitations. In order to obtain insight into the reason

for the success of the hydrodynamic arguments of Feynman , we t ranscr ibe the semi-classical theory into hydrodynamic form in Section 5. The main

differences f rom the usual equations of compressible flow are the non-local pressure-densi ty funct ional relation, and a quan tum mechanical pressure term.

The la t te r is responsible for the vor tex core structure. Provided the cores are

well separated, complicated solutions of the equations can be obtained in the

same way as in usual hydrodynamic theory. Each pa t t e rn of flow represents a wave packe t of macroscopic durat ion. This opens the way to a consideration

of general hydrodynamic flows of a sys tem of bosons, on a basis which has a clear and definite quan tum counterpar t .

2. - S e m i - c l a s s i c a l t h e o r y (4,,).

Consider the boson fluid governed by the Hami l ton ian

h ~ :, ) V ( l x - x ' I ) ~ ( x ) v 2 ( x ' ) d S x d ~ x ' . (2 .1 ) H = ,2~i f vv,+ VWa3x + 1 "

The equation of motion of the field ~o is

(2.2) i h ~ - - ]12 f 2MW~ + q,(x) V(Ix-- ,,'L)/IV,(x')l~d~'x ',

and will be studied as a classical field equation, of the self consistent field type in this section.

Le t us look for special exact solutions of the equations of mot ion possessing cylindrical symmet ry . We pu t

(2.3) g,(x) = f(r2) exp [itcO ] e x p - - i )7~ ] '

where ~, 0, z are the cylindrical (.o-ordinates. The z component of tile field angular m o m e n t u m associated with sueh a solution is

S= = , ~ j \~p ~ . ~ - - -~.~- ~v dSx = hla [ ]12(o)dSx .

But the to ta l number of part icles is N = f V ~ v d ~ x = f l / 1 2 d ~ x . Thus the to ta l angular m o m e n t u m is N h t z , i.e. lilZ per particle. In addition the local angular

S T R U C T U R E ~1" A QUANT1ZEI) V~HYrEx IN B(~SON SYSTEMS 4 5 '(̀ t

m o m e n t u m densi ty is constant in t ime and equal to Ili2(~)h/~. The component of the veloci ty of the field a t a given space point m a y be defined as

l 1 h{ I ~ F ] ] W ~f '..' M i o P O ~ ?.0 ~' '

and is equal to h # / M e. I t exhibits the characterist ic radial dependence of the

flow pa t t e rn of a classical line vortex. We shall see tha t in contras t to the

usual incompressible (or even compressible) fluid the densi ty ~o+~p----II(o)l ~ tends to zero as ~ - ~ 0 (at the vor tex line). However, as ~ - + oo the den-

si ty tends to a constant as is the case for the nsual vortex. The vor t ie i ty w~= (curl v )~= ½(1/~)(~/3Q)(QI'~)= 0 everywhere except a t the singular line. Bu t the circulation is F = j ' % ( d O . p ) = # h / M ¢:0 , so tha t we m a y write

v o = F/2:~o~. The radial function ](o) mus t satisfy the equat ion

(2.4)

Since

El-- , 2 ~ \ ; ~ ( e ~ ) - - ~ I(e) + ](o) V(!x-- x' ip! / (e ') l '~d~.~ ' ' .

V(Ix - x ' l )= v ( v ' ( z - :')~+ o-~+ e ' ~ - 2e~' (~os (~ -~9~),

is invar ian t to the ro ta t ion 0-+0~. . ~, ~ ' -+ 0 ' + ~ , the assumed separat ion of

variables is indeed consistent. To examine the behaviour of the function ]Qo), we consider first for sim-

plicity the case of a short range potential , in fact Y ( I x - - x ' l ) = V D ( x - - x ' ) . We have in mind, more precisely, a pseudopotent ia l (% At this stage of the

theory it can be t rea ted as a h-function. Then

(2.5) ~-:I --._,M\~o. ~)- 1 + l'lll(~o)i ~,

with

f :l(~,)!2d.~x =_ Y .

(For the s-wave pseudopotent ia l V = S,~x(h~/M), where a is the radius of the

hard spheres.) ](o) m a y be taken as real. For small Q, the centrifugal force t e rm domi-

nates and

(6) K. }tUAN(~ an(I (~. N. YANk]: t~h!l.~. Rev., 105, 767 (1957); K. HUAN~', T. D. L ~ and C. X. Y~'*;: Phy,% Rer., 106, 1136 (1975).

460 E. 1 ~. (;ROSS

As e ---> c~

(2.7) ](e) -+ ]o = c o n s t a n t .

W e m u s t have E = V ] ~ . The last r ema in ing cons tan t , ]2, is fixed b y the

no rma l i za t ion condi t ion f l]~id3x= 5". This has a small finite con t r ibu t ion

f rom the core of the vo r t ex as N -+ ~o, ~ -+ c~. I n this l imi t

X Y ]2 --~ .(2 = L ~ R ~" '

where L is the ex ten t of the sys t em in the z direct ion and R is the radius of

t he cy l inder of (( quan t i za t ion ~>. Thus ]~ is the m e a n n u m b e r densi ty . The

behav iou r of ] as e ~+ c~ is qui te remarkable . I t is b r o u g h t abou t b y the

non l inear t e r m VII/12, i.c. the self-consis tent field. I t is phys ica l ly clear t h a t

t he repuls ive forces should force the sy s t em to un i fo rm dens i ty a lmos t every-

where.

The energy of the fluid is ob ta ined b y subs t i tu t ing the solut ion ]"(e) in H

(2.s) H - - ~ 2 " VM)/,,(o) d ( d I f' I

F o r the vor tex- f ree case ( # - - 0 ) the lowest s ta te is / o : ] 0 = e o n s t . , wi th an 1 4 :~ energy H : ~]ofI (s)d3s. W h e n ~ vo r t ex is present (#0~ 0), f'(e):/= ]o every-

where and there is a finite correc t ion to the po ten t i a l energy per un i t length

of the vor tex , arising f rom the core. The behav iour of the k inet ic energy is,

however , more i m p o r t a n t . I f the kinet ic energy is eva lua t ed wi th ] ' ( e ) : / o

everywhere , the resul t is d ivergent , as o -+ 0. However , the ac tua l ]t'(e ) -+ 0,

as e -+ 0, so t h a t the dens i ty of fluid tends to zero as the ve loc i ty tends to c~,

in such a w a y t h a t the kinet ic energy per un i t length of the vo r t ex core is

finite. The region outside the core (o > a) cont r ibu tes a kinet ic energy

g'/j •'= o de" 2•L ---- 2 M /~' 2~L 1,1 -- ]8 in a (2~L),

i.e., t he charac te r i s t ic logar i thmic dependence on the outer radius of the vor tex .

F o r a more general , b u t still essent ial ly shor t - range potent ia l , ](e) still t ends to a cons tan t f0 wi th

(2.9) E : ]o21 V(S)d3S . J

> ' I ' t l I ' ( " I ' U I / I : (~I' A QI A. N I'IZEI) ' , ' l ) l~ ' I ' l ' :X IN l~ll~,l~N S Y ' T I : M S 4- t i ]

T h e b e h a v i o u r for 9--> 0 is a g a i n the same as for a & f u n c t i o n p o t e n t i a l , a n d

is d e t e r m i n e d e n t i r e l y b y the cen t r i fugM p o t e n t i a l .

I t is of course n o t e a sy to f ind e x a c t so lu t ions of t he self c ons i s t e n t f ield

e q u a t i o n for ] ' (o ) . H o w e v e r , i t is easy to f ind an a c c u r a t e e s t i m a t e of t he

size of t h e v o r t e x core. W e use our k n o w l e d g e of t he e x a c t b e h a v i o u r a t smal l

a n d l a rge d i s t ances a n d a s s u m e t h a t

(2.10) ] " (< j ) = Aj~ D= a , o < a ,

= / o , 9 > ~ .

A is f ixed b y the c o n t i n u i t y r e q u i r e m e n t a t 9 = a,

(2.]: i) A = t<,!),, D_> a .

T h e c o n t i n u i t y of t he r a d i a l d e r i v a t i v e s of ] r equ i res

) (2"]'°) ~ t t k~ o'o , , = o .

T h i s f ixes the core size a~,

W e h a v e

in t e r m s of the pos i t i on of the f i rs t zero of dj" td O.

D /~ = 1 a] - -

"~ 2 M E

/~ - - 2 ( % =

/~ la rge a,, =

.~( .59) .

.~(.'57),

( p - - . 2 6 ~ f f i ) .

:For # = 1 t he core size is of the o rde r of t he de Brog l ie w a v e l e n g t h a s soc i a t e d

w i th the m e a n enerody of i n t e r a c t i o n per p a r t M e E----V]~. F o r a d i lu t e gas

of h a r d spheres of r ad iu s a, mass M, V S~(~D"-/3I) and we have

. ' = {.5UT ~ .~) ~:

"i.e.: the v o r t e x r ad iu s is l a rge r then the h a r d sphere r ad iu s b y t i le f ac to r

1 ( p a r t i c l e sep., .ration/~

? t s p h e r e l ~ f i i l i s - - ! "

T h i s i n d i c a t e s t h a t in a m o r e dense gas, t he v o r t e x w o u l d h a v e a size a b o u t

e q u a l to the i n t e r p a r t i e l e spac ing . B u t we c a n n o t be sm'e for t he zero p o i n t

m o t i o n s of t he phonons are as i m p o r t a n t as the se l f -cons i s t en t field.

3 0 - I I N u o v o Cimento.

462 E.P. GROSS

The energy associated with this app rox ima te solution m a y be ob ta ined

readily. We define the pure numbers

(2.13)

y:i~Sd,~ IgSdS e f t = o J ~ ( G ) , D r = o

"4 ~ ,

z~[ jo(S.)j,(s,)]

where S" are the roots S I = (.59), $ 2 = (.97), ....

fl2dZx = N yields

(2.1~) x _ fo/"~ ~' R ~ - , ,q / ~ - . + - - 2~L ' 2

The energy is

(2.15)

Then the normal iza t ion of

H h ~ ( D, 1"/~ h2 1 R

I n addit ion to the potent ia l enerKy of the quiescent fluid and the logar i thmic

vor t ex energy there is the finite correction (as R -~ cxD) per uni t length of the vor t ex core. I f we write In (R/a)= In (R/b)--ln (a/b), we can make the cor- rections vanish by taking

C~ + D,~ 1 In a a 1 . 4 - 3; b l - ~ - .

Then b is a ((pseudo ~> radius of the vor tex core. We might improve the above es t imates by taking a more refined t r ia l

funct ion / ' (~) and using the var ia t ion principle for the energy with ] subject

to fl2d3x= N. But the results differ f rom the above in only u n i m p o r t a n t ways. I t is more impor t an t to consider the possibili ty, t ha t for a given #,

there exist other solutions, with nodes, of the self-consistent field equat ions. This requires more detailed study. I f they exist, they would have to be s table to be direct ly re levant .

We note in conclusion tha t the results of this section m a y be obta ined direct ly in configuration space. The Har t r ee approx imat ion is k~'(x, ..., x ~ ) =

dV = (1/~/~) 1-I ]"(~,) exp [ i#~] , and represents a fixed number of particles. The

j=l funct ion ]'(Q~) is the same one tha t is used in the field picture. The advan tage of the field point of view appears in the introduct ion of quasi-particles, as dis- cussed in the next section. This is done a t the expense of working with an indefinite number of particles, and an indefinite angular m o m e n t u m .

S T R U C T U R E OF A QUANTIZ]£I) VERTTLX IN BOSO_N SYSTE318 4 6 3

3. - Small oscillations of the fluid about a vortex motion.

We have shown tha t there is an exact solution of the classical equations of mo t ion involving circulation. I t differs f rom a hydrodynamic vor tex in

t h a t the s t ructure of the core is fully determined, and the densi ty goes to zero a t the singular vor tex line.

In the classical theory of the wave field, the nex t step is to examine the small oscillations of the sys tem about the exact solution. We pu t

(3a) v, = exp [-- i ,:t] t, J {/~(e) exp [ i ~ ] + ~(x, t)},

and linearize the equations of motion. One finds for a &function potent ia l

-- E)m ÷ V/2(O) exp [i#va] •

• (exp [--/tr~]q~ + exp [i/~va]qJ+},

together with the complex conjugate equation. I t corresponds to the Hami l ton ian

(3.3) h: f(p ' ~ j f ~ ( 9 ) { 7 + e x p [ - - i / x ~ ] } ~ d S x , H.2 -- 2~1 + V ~ d a x = I exp [i#~] ÷

which is posit ive definite. The same result holds for more general posi t ive

interact ions V ( I x - - Y I). To analyse the normal mode spectrum, we define a complete set of ortho-

normal functions as the eigenfunctions of the linear opera tor

(3.4) Z ' - - 2M v~ + VI I (~'! -- E .

We shall continue to work with the &function, in spite of the fac t t ha t one m u s t use a pseudopotent ia l to avoid a divergent to ta l zero-point energy f rom the shifted no rma l modes. The spec t rum itself is quite definite and finite.

We wri te

(3.5)

464 E . p . (;R<,s~

where

(3.6)

1 V ...... - - - exp l i~:] exp [imO]g~., .(o) ,

"V 2 ~ L

j .q*:"A~)9~,.~(o)o d,, = ~ . +

The superscr ip t f, is to emphas ize t h a t the set of func t ions depends on the

s ta te of vo r t ex exci ta t ion. The eigenvalues E~ depend on lint because of the

cyl indr ical s y m m e t r y of the potent ia l V i]'(.o)j'-. We will work with s t and ing

waves . The funct ions g~,,~(Q) can be t aken to be real and defined in a large

cyl indr ical region, with the b o u n d a r y condit ions

( 3 . 7 ) a ~' (o = R ) = O . • J o , m ,

W e note t h a t

1 t, E" = E go , , , - f~'(e) , o,~ ,

where we write a = 0 for the lowest possible value of a. W e also have

g"o.., = g"~.-,,, ; ¢ . . . . . . . = 7 - . . . . - , , , "

Le t us now expand

(3.8)

Then eq. (3.2) is equ iva len t to

(3.9) { ih ~ - - '

where

E + 2 M ]~ a ....... =

= ' - - m ) a . . . . ,.~,_,,,} , ~ {F" (,~,~ [ ,~ ,~),~.,o,,,,, + F (,~m [,¢, 2~, + o"

f l(~'m') = V tg~. , ,g~, , , , , ( /")2edo ,

I t follows t h a t

F * = F .

(3.10) h,~'~ql +

(E~,2, ,- , , , - - E + - i l , ~ - ,,TYl}J ~ . . . . . = " - = =

= ~ {F"(,~, ~/~ - m l~', 2,u - mlaL,.+.~,,_~, + f " ( ~ , 2~, - m l~ ' , m)a,.o,.,,,}. ty'

STI (U(" I ' I ' I IF ()I" A QU.~NTIZI: I I VIq' , ' I ' I :X IN BI~S')N SYS'fl ' ;MS 4( j5

The dominant feature of these equations is the strong coupling of a ....... and

a+- . . . . w- , , ," There is a less impor tant (.oupling of the radial modes. In the

absence of vortex motion (/* = 0), the coupling is tha t of equal and opposite

linear nmnlenta, i .e . Bogoljubov's theory, here desmibed in cylindrical

co-ordinates. When there is a vortex, the annihilation operator of angular

m o m e n t u m m --/* relative to the vortex ang'ular momentum is strongly coupled

to the creation operator of angular momen tum - - ( m - - / * ) relative to the

vortex. This actually involves a net loss of ang'ular momentum in the smM1

ampli tude approximation. In the quantum version the state vector for the

vor tex contains pail's of particles of equal and opposite angular momen tum

excited out of the vortex ((reservoir)>. There is a loss of 2/* units of angular

m o m e n t u m for each pair~ since it comes from the moving subst ra tum which

is t reated as fixed. Tile state is one of indefinite angular momentum and is

analogous to, but in addition to, the description in terms of an indefinite number

of particles in the usual boson theory.

Let us consider only the terms in which a ' = (~, thus neglecting the radial

mode coupling. We look for solutions where a and a + have the t ime behaviour

exp[- - ie t /h] . (The fact tha t this violates the requirement tha t 9)+ be the her-

mit ian conjugate of 7 is taken care of by the fact tha t there is a corresponding

solution exp [ + iet/]i].)

• , + * 4 + . (3.11) a ....... = A . . . . . exp [-- i e t / h ] , a ....... = exp [-- ~et/ti]. ~ .....

The energy s is given as

+ A..~,~._ ,,, - - d ....... ,

(3.12) e 24- s { F ( a , 2/, - - m) - - F(am jam)} =

- - ) . , ~ ,<2 t* - ,,~ T 2 / * 2 / * - -

- - F ( a m ]cr, 2 / , - - m) F(a~ 2/* - - m [ a , m) ,

E .. . . . - - E @ 2 M "

The eigenveetor is

(3.13) A ...... F(a , , I a. 2t, - - m)

In complete analogy with the usual boson theory, the variables

(3.14)

b . . , . = cosh y a ....... + sinh y a + . . . . 2.-, , , ,

a, + b+- ~.o,2,, -,,, = sinh 73 ....... + eoshy .... 2#-,,,,

y = y ..... ,

466 E.P. GROSS

are connected by a canonical t ransform to the a, a +.

ically as

i/i b .. . . . ----- e~a ~ b . . . . . .

They oscillate harmon-

(Compare eq. (4.12) to (4.16) for the values of ~ .~ . . ) Fo r the case tha t there is no vortex, we should recover the results of Bo-

GOLJt~OV. Le t us review how this comes about. Our t r ea tment is unfamiliar, first because we are using cylindrical co-ordinates, and second because we have expanded in standing waves. The first step has been the determinat ion of the self-consistent field fl'Qo). The function ]"(Q) obeys equation (2.5) with # = 0. I t is independent of ~ as ~ -+0 . We have also E : V ( J ° ) ~. Bu t with the boundary condition (3.7), as ~ - + R , J° must adjust to our demand ]o(q = R ) = 0. The function drops to zero within a distance ~ / ~ / 2 ~ of the cylinder of quantization. This is the same characteristic length tha t occurs

in the core size argument . Thus the ground state contains a finite fract ion of the particles in a single-particle state which is essentially a constant except near the boundary . This is substantial ly the starting point of the usual theory as developed with periodic boundary conditions. The details of ]o(~) near the boundary complicate the analysis, bu t are not essential for ma n y purposes. I t is clear tha t the semiclassical starting point offers a natural way of t reat ing general boundary conditions. The highly occupied single-particle state, as de termined by a self-consistent field equation, tends to uniform density except near the boundary. One can impose the condition tha t the wave function vanish, just as one can impose periodic conditions, wi thout creating artificial difficulties which occur in other methods. These difficulties arise because the standing-wave solutions of the SchrSdinger equat ion for non-interact ing par- t ides have large density variat ions throughout the box.

When there is no vortex, the functions g°o.m(Q) are ordinary Bessel functions i,~(a~), with the quasi-continuous values of a determined by j~(aR) = 0 and with . E ~ - - E = J~2a2/2M. The energy normal-mode-frequencies are

(3.15) ~ / h2 } e2 = ~ ( ~ + ~ ) [ .~] / (a~+ ~?) + 2F°(am [ ~ ) .

Now the matr ix element

,~(o) g°,m(J°)~e d o ,

is just ~,.o,(/o)2 and is independent of m. Thus the f requency spectrum of the excitat ions is degenerate with respect to the azimuthal quan tum number . Eq. (3.15) is of course Bogoljubov's spectrum, as wri t ten in cylindrical co- ordinates. In the special case of no vortex, the entire Hami l ton i an / /2 is made

STRUCTURE OI" A QUANTIZEI) V~)HTEN IN B,tS{)N SYSTEMS 467

diagonal by the simple normal mode transformation, i.e. there is no radial

mode coupling. Thus we see how the usual theory is contained in the present

description.

When there is a quantized vortex, there are modifications of the normal

mode frequencies and eigenvectors. The most striking feature of eq. (3.12)

is the removal of the m degeneracy. The only remaining degeneracy is tha t

eq. (3.12) is invariant to the substitution s --> - - s, m ~- 2 # - - m. This reflects

the possibility of forming standing waves relative to the vortex. The modes

with different m values represent different motions relative to the vortex and

are therefore per turbed differently.

A more detailed s tudy of the normal mode spectrum may be made if one

takes a crude approximation to the (~ self-consistent ~) potential

(3.16)

f U(x) = i V ( I x - x'l )IF(,.,')[-' d3x ' ,

U(x)--~ 0 for ~ < a ,

E for Q > a ,

where a depends on /~.

Thus a cylindrical well will be used to generate an approximation to the

or thonormal basis ~ . ~ . . This leads to

(3.17)

where

I g~.m = ~],,(a'a) , ~ < a,

= \ ] ~ - ] ' " = \ 1 ~ (E~--E) ,

and ~, fl, y depend on a and m. The conditions tha t go.m and its radial deriv-

ative be continuous at Q----a fix fl and ~ in terms of ~, i.e.

(3.18)

. ! L~,(cr a) = fl?,~(aa) + 7n,,(aa) ,

tl(ff a ) (l(O'a) d ( ~ a )

In turn, ~ is fixed by the requirement tha t g,.m be normalized. The possible values of a are determined by

(3.19) go.re(R) -= fljm(aR) + 7n,~(~lR) = 0 ,

468 E. i,. (;R(~s,-

There are of course shifts f rom the vor tex f~ee values in the pe rmi t t ed values

of a of order l / R , and associated shifts in E , = E + (h~(r~/2M). There will

be addit ional shifts in the normal mode frequencies e ....... . This gives rise to

a change in zero point energy. We can compute the difference in zero-point energies with and wi thout a vor tex as

(3 .20) ~ ~ (~ , ~ - o y.cr 11l

using eq. (3.12).

The examina t ion of the phase-shifts is closely related to the theory of the scat ter ing of a phonon f rom a vortex. I t is sufficiently interesting to wa r r an t separate t r ea tmen t , which will be carried out in a separate paper.

Le t us now sketch the modifications to be expected for a more general potential . The opera tor L" becomes

(3.21) b~ /

v~-~ly(Ix- x'l)~j ~ ) , ' d3x ' - - E L I , = 2 M j

The (~ potent ia l ~ is again cylindrically symmetr ical . We expand

(3.22) V ( [ x - - x' I) - -

The ~ potent ia l ,~ is

1 ) , ~ W~, ,,r(o, o') exp [ix(z -- z')] exp [im'(O -- 0')].

(Wo.o(_o, ~')l/"(o')] 2o ' d o ' .

The equat ion for the small oscillations is

(3.23) (il~tm~V~)¢t={fY(x--x')[f'(,,')l~d~x'--E}qj+ + r(Q) exp [ i~ ] f ) ' ( ~ - ~')/,'(~') (exp [ - i~,~'] ~,(~') + exp [~,~,] ~+(~,)) d,x'.

E x p a n d i n g in te rms of the basis ~v ...... , we find eq. (3.9) and (3.10) with the rep lacements

(3.24:) f f , r p /~t ~ = .od,,e. d o / (o)g,,m(o) W~.,~_,,(o, o')/~(e ')g.,..,,(~o').

The rest of the theory is then developed as before.

Yaq'Ru("FI_IH; Ill." A (?UANTIZI; I ) \ ' ,~I,"I'LX I x̀ !:,}-,~N -Y< ' r lq 'q~ 46!)

4. - Quantum theory.

There are many, essentially related ways, of developing tile quan tum theory of weakly interacting bosons which yield identical results in the lowest ap- proximations. We adopt tile general approach of Section 4 of reference (ab), which is in part icularly close correspondence to the semielassieal theory. In this approach the main new point in the quantum theory is tha t the approx- imate eigenstates no lon~'er correspond to a sharp wdue for certain constants of the motion. Thus the classical solution exl)[--iEt/l~]]~'(o)exp[i/~v ~] corre- sponds to fixed, t ime-independent values of the number functional ~Vo~--jv,+wd~. and the angular momentum fun,.tional ,L (b/2i) I'(V0+(~V,/~,9)- (~0+/~0)W) d~x. In the quantum theory, the approximate eigenstates of H have g'iven expec- ta t ion values for the operators No,, aml J: , but are not exact eigenstates. This is part icularly apparent in the case of the number operators. The phase factor exp[- - iE t /h] may be removed from the field operators by the time- dependent canonical t ransformation U = e x p [ i ( E t / h ) X J which changes the Hamil tonian to H - - E N o s . This is equivalent, in the quantum theory to introducing the Lag'range multiplier E and determining it by the condition tha t the expectat ion value of Noo is N. To find approximate eigenstates of the Hamil tonian which are also exact eig'enstates of X and J:, one has to apply projection operators in the present formalism. For the vor tex state, they are of the order of N partieles in a single-particle state of angular mo- men tum /~# with a vor tex- type radial dependence. This amounts to a total angular momentum of Nhtt. In the quan tum theory we oug'ht to introduce another Lagrange nmltiplier o9, to be determined by the requirement tha t the expectat ion value of J~ is Xtt#

We shall therefore s tudy the effective Hamil tonian 24' =H--ENos- -O)( 'L)o ,a The quantized field is expanded in a complete basis similar to tha t of Section 3. The Hamil tonian is

(4.1)

Eere

(4.2)

j/t o = ~ , { ( K ( , m [ T I K C m ) - - ( E - - f w > m ) b ~ , , } <~,,~,,~%~,,,,+ +

q- ~. (K~m; K 'Cm' IGIK"a"m" ; K " a ' m " ) + + . ~Kam ({K'a'm a~i"a'm" ~K"t~"m .. . .

* L

(KamlT]K 'a"n ' ) 2 M )%""'V-~cK'~"c d:~x

6 ...... ,aK~,(-- b - ' \ F , K ~1 r,~,,,~odo _ . . _ . ~ ' J I ) J g ° ' " ( " ) { D " m"-o 2 . _ ~ ,

1 (d) / ) o - - - - O

470 E . r . c~Ross

and

(4.3) <.Kam; K ' a'm' ]G I K"a"m"; K ' " a ' m " ? --

- - ~- do o' do ' WL,,,(o, {J )go~(o)g~",~,(9)go,~+(~ )ge',~=(9 )"

G conta ins fac tors 5 + . ,+ ,,6~+x,~,+x~.

W e n o w in t roduce the u n i t a r y t r a n s f o r m a t i o n

(4.4) W~ exp + * = [Coo/, aoo~, - - coo~, ao+,]

expres s ing the pr iv i leged role of the single par t ic le s ta te . T h e n

(~.5) W 1 aoof, W-1 x = Coo~, -4- aoo~, .

:For s impl ic i ty we wri te Coo ~ = c and aoo ~ = a s. T h e t r a n s f o r m e d H a m i l t o n i a n is expressed in the f o r m

4

(4.6) W I . ~ W ~ 1 ----- ~ . ~ , i = 0

wi th

gfo = ((00g T T l 0 0 g ) - - E - h~o /~ ) Iv?÷ (00/~; 00~ [V 100/~; 0 0 g ) I c ? l e l 2 ,

g l = E ' { ( 0 a g I T 100#) - - (E -4- he)g)} a + -4- h. c. -4-

+ ~{(o~g; oog[o[oog; oog) + (oo~; oa~Io[oog; OOg)}~+e*e~+ h. e.,

~ = ~ { (gain I T I K'o'm') - - (E + ~ )6~o)a+~o , ~ + ~ +

4- 2 ]e [2 ~_ { (Kam; 00# ]G ]ga 'm; 00#)-4- (gam; 00g ] G [ 00ju; K'o'm'))a+a~a~,,,~÷

4- c ~ , ( K a m ; - - K a ' , 2 / x - - m ] V]00g ; O0#)a+ a_~.,,2~,_,~.4, h. c. ,

• ~ f s = 2e ~. ( K a m ; K ' a ' m ' ] G [ K ÷ K ' , a", m ÷ m ÷ m ' - - g ; O0#)a+a.,a~,.,~, •

• a~+~,,..,,~+m,_l, + h. c . ,

~ , = H , .

I n the l imi t of w e a k l y i n t e r ac t i ng bosons ]cooj, ]2 is N , b u t for finite in te r - ac t ions the re is a dep le t ion effect which reduces the va lue to a f inite f r ac t ion of /~. I n the l imi t of w e a k in te rac t ions the func t ion is d e t e r m i n e d b y the

S T R U C T U R E OF A Q U A N T I Z E D V O R T E X IN Y~oSI)N S Y S T E M S 471

condition tha t d(f 1 vanishes. There will then be no constant te rm in the equat ion of mot ion of the a~o m. This leads to the requi rement

h ~ V o f (4.7) 2M "%0, + [ c]~%o~,(x) V ( x -- y ) ]Cfoo~(y) l ~ d3y = (E + tio~tt)cfoo,(x) .

This is substantial ly the same as eq. (2.4) of the semielassical theory. I t is the same condition as 8~o/SCfoo~,= 0 for fixed [c[ 2, subject to the multi- plier conditions

Cfoo~l 2d'~x = 1 , i]~ . Cqo,~, oo~ ?,~ d*~x = It#.

The operator which takes the place of the L ~' of eq. (3.21) is

(4.8) L ~ = - h~ / 2-Jl V~+ [cl~ V ( j x - - Y[ ) l%°"(Y) I~d~Y- - (E + hcott) .

I f we use the eigenfunctions and eigenvalues ~, .~, E~.o of L ~', and mult iply

(4.9) Lt'cf~,~ = E~oT~ m ~= E ~ - E + ]to):~ + 2 M ] 7:E~,~,

by * ~,~, , and integrate over space, we find

(4.10) ( K ' a ' m ' ] Y ] K a m } - - (E 4- ]io)m) ~,~, 5~.o, ~m.m' 4-

4- 2 ]c]~(K'Cm'; 00# [G [ K a m ; 00#) = E~o~,~,5. . . ,(5 ... . , .

~F~ then takes the simple form

= ' + a (4.11) jtf~ ~ E ~ a + m a ~ . , ~ 4 - 2 ~ . l c ] 2 ( K a r a ; O O t t ] G ] O O t t ; g a m ) a , ~ , ~ : o , m 4 -

2 + 4- ~ (Kam; - - K , a', 2 ~ - - m I G ] O 0 # ; O0#)c a~oma_~:.., ,t,_~ 4- h. c.

The main point is the strong coupling of a~.om to a_x...~._ ~. We ma y again part ial ly diagonalize ~ (neglecting the coupling of the radial modes) by intro-

ducing the normal mode t ransformat ion W2.

W ~ a ~ . . . . W-~ 1 ~ '

(4.12) W~a+~.o.~t,_m W ; ' ~ '+ a +

4 7 2 E . P . (;ROS, ~

W i t h

(4.13) { d~o,~ = E ~o,,, - - '2 ' c ]" (Kcrm : 00u i(; ; ()0tt : K a m ) ,

we find for Yxo~

(4.14) (1 + tgh 2 y~,,~) + ).eo,,~ tgh ),a. .... = 0 .

The zero po in t ene rgy is

(1.15) ~o = ~ g~ .... { sinh~ Ko~n

1 / -- . sinh :27K~,, . r

J

while the s p e c t r u m of exci ta t ions is

(4.16)

where

Thus , we have

+

sinh "z 2),Ko,,, /

(4.17)

[+ ! ~ W 2 ~ 2 W ; 1 = SO+ ~ ( A . ~ r a a K ~ , n ~ l £ . m = , '

+ ~: c~(K~,,; Ko', 2~-- ,~ I GIo0~,: 0o,)(%,,/~o, ~,_~)' + h. c. O~rl'

I n the lowest app rox ima t ion , ne~'leeting deple t ion effects and the modi -

ficutions of the self consis tent po ten t ia l arising f rom elose collisions, I c0o,l~= 37 and the L a g r a n g e mult ipl ier o) is zero. The results are then exac t ly the same

as for the semi-classical theory , and ma in ly provide a formal just i f icat ion for

add ing the shift zero-point ene rgy of the oscillations to the self-consis tent

field energy of the vo r t ex state. The s ta te veetors in this a p p r o x i m a t i o n are

~ffnew: W][ W2~old, where (i~ol 1 r u n s throuo'h a set of s tates consis t ing of a v a c u u m s ta te ~o such t h a t a~,crP o = 0, and a set of s tates ob ta ined b y oper-

a t ing separa te ly on ¢o with the crea t ion opera tors

~,~,,, --- ,*(x)q'Ao,,,(.r) d~x ,

~°o.: exp [2 (e<--~,%)].exp t~- ',+ ~+ l,. c.)]-~o,~ LZ--, [ z o m [ /~'am x,a,2~ m - - (l

~ ,TR/ ' I 'T I t l J c ; tpl" A t , ~ IAN ' I ' IZEI ) V~)tI ' I ' I :X IN BI)SI~N ~Y>'I 'I ' ;M < 4~73

I t is of some interest to examine the exi)ectat ion values of phys ica l quan-

t i t ies to see the modif icat ions of the self consis tent field a p p r o x i m a t i o n induced

b y the zero-point mot ions of the field. The expec ta t ion va lue of the dens i ty is

(4.18) ~'~ ( X ) : ( ~ 0 , ~9+( 'p) ~ 9 ( x ) % " = ( , ( / )0 , ( l I 1 1 ~ 2 ) - - 1 V ) + ~ / ) ( ~ 1 ~ 2 ) ( / ) 0 ) '

- - ]q-oo,,(x) i°-] e '~ = ~ sinhe 7~ , , , "g~ , , , (9) I : .

T h e ma in interest is when o-->0. Fo r m ~ 0 all t e rms t end to zero. The

on ly con t r ibn t ion is f rom m = I) and is ~ sinh *yA-.olg~o(o)]~-~ 0. The zero- K~

poin t f luc tuat ions of the s-wave phonons give a non-zero dens i ty a t the vo r t ex

line. We m a y define the expe(. tat ion value of the ve loc i ty as

__ 1 ih ,/~rj ( (4.19) vo - - , (x) 2 ,.\, o, ~p-

The dens i ty is now non-zero at o = 0.

]tH (4.20) leo,),, ~ " " - - ~"o',,( :, ) "

1 7~' i ~'~/'- '\ "'\

o ;9 ,5 ~9 ~') % ) '

Bu t the n u m e r a t o r is

]t ))1 i M - ~ sinh2 7~,,~ - - ' go , , (9 ) ! ~ " [) ( )

I t tends to zero as o ~ 0, since the m = 0 t e rm conta ins a f ac to r lira, and

is thus absent f rom the sunl. Hence , the expec ta t ion value of the ve loc i ty is

finite, and is in fact zero at 9 = 0 . F o r , u = 1

, ! i (~001 2 i ,(,I01(/)) ! 2

( 4 . 2 1 ) ~ ~-+o ~ ('2 = 0) - '

! since gol(O) goes as ~)~(~ ( 2 M E / ] P - ) o ) , where :¢ is a no rma l i za t ion factor , we have

(4.22) G - ~ i('""~2 ( 2 3 I E I ,.(c, -- iii /' \ /

Thus the vortieity w= 1(I/~2)(~/~o)(o~',, ) has the limiting vMue

T h e vor t i e i ty has a finite value in the (.ore. I t is clear t h a t these results depend on the general fo rm of the vo r t ex

w a v e - f u n c t i o n l l l l I f l )o , and are i nde pe nde n t of the precise values of IC0o,I 2 a n d y ~ , . As descr ibed in ref. (,b), ill a h igher a p p r o x i m a t i o n these pa ramete r s

are shif ted f rom the values given by the t h e o r y t h a t considers only d f o + d f l + d # 2 .

One m u s t consider the modif icat ions i nduced when the creat ion and annihil- a t ion opera tors are reordered af ter the no rma l m o d e t r ans fo rmat ion . This

474 z . e . (~ROSS

includes a diagonal contr ibution to the ground state energy from W2e%f4W~ 1. The best ground state for the chosen type of approximate s ta te vector is ob-

ta ined variationally. The parameters coos, ~0, and E, are determined by mini- mizing the ground state energy expectat ion value with respect to c~os, and imposing the two Lagrange multiplier conditions. Since we are not working with a plane wave basis, the reordering of W2~3W~ 1 leads to a modified self- consistent potent ial because terms linear in the creation and annihilation oper- ators appear. W~3/~4W: ~ also contains terms tha t lead to a modified normal mode problem, i.e. a redeterminat ion of the y ~ , . Following these lines we may obtain an improved value for the energy of a vor tex state wi thout phonons. One main effect is the replacement of the total density N/~2 by the superfluid density I coos [2/Y2 in the estimates. Fur the r improvements must consider a more general normal mode t ransformation to take into account the radial mode coupling, and also off diagonal contributions of ]t~(Jtf3~-$ff4)W~ ~. I t would be worthwhile to under take a systematic calculation to give u clear develop- ment of all quantit ies in the strength of the interparticle potential . A trust- wor thy calculation for the actual case of liquid helium would be difficult. I t is likely tha t the zero-point oscillation-spread of the core density is compa- rable to the self-consistent field contribution. I t is well known tha t the Bogol- jubov normal mode t ransformat ion is too restr icted for s tudying the phonon spectrum in higher approximations. The off diagonal elements of W~(.~ff~-+-~4)W; ~

must be considered to avoid spurious energy gaps. I t is, however, not our concern here to s tudy these refinements. Fu r the r

development might be preferably under taken by other methods (7-1o). The m~in point, however, already appears in the simpler t r ea tment presented here. I t is the specification of a special, highly occupied, ~ vor tex ~ single-particle state (Voo,(X). For weakly repelling bosons, where the self-consistent field con- t r ibut ion to the s tructure of the vortex dominates quan tum fluctuation effects, the picture is part icularly clear.

The wave function representing an e lementary vor tex state has an ex- pectat ion value ATh# for the angular momentum. The projected state with the exact N~/~, is of course orthogonal to the ground state wi thout a vortex. In our approximat ion the orthogonali ty occurs as 2¢--> c~, essentially be- cause the inner product

<e.,(,./.'(x)'°.--....l'o/.X, lk"(.),'(.)".--....]Oo)--o.,--',,.

(7) K. A. BRU]ECKNER and K. ~A~'ADA: Phys. Rev., 106, 1117 (1957). (s) S. T. BELYA~,V: Soviet Physics JETP, 7, 289 (1958). (9) R. AB]~: Prog. Theor. Phys., 19, l, 57, 407, 699, 713 (1958).

(~0) •. 5[. HUGI~NHOLTZ and D. PIN~s: Phys. Rev., l i6, 489 (1959).

STRUCTI~I~E OF A QVANTIZED V(-)IYI']~X IN BOSON SYSTE~IS 4 7 5

Cf. the similar discussion of the periodic ground states in ref. (4b). This argu-

m e n t holds when we consider corresponding states wi th a small n u m b e r of excitations. Bu t it breaks down when there are of the order of N excitations,

for example a t a finite t empera tu re near the ),-point. Fur thermore , the t e rms W ~ s W ~ ~ and W25f4W~ ~ create and annihilate only a small n u m b e r of par-

ticles in a single act. Thus an e lementary vor tex can only decay b y a ve ry high order process in pe r tu rba t ion theory, i.e. it has an essentially macroscopic

lifetime. We do not t ry to es t imate this lifetime here. Wi th a large n u m b e r of exci tat ions the vor tex s ta te becomes seriously depleted and the core grows in size ~nd fluctuates. There are then rapid transi t ions to ~ circulation-free s ta te with product ion of phonons. The problem appears difficult to discuss

precisely f rom the present point of view. We have an overcompleteness of

s tates characterist ic of self-consistent field theories. Each ]"(x) and the asso- ciated exci ta t ion spec t rum presumably spans the same space of s ta te vectors. The approx ima te wave functions can then be accurate only when are there is a

small numbe r of excitations.

5. - Further d iscuss ion of the s e m i c l a s s i e a l theory.

The equat ion of mot ion for the classical field ~t,(x, t) m a y be rewri t ten in hyd rodynamic form. Pu t t ing ~0= E e x p [ i S / ? @ where R and S are real, one

finds

( ) (5.1.) ~ (R -~) = -- div R ~ VS ~ - ,

PS (VS) 2 h~ V-'R /

(5.2) - - ~- t V ( x - - y ) R ~ ( y ) d 3 y . 7_'t 2 M 2 M R '

We introduce the velocity field v = V S / M and take the gradient of the

second equation. Then

(5.3) M ~ - + ( v ' V ) v : - V H = 0 ,

H is given by the functional

(5.4) H -- 1~2 V2R

2 M R + ) V ( x -- y ) R 2 ( y ) d 3 y •

consisting of a <~ q u a n t u m ,> contr ibution and a contr ibut ion arising f rom par-

ticle interactions. I t has the dimensions of energy. In the usual h y d r o d y n a m - ics one assumes the pressure p is a function of the densi ty and H = f d p / n . But the present hydrodynamics is quite different. I t is useful to introduce

476 E.P. ~m(,ss

the density ~ = R ~ and to write (summation (.onvention)

(5.6) ~',l ; (~tt',) : o , ? t 7,rk

; ~ 717 (5.7) 31 ~ (~r,) + Jl =-- (~l', r,) -- ~

C.I'~. ( d',

n(OH/Ox~) can be expressed as (")

(5.8) , . - ~l = - J V(.;'-- ~t),~(~/)d:~/ ( ( 1 " k ( E l , ' , • ~ ~ , 1 , ; .

with a stress tensor

(5.9) a,~, -- 2UII '~ ~'~-):~).,', " - ~ (lI~ V : F 1 (,1'~ ( J ' l

This differs from ordinary hydrodynamics in tha t the stress depends on deriv-

atives of the density, rather than velocity.

The elementary line vortex has tile propel ty that V~,~- 0 and VS.VR = 0,

and that R = 0 on the vortex line. The quantum pressure - - (li2/2M)(V'2R/R) ---->

-->-- oo as p ---> 0 and cancels (V,u):/2M which --> ~ oe, leaving a finite quan-

t i ty. I t is clear since the vortex core is small in extent, that we can generalize

the argument , and find approximate solutions representino' steady patterns of

vortices separated by distances greater than the core diameter. One way is

to look first for solutions of V°-S 0, e.g. ring vortices, sets of line vorti-

ces, etc. Then one finds the appropriate behaviour of R near the lines of

singularity. This brings into play the type of classical hydrodynamic argument

already used in interpreting experimental properties of superfluid helium. The

main role of the quantum meehani(.s, at least for weakly interacting bosons,

is to provide a foundation for this procedure, and to ensure tha t there is a definite theory of the structure of the vortex (.ore. This type of consideration

is foreign to classical (even compressible) hydrodymmfies. The structure de-

pends on the quantum pressure term, as is evident from the fact tha t the

characteristic size depends on a De Broglie wavelen~'th. In addition, the s tudy

of the quan tum state vectors associated with a flow pat tern should show that

decay is possible only in a macroscopic time. Formally, each solution of the hydrodynamic equations defines a basis

funct ion which is occupied by a finite fraction of the particles. Such a state

is not as symmetrical as the elementary cylindrical vortex and would prob-

ably not be considered from the point of view of ordinary quantum mechanics.

The complete set of functions orthogonal to the basis function permits an

(11) T. TAKABAYASI: Prog. Theor. Phys., 8, 143 (1952); 9, 1~7 (1953).

5 T I I [ ' I ' T I H] ; ~)I ' A Q I k N T I Z I ' ; I ) V ( ~ I ' T E X IN B I ) S q ) N S Y S T E M S 4: '~7

a n a l y s i s of the smal l osc i l l a t ion spe( , t rum c o m p a r a b l e to t h a t of Sec t ions 3

a n d 4.

P h e n o m e n o l o g i e a l s ingle f luid h y d r o d y n a m i c equa t i ons h a v e been used to

s t u d y processes such as s c a t t e r i n g of phonons a n d p r o t o n s f rom vor t i ces (~").

These theor ie s a s sume a p re s su re d e n s i t y r e l a t i on such t h a t (dp/d~) ½ agrees

w i t h t h e o b s e r v e d f irst sound ve loc i ty . I n c o n t r a s t t he p r e s e n t t h e o r y con-

t a i n s t h e q u a n t u m pressure t e r m a n d is d i r e c t l y r e l a t e d to bas i c q u a n t u m

mechan i c s . H o w e v e r , the de fec t of our p r o c e d u r e is t h a t on ly w e a k l y i n t e r a c t i n g

pa r t i c l e s can be t . reated in a c lear m a n n e r . I t shou ld be possible , in t he sp i r i t

of a p s e u d o p o t e n t i a l m e t h o d , of B r u e c k n e r ' s ideas , or of L a n d a u ' s t h e o r y of

F e r m i l i qu id t.o j u s t i f y r e p l a c i n g the i n t e r a c t i o n p o t e n t i a l ab in i t io to t r e a t

s t r o n g i n t e r ac t i ons . T h e n the semie lass iea l t h e o r y i t se l f w o u l d b e c o m e usefu l

for q u a n t i t a t i v e a p p l i c a t i o n s to supe r f lu id he l ium.

The a u t h o r is i n d e b t e d to P ro fe s so r I)AVID Bol t s i for m a n y v a l u a b l e dis-

cuss ions . H e also wishes to express a p p r e c i a t i o n for t he h o s p i t a l i t y of t he

U n i v e r s i t y of Br i s to l a n d C E R N , a n d for f inanc ia l s u p p o r t f r om the N a t i o n a l

Sc ience F o u n d a t i o n , a n d the office of Scient i f ic Resea rch , U.S. A i r Force .

(12) L. i ). PITAEVSKI: Not'iet .Physic,s' J E T t +, 8, 888 (195s).

R I A S S U N T 0 (')

Sviluppiamo una teoria delle eccitazioni dei vortici lineari elementari, per un sistema di bosoni debolmente repulsivi. Lo stato di vortice b, carat ter izzato dalla pre- senza d i u n a frazione finita di particeiie nello stato di part icella singola con Inomento angolare intero. La dipendenza radiale dello stato densamente occupato segue da una equazione di campo autocongruente. La funzione radiale e la densit'£ di particelie associata sono essenzialmente costanti dovunque tranne che nell ' interno di un noc- ciolo, dove si annullano. La dimensione del nocciolo ~ uguale alla lunghezza d 'onda di de Broglie associata con l 'energia media di interazione per particella. I1 valore previsto per la velocit,~ ha la dipendenza radiale del vortice claasico. In questa approssimazione di Hartree la vorticit~ ~ nulia dovunque tranne the sulla linea di vortice. Quando si raffina la descrizione dello stato sino ad iacimlere le (~scillazi<mi nel punto zero del campo fononico, la vorticit'~ si estende a tu t lo nocciol,). Questi r isultat i confermano nei punt i essenziali gli argomenti intui t ivi di ~)nsager e Fevnman. I fononi etm si muovono normalmente alia linea di vortice hanno eccitazioni congruenti e~)n quantit'~ di moto uguale e contraria r ispetto al substrato di partieelle in movimento ehe eostituisce il vortice. I1 movimento del vortiee risolve ]a degenerazione dei fononi di Bogoljubov l ' ispetto al numero quantico azimutale .

(*) Trox~uzio~e a cura della lCedazio.~.

3 1 - I l N u o o o Cimento .