The principal device eml)loyed, :ts in lxlrt I, is 1IIC project ion olwrator which selehs the open channrl components of the W:IW function. It is found that the forrnd structure of prt. I providing a unified tlrscriptiorr for dirwt :md conpound nuclear reactions including the cor~plrtl rquation clewription for direct rewtions renmins valid in this wider contrxt A Ik~pu1~~l’cirrls espmsion m:~y dso be rr:ttlily ot)t:tincd. The concept of ch:~nrlel rxtlii is not nretlrd nor is any deconiposition of the LS:ive function for thr system into :mgul:tr momentum eigenst,:kt.cs required, so that t hi esprwsiolw for I ransition :tmplitudes and widths are invariant wit,h rrspect to the angul:~r ltloriientrrn~ coupling schcmc. Since thp OIIPII channels can only Iw ~lefirl~tl in XII asymptotic sense, the corresponding projwtion opernt ors :IKP not unique. As :t consequence the projection opcator method has :t flcxihility lvhich in thrs first place is consonant with the \vitlr r:mge of I)hcnomcln:l \vhich c:m occur in nuc.Ic:Lr rewtiow :tnd in t hc second ~)l:tw can c~f’l’e~tiwly rxploit XI intuitive nnclrrst:rnding of the phenomrnx. li:\-:unplc~ of pr”.jwtion opcr:~tors :II’P obtained inclu(ling one which lexls to t hr \2ii~npr-15i~p~Il,llti foun:~lisnr. :mot hpr \vhich is :tppropriate for the strip[)ing wwtion. :rnd, finally. one \I-hich t alies the I’nuli rxclrision l)rinc.iplc into account Note, that clxl)licit r.cl)~.(~~(~~~t:~tiolls of the projwtion oprr:ttors XI’P not required for t hr tlrvelopmrnt of general formd results I)rlt :ir(’ rrwws:try if, rvrnt u:lll~,, ct~wntit:ttivc c:llcrll:~tiolis :~rp tiiatlc. I. INTI~OI)~~CTION
In this p:tpw ltesctions” (I )’ with developing and compound 1x1 both cxsily
the forrnalisn~ dcvclopcd in “?I I’nificd Theory of Swlcur is gcnernlized and irnprovcd. In I wc wcrc principally conwrncd :L theory of nwlcar reactions from which :l description of direct nuclear proccssrs, as ~11 a:: tht mnplcs potential model, would and nat~uxlly alwtractcd. This inwcnpahly led to :I dcri\-ation
* This Lvork is supported in part through .4Is:C (‘ontrwt vided t)?; t.ho IV.9. .4tomic b:nergy Commission, thr ( )Iiire Force Oilice of Scientific I
AT(30 of T:,v:d
I)-2098, by funds prw I?ese:~rch :tntl thr Ait
of resonance reactions which did not involve the concept of the channel radius, nor did it require t,hc decomposition of the wave fmlc+ion of the systrm into orbitma angular moment,um eigenstatcs. although our treatment was successful, t,here wrrc a number of limit,at,ions. We considered only situations in which the mass nunlbcr of the incident projectile and that of the emergent light part’iclc were the same as they would bt in an (n, p) or (n, n’) process. St’ripping and similar prowsses were not treated. Secondly, although it was possible to show that t,hc effects of the Pauli exclusion principle and exchange scatt’ering would not change the structure of the formalism, the discussion was rather indirwt and did not provide a method for the evaluation of these effects.’ Finally the connertion of resonance formalism in I with the boundary condit#ion methods of Wignrr and Eisenbud (2) and I
hy J. S. Bell, the discussion in I was incomplete. are given in Appendix 13 of “Radiation of LOUwritten with I). Uennie and submitted for publi-
NUCLE:AH
ItEACTIONS
289
the direct rtact,ions, for potential scattering and for “single particle” resonances. The rapidl,y varying part, gives rise to t,hc narrow compound nuclear resonances; t#hr widt)hs for these are expressible as matrix element,s involving the “single part,irle” wave funct,ions of the dirwt rewt#ion Hamilt’onian. In another treutment the effcct,ive Hamiltonian is not, broken up into a fast, and slowly varying parts. Instead the transit,ion matrix is expnndcd in t,erms of the eigen solutions of the effective Hamiltonian for the closed channels ohtaincd hy eliminating the open channels in the ahsenw of any incident, wave; the open channel part of t,hew eigcnfunctions sat,isfy an outgoing waI,r 1)oundary condition. The cigenfunctions have complcs eigcnvnlues and are clearly of the Kapur-I’rierls t,ype and t,he rrsulting expansion of the trnnsit,ion matrix hccomes 311 expansion OVPI resonanw::.” There are howcwr several signifirant. differences between this reslllt, and t,hat of the standard Iiapur-J’cierls formalism. These all stem from the fact t,hat in the present work not even the concept, of nuclear radius is used. Our resuli s hold thcrcfore even when t,ht> pot#ent’ial involved does not have a sharp rutoff. As a conse(luerwe t,here is no need to dccompost the wave funcAtion int,o eigenstates of t’hc orbital angular momentum. IGnally, thr widths art expresstd in terms of matrix elrments of the interabon rather than in terms of overlap int,cgrals hetSween inside and outside wave functions at, the nuclear radius. These t,wo expressions for the transit,ion matrix, that of I which we shall call the cffcc%i\-c Hnmiltlonian method and the method of expansion in complex cigenvalues. are of course equivalent,. However it, seems to us that the efiect,ive Hwniltonian method is most conwnicnt for nuclear reactions in that it differenGates hetwren the narrow compolmd nwlcar resonancw and thr “single particle” giant, resonances ; i.e., brtween compound and dirrrt’ nuclrar reactions. In t’he romplrs eigrnvnlur expansion both kinds of rrsonanws arc lulliped togrt.her indiwriminat~ely. The projection operator which srlrcts open channrls is not uniclue sinw it is possihlr to define oprn channels only in terms of the asymptotic behavior of the wave function mhcn the rrwtion products are far apart. This gives thr formalism an additional flexihilitly which is extremely usrful since it allows one to choosr t,hat projection operator whirh is most convenient, for the problem under invrstigation. In the present paper wr arc for the most part, cwncerned with thr type of project’ion operator employed in I in which the open channel war-c flmct~ions are defined by ext,rnding the form of the wave function lvhich is valid asymptotically to all of (*onfiguration spaw. This mrans that ollr open rhannel J These
eigenvalues
twrisit~ion
elnployirlg
*Sieged -H:m~ldet series. ~liscussion see IIumhlet
are functions
of the cnerg,~-, h’, of the systenl. i\n e>-p;lnsion of the which are independent of E is referred to as the Such :I series is derived in the appendix to this p:~pe~. nor :, yevent and I~osenfrld 151.
vollll)les eigcnwllucs
wave functions are linear superposit,ions of the possible residual targrt, nwlenl states. The class of these states arc rest’rict,rd of course by energy c*onservstion and ot’her conservation nlles which may be pertinent. We also briefly discuss another projection operat#or which introduces the channel radius into the theory. The projection operator here is unitmy for regions in configuration spaw whew the partirlcs are separated by distanws great,er than t,he channel radius, and is zero elsewhere. if t’hc channel radii are chosen so that cscept for Coulomb or ot,her long range pot,rnt’ials the interaction vanishes where the projection oprrator is unity, we are directly led to t#hc Wigner-ICisrnbud (2) boundary conditions. These two types of projection operators have essentially complementary domains of usefulness. The Wignrr-Kisenbud projection is most, convrnient when details of the internal region are not under scrutiny. Thrre is no difficulty with the Pauli prinriplr and tschangc scattering. The resonance energies have a definit,ion indcpcndrnt of the incident, energy. The project,ion operator used in I is appropriat’e when drtails of the int’eract,ion arc of inter&. It is the natural ext,ension to thr ront~inuum of methods employed for bound statr problems and at, t,he same time it can be easily joined on t,o t#he high rnergy multiple scuttrring limit of Krrman rt al. (6). Therr is t,hus no nwd to cahunge thr formalism as one’s at#tent,ion changes from negative to pos;iGvc and finally to high rnergirs. The closr relat,ion of t,he shrll model pot,rntial and t.he real part of t#hr complex potenGal hecomrs manifrst and one rralizes t,hat# the residual potential is rrsponsible for t,hr variegat,ed phenomrna which occur in nuclear reactions. On the other hand with t#his projection oprrator type t’hr trcatmc~nt of the Pauli principle (discllssion Section III) is not’ simple although it’ dors srrm mnnageablr. Srvrral writrrs havr cont~ributetl to thr eclui\-alent Hamiltonian method since the publication of I:’ Wr me&on Hwnig (7’), Srn-ton and Iconda (8), &Ygodi and Eberlr (9), and most rrccnt,ly Lipprrheide (IO). Each of thrsc havr developrd material whirh overlaps somr of our work as n-e shall duly note Mow. Lnnc and Thomas (11) and G. Rreit (12) give ewrllent, reviews of thr Wignrr R matrix theory and reacation theorirs in grneral, whilr the most recent, treatment of t,hc rspansion in resonanws is givrn by Rosrnfeld and Humblrt (5). Wr conclude this introdw+on with a hricf description of the rontrnts of the paprr. In Sec%ion II t,hr general throry is drveloprd in terms of projr&on oprrators. Specific forms of thew operat,ors arc deli\-cd in SecLtion III for the stripping reaction, for incorporating t,he Pnuli principlr. and for thr Wigner formalism. In Section IV, rrsonancc t’hrory is devrloped and an expansion in terms of rwonances of a Kapur-Peiwls type is dcri\.rd. Section IV dors not drpend upon t,he d&ails of S&ion III so t,hat it can br read dirwtly after II. In an A1pprndis, t,he Scigrrt,-Humblet, description is dwivrd. Thrw will hr some re\-iew of the 4 See I for
the c:trlisr
refrrsnces
earlier paper on t,his subject, 1, bot#h for tht purpose of completeness as well as to pxmit some further remarks which round out the earlier discussion. The interaction of gamma rays with nuclei will br discussed in another comnmnication. II.
THE
WFIXTIWS
HAMILTONIAN
When a projectile a st#rikes a target nwleus X, a \-xi&y of reactions can occur. Ela4ic scat,tering in which the final product,s are agzlin a and X will always tak.e place. In addition t#hcrr might be various transmutations in whic&h the final product’s h and I’ (b can itself be composite) differ in some int,rinsic* respect’ from a and X respectively. Those reactions (including elastic swt,tering) which are encrget’irally allowed are referred to as open channels; the others are closed cahannels. Let t’he wave functions dew-ibing t#he energetically possible’ residual nllc~lei hc drnot)ed by c$, . Note that, these functions are generally not mutually ort’hogonal nor do they nwessarily involve the same number of coordinates. Asympt80tically, when the rrac.t8ion prodw+ are well separat8ed, t,hr wave fun&)n for the spstem will take on the antisymmetrized form:
whcre,f, arc functions of t,hc wordinnt,cs not contained in +i and of course satisfy appropriat#e asymptotic bolmdary cwndit8ions. @ is the :~ntisymmrtrization optrat’or. It is now possible to define t,hc projection operutor which sclrct,s t,hc open channels. It. is any projection opcratx f’ which, operat,ing on any ant)isymmetrical fun&m x, satisfying t,hc same asymptotic boundary wndit,ions as q, b111, otherwise arbitrary, yields a fuwt~ion which asympt~otically is of the form ( 2.1 ) : 12.“) The quanGtirs 11,are fuwt~ions of the (wrdinates not wntained in 4! satisfying the same boundary conditions as ,/“, , as well as c~onditions imposed by invariant principles. A$ simple example of such an opc&or is t.hc one PwE which, as we shall show in Swtion III, leads to t,hr migncr-F:iscllt)lltl formalism:
292
FESHBACH
Another example is furnished by the projection operator P1 employed in 1 which we shall now suitably generalize. Consider the set of all antisymmekzrd wave functions Q, satisfying the same asymptot,ic boundary conditions as q which have t,he form: a = @c
ZL$#JL
(2.41
The uL approach t,he Ui of Iiki. ( 2.2) axymptBotCirally. The set (a] subtends a portion of Hilbcrt’ space (let’ us call it the “open-channel subspace”), associated wit.h the Hamilt.onian and the other pert,inent’ operators of the system. Moreover a projert’ion operator I’ I must exist which project#s on t,o t,his open channel subspace. In other words if x is an arbit(rary stat,c vector, P1x helongs to the set @‘; that is, P1x can he expressed as a sum over the open channel residual &ate C/J~. We shall obtain explicit expression for PI in Section III. Sate that, Pw, and P, are only two examples of posnihle projection operat,ors which satisfy condition (2.2). IJIany others are possible; wh.ich one is to be used will be determined by t,he physics of thr phenomenon under investigation. It, is important to realize t’hat, t,he remainder of the discussion in t,his section, as well as t,hat in Section IV, requires only th.c exist,ence of t’he projection opcrator. If 9 is t’he wave function for t#he system it is clear t,hat. we need only ransider FP in order to obtain t#he asymptot,ic behavior of + and thus the scattering and t#ransit,ion amplitudes. It, was t.his fact’ which was employed in I for the case considered there and it’ will be useful principally for completeness to rephrase some of t,he results of I in terms of an explicit’ project,ion operator formalism.” We notIe t#hat’ we have already oht#ained the generalization of I to situnt’ions in which the final reactions products can be arbit8rarily complex. There is no limitat,ion on the number of open channrls or on t’ht nat.ure of t,he emergent part#icles. All we need to do is construct t,he appropriak projeck)n operator. Let us rephrase the calculation in I up t)o t#he point, where we obt8ain the cqui\ralent, Hamiltonian Ccalled generalized optical pot’cnt,ial in I) . q sat.isfietl t,Ee equation
6 This option wvas n~entioned in refs. 8 and 9.
in I. w:ts partially
described
in ref.
4 and hats been emplopcd
Since @ f0110\vs8
projeck
int#o t,he calosed channel
suhspace
we can solve
(2&b)
as
(2.8) SubstitAng
Eq. (2.8) in likl. (2.(k)
yirlds
t,hc Schroedingrr
(,B - H)Z% = 0 where N., the effect~ivr Hamiltonian,
cquat’ion for Pq:
(2.9)
is
Eyuat~ion (2.9) is identical in form t)o Eq. (17) in ref. 4 and Eq. (2.15) of 1 but has a wider applicability as a consequence of the generalizat8ion in t,he tlcfinition of 1’ given above. ?clortover, bccsause the form of 1’:q. (2.9) is identical to t,hat of Ect. (2.15) of I, tht deductSions in I, including t,he resonance formulas, the complex poknt,ial model, and the direct, rea&ons based on Eq. (2.15) of I, remain valid in the present contest. We shall discuss tbcse at grcukr length in Section IV. For the prrscnt let us render Eq. (2.10) less formal by determining I’ for some csamples of rnlclrnr reactions. However let. us emphasize that the formal rcsl&s of Sect,ion I\’ do not8 rrquire explicit, expressions for 1’. The need for such expressions o~urs when cvaluatSions of t#hr various matrix clcments for various ;~ir~clear models or cvent~llnlly from nn~lcon~nll~leon forces arc being made. I I I. 1’13 )JE:(‘TION
OI’ISIL4TOIW
In this scvt’ion the projec%ion operators of t#hr type I’, corresponding t,o various possible final st.:lt,es arc esplicitl? T given. Wr have chosen to csprcss t#hese in conliguratioii spac*cl. How-ever it is also possible, and it is often more convenient, to use t,hc language of second cluniit~ization. ’ Thr translat.ion, configuratlion space to SWOIK~ cluantization is cluitc straightforward for t#hc cases considcrcd hrrc, so that, w will lravr it to the Icader to rcwritc our resuks in the latter form. We c*onc*ll& this svction wit,h a tliscsllssion of I’ WE and t’hr ~~ignc~r-I~~is~nhud for malism. ( a) WC considrr first the vast whcrc the inc*ident, and cmtrgcnt, part,iclc art sam(‘ 1~11:arc not identical with t)he particles in the target nuc~lcl~s. l,c,t8tht, wnvc
294
FESHBACH
function for the ground state of the target nucleus he do and suppose that the incident particle is sufficiently energetic, as to he able t’o excite t,he first, II excited states of the t’arget whose wave full&ions are +j , j running from one to n. Then (3.1) Inserting
Ho, = ($dWo) IO The particle to he “picked up ” is not antisynlnletrized with respect to the particles the residual (-1-l) nuclear wave funct,ion & This :rns:ltz implicitly assumes that particles act essentially as :t spectator particle, not particip:Ltiny in the reaction.
in :l-2
This is the form from \vhkh most, direct intcrx%ion theories of t,he picak up process &art. Tbrre is one import,ant diffcrenc*c. lfquation (3.7) cor&ins t,he resonance phenomena as dell for t#he incident neutrons or if me are disc:lSng t,he inverse process, &ripping, the resonance phenomena for the incident dcuteron. We shall discuss the rcsonanc( terms in Section IV. Our discussion is not yet compl&. We should continue t’hc proj&ion proc~lurt~ one st#tp furt’her so as to isolate t,hat, part, of t,hc have funct,ion w(r,, , r,) I\-hich describes t,he process in \\-hich tbc residual nuc:lc~~s wave’ funckion is 4,)~rl t r.,) . l’omarcl this rnd n-c write wtrcl , rJ \vhcre I’,v is t,hc projec+on
= P.du + QNw
operatot 1’3 = .fX.T
:md QN
‘I’he fun&m
=
1 -
I’.%J
IU t,hrn assumes :he dcsirecl form zdr,, , rI) = u(r,,).f(r,)
+ t&w
(3.8)
whrrc ‘I = (,f(r1)2u(ru , rd Sllld (.f,
QNW)
=
(3.9)
0
Sotc t,hat 21 will asympt~ot~i~ally clrscrihe t’hc iwidcnt ( if any) and emergent, nac~lcon waves, while Q,“w, thtx remainder of w, will provide a similar description of the dcwtrron. Tht fmwtions u ant1 QNw sat,isfy t#he following 1,G of couplet1 f~clliations [E - H’(rJ]u(r,,)
11:quat~ion (3.10) is wry similar in stxuc%tuc~ tSo13~1.(3.3) and suggests that as a gencml princ4plc it, will lx> possihlc t’o dcwrihc a rcxc:tion invoking t,hc> amptit,llcles for vnriolw opcw ~hnnnrls iu trrms of c~~uplecl S~hrocdinger type equations involving cfic~ctiw HaInilt,oiiialls. We consider nest the> more realistic* situation in \I-hkh 4,) is not proport8ion;t1 to +bo; i.t,.. 1<(1. (:<..i) is not valid. Then P == &‘(rl
. r,j)(4,,‘(r1
r.,) + $dr?
r,))($drt’
.
r41
(3.11’)
whew 4”’ = N[qh - $,I(+” , &)I;
(Ql,,‘, $h) = 0
(3.13)
296 N is just
FEHHBACH
a normalization
chosen so that (40’ , 40’) = 1
Kate that projection operator (3.12) reduces to (3.6) when +0 is proportional to $0 . With (X13), and with suitable rearrangements FCq. (2.9) becomes an (quation for
I*
= u(ro)40(rl . . . rJ
+ QNw(fO, rl)lC’0(r2 . . rA)
(3.14)
where (3.15)
QN = 1 - 40)(40 The equations
for u and QNw are identical w -
(4” , I’HP4,)lu
in form wit,h Eq. ( 3.10) : = (4lJ’Hl’QNW~O)
where H is given in I’,({. (2.10). (c) The inclusion of the Pauli principle into the theory of rract,ions has been the subject of several papers (10, 11, 14-16). In t,hr prwent, treat,ment \vc avoid the subtleties and complexities of the t,imc dependent, treatment, in which t,he behavior at infinite tIimcs must, be examined with considerable care. Our cffect,ive potSential will be hcrmit,ian; some t’reatme& drduce nonhtrmitian effective potentials. For simplicity we again consider the cast n-here the incident, particle is a nucleon. Moreover we assume t#hat’ it is not energetic enough t#o excite t,hr target, nucleus so t#hat the only process cncrget’ically allowed is elast,ica scattering. If the ground state of the target nucleus is t’hc antisymmetrized wave flmction operator we require has the following property 4drl . . . r.4) t’he projectIion
P*(ro,
rl
. . rl)
= @u(r,)4,dr,
(3.17)
ra)
where C%is the antisymmetrization operator and + is antisymmetrized. If 9 is the wave function of the system, t.hen t)he function u( r,,) asymptotically yields both the direct and exchange scattering amplitude. The operator 1’ selec+ ollt of JP that part which describes the target’ nucleus in t#he ground st,atc. The equationI determining u is
(40(r1 . . r.,), [Wrorl . . ra) - @u(r0)4dfl . . . rl)]) where t’he integration is over the coordinates of the inner product in (3.18). That condition as follows. From (3.18) we have I1 In the discllssion to follow, 4 is :tn arbitrary nte boundary conditions at infinity.
= 0
which are common (3.18) ib sufficient function except
I:rom this last ccllutSion \ve SW t#hat# ~rwe (i2.18) is solved, WC can hrcak up q into t.wo mutually ort8hogonal parts of which one is t,he desired form (i3.17). It’ is &ill ncwssury to show that the rrsult,ant, form R( u#J,,) can hc writ,trn as a projrboll opcrut#or 011 \lr. But for t’his we nerd first. to sol\-c 1’:(1. (3.18). I’wforming t,hc indicated opclrat,ions Eve oht,ain
‘l’hc krrnel K has several imporbant, properties. wave function for a hound st,ate, K decreases hwon~c~ infinite. K is hounded:
type (3.‘~‘2) at
( 3.21’)
It, is hermitian. Since 4” is the cxpownt~ially as either rl or ru
‘l’hc traw of K is finite and equal to 11. It will he useful to keep in mind as an r~.u~wple t,he lwrnrl K WC oht’ain when c#,) is a Slntcr dct,erminant composed of mut~unlly ort,hogonsl rlcments ‘wi Then
.A K = c w,*(t)w,(rO) 1 Bcfor(~ solving t,he inhomogencous tquat,ion the eigrnvaluc problem of the wrmsponding
(3.X)
(3.21) it, is wnvenient homogeneous form :
u,(roJ = A, (K(ro j rl) u,(n) ) From
the proprrtirs
of K it follows / X, 1 1
t,o consider
(:r.ar,,
t)hat# X, are real and 1 and
c
1
a
= A
(3.26)
298
FF:SHH.‘.(‘H
The eigenfunc%ion complete. However,
u, will form a11 orthonormal stlt, which is not necessarily K can he expanded in terms of normalized 21, as follows (3.27)
For example, (3.X), { Us} = (,w,} and A, = I for t,ht cnt,ire set. The case where t’he X, eyual u&y requires some special considerations. Denoting these eigenfunctions by 71al , wc note that they sat,isfy IL,I = (K(r0 I rdud) 01
($O(rl ... r.4), ~~~,l(roi40(rl From this equation
. h))
= 0
we have
(~t~d(rO)+O(rl
r.<), Ckl(rdddrl
.
r,,))
= 0
or @zh(rO)+drl
(3.28)
. . rA) = 0
For example (3.24) ; t#his is just t,he determinant,al t8heorem which states that the determinant vanishes when two columns are identical. Marc generally it, may be shown t.hat# if condit.ion (3.28) is satisfied, $0 itself is given by hl(rl
. . . rA) = CkI(r1)1c/(r2
. . r.d
where J/ is antisymmetrical. Hruce by examining 4,) OIlC can pick out those u, , u,, , whose eigenvnlue X, = 1. Auother consequence of immediak interest, is that, ( 11,l ) I’)
= 0
(3.29)
To prove this, simply note t,hnt t#he uhore bracket (uaI(rOb(rl and since \Ir is antisymmctrkal
. . r.$), *)
it, is proport,ional (@u,,(hb#dr,
is ident8ical wit,h
to
. r.4), *)
which in virt,ue of (3.28) is zero, proving (:‘,.39). We now return t,o Eq. (8.21’). Employing t,he expansion the orthonormality of t#he set, lu,} we find ( lLa, 11) =
(3.27)
for K, and
A, # 1
(3.30)
l“illtlllJ',
(3.81)
As indicated, the terms in u,l--tShc amplitSude of t’he eigenfunctions whose eigcnvalw X, are unity-arc not2 dct,ermined hy IQ. (3.31). However, hecar~sc of ( 3.28) t,hc function of interest, a( 2~4”)) does not c*ont,ain these terms. In fact : U7c(rfl)$drl
.r.,)
= Qu’(rll)+O(rl
14nallp, making 1~ of tht> antisymmc%ry
h)
of q TVPha~c
I’ is th.c dwirttd projection operator which projc&s out of a fun&m t#hat, part. which cm tw writkw c~u+~ (cf. 11:(1.(3.17) ). To prow that it is a projcc%oll opc>rator \vo nwd only note that I’ is hcrmitiau , and from I~Zq. ( 3.20) that. (I’xv, ( 1 - r’)*)
= 0
‘I‘htw two propert,ies haw t,hc c’onsequrnw that, II” = I’, proving that I’ is a projwtion operat80r. It is amusing t,o note that when 9” is a Slater dekrminant @(u+~) is just, ~t+~)(+, awording to (3.35). This result’ tells us that in that. cxamplc, t,o project ollt the ~dcpendenw on +, we need only employ t,hc naive praj&ion opcrat,or This resldt, is no surprise from the swond +,J(+,, and thw antisymmctSrizc. cluantization point8 of \-iew sinw t#hc appropriat)cl projt4on opcmtors in this cxsc can tw writt,en simply as a product, of crration operators for the st8at8rs c*orrrsponding to w, ac*t,ing 011 thcl “vacuum. ” Thr mow general operator (3.36) cwrresponds to the errntion operator in which t,hr A psrticlcs arc creat#ed in the stat,c &, AUthough tbc simplicity of this rwnlt for the simple dcterminant~al 4” culllot. I I? maintairwl for mow complicat~rd 4,) there appears to he very litt,lc difficwltp in solving Eel. (32.25) for uu and X a as long as $3” may he writt,en as a liner cwmhinat~ion of Slatcr dtt~erminants. WP have thrwforr provided a dirwt and prwt~iwl met,hod for tbc det crminat.ion of the projwt,ion operator which
300
FESHBACH
selects out that’ part’ of * which represents the t,arget nucleus in it,s ground statt,. Equat~ion (2.9) now becomes an equat~ion for B( u+,) . The existence of sucsh an equat#ion is all t,hat we need for the derivat,ion of t,he resonance formulas, tht optical model, ct#c. The antisymmctrizatio11~11 operators give rise t’o exchange interactions in the cfiect,ivc Hamiltonian. Kot,r that the cffwtivc Hamiltonian operat,or is hermitian and that from the asymptot,ic behavior of u(rO) me can obtain the ela& scatCtcring amplit,ude which inc~ludcs both t,he dirwt and exchange scattrring amplitudes. It may he convenient, to derive an equat,ion involving only ON wordinutr, say r. . This quat~ion can be obt#ained by mult’iplying 1%~. (2.9) from t,he left over coordinat#es rl . . . r.i . The prop&k of by +O*(rl . . . r,,l) and integrating the projection operat,or permit some import’ant simplifications in t#he resultant equation. Xote that Eq. i3.18) can be ren-rit,ten 7’(r0)
=
(40(r1 ... rA), *)
= (40(r1 ...
r.,),
This equation is valid for Amy antisy~~~wtrical function most convenirnt~ t,o consider t,hc equatJion for C’ which, the symmetry of H hccomes:
T’*)
(3.18’)
q. Rwarw of this it, is in lirtuc of (3.18’) and
This equat’ion has t,hr important virt,ue that, t,he modifications arising dirwtly from the inclusion of t,he Pauli principle do not, intSrodrwr any additional explicit energy dependence. (d) In a cert,ain sense t#hc simplest projection operator, which projects outS from the exact solution q a part, which has t,he same asymptotic behavior as \Ir, is one which is unity outside the region of interaction and zero inside said region. In other words, I* is just the asymptotic dependence of Q outside of the region in configuration space in which interactions occur. I3 As we have asserted earlier this is the proj&ion operator employed by the Wigner-Eiscnhud (2) formalism for nuclear rcact)ions as we shall now proceed t’o show. To keep t’hc discussion simple wc shall only vonsidcr t#he sit#uations in which the only possible reaction is elast8iv scattrring. The procedure is easy enough to generalize and we shall indicak how this can he done but shall not carry out, the details. In the clustic scattering cast I’( r,, , rl