Completeness, unitarity and dispersion relations in potential scattering

Nuclear Physics 87 (1966) 267--272; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permi.~sionfrom the publisher

COMPLETENESS,

UNITARITY

AND

DISPERSION

RELATIONS

IN P O T E N T I A L S C A T T E R I N G I. SAAVEDRA * International Atomic Energy Agency, International Centre for Theoretical Physics, Trieste

Received 12 April 1966 Abstract: It is shown that the assumption of completeness of the set of scattering states implies the unitarity condition for the total scattering amplitude and dispersion relations for the S-matrix for each value of the angular momentum.

1. Introduction In this note we consider the non-relativistic elastic scattering o f spinless particles by a potential of finite range, that is, a potential such that we can define asymptotic solutions Ok(r), say, o f the Schr~dinger equation

= Ek k(r)

(1)

satisfying the boundary condition e ikr

$k(r) ~

e; k ' ' + f k ( k " P ) - - .

(2)

As is well known, the entire theory is based on the assumption that these functions form a complete set; otherwise the wave-packet description of the scattering process could not be given. Now, it is also well known that the p r o o f o f the completeness o f any set o f eigenfunctions is based on the analytic properties of the resolvent or Green function of the eigenvalue problem under consideration. Completeness is thus, in this sense, a statement o f analyticity; it is clear of course, that this argument certainly does not imply that further physical information cannot be extracted from it. In this note we study the consequences of the assumption of the completeness o f the set {qJk(r)} defined by eqs. (1) and (2). To be more precise, this assumption implies an orthogonality and a completeness relation, and we look here for the physical information that can be d i r e c t l y extracted from them. We have considered this problem in a previous paper t), but in terms o f the scattering matrix (or phase shift) and for s-waves only. We proved there that from the * Permanent address: Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Santiago. 267

268

1. SAAVEDRA

orthogonality relation the so-called causal inequalities (Van Kampen's and Wigner's) can be derived, thereby showing that this relation (for a given value of the angular momentum) is connected with the conservation of probability at a given time (in contradistinction with the unitarity condition &(k)St(k)* = 1, which is connected with the conservation of the time independent total probability). We also showed in ref. a) that the Van Kampen dispersion relations follow from the completeness relation. We study here again the same problem, but this time in terms of the total scattering amplitudefk(~ • ~) defined by eq. (2). Our conjecture is of course that from the orthogonality relation it should be possible to derive a statement concerning the conservation of total probability, and that from the completeness relation dispersion relations should follow. This we prove in the following section, where from the former condition we derive the unitarity condition for the total scattering amplitude whereas from the latter we obtain dispersion relations for the partial transition amplitudes (or the scattering matrix) for each value of the angular momentum. 2. Consequences of the Completeness of the Set {~,k(r)}

We assume that the set of functions {~k(r)} defined by eqs. (1) and (2) is complete, i.e., that these functions are orthogonal,

= 6(3)(k- k')

(3)

f ~-~3 dBk ~,(r)~b*(r') = 6(3)(r- r').

(4)

(2~) 3 ~k(r)~J~,(r) and satisfy a completeness relation,

For simplicity we have assumed here that the Hamiltonian H has no bound states. We consider first the orthogonality condition (eq. (3)). Choosing a sufficiently large Irl = a so that eq. (2) is satisfied we can write

(2n)a(5(3)(k- k') = f.~or"d,'fdO4,,(r)~4(r)+ f.(or 2dr f d~@,(r)~4(r).

(5)

and using eq. (2) for the second term in the right-hand side we find

. fZ d" f dO~'k(r)t~:'(r) = (2rO36'Z)(k-k')+

I(k, k'),

(6)

+ fk(l~',.'~)e-ll"'"+ik'+ ~Tfk(i~.~')f~,(l~''')ei(k-k')'} .

(7}

where

l(k, k') -

f [ ~= f : d r f

OQ Ifff'(( " ?) e i k ' r - l k ' r ~

r

269

POTENTIAL SCATTERING

To evaluate these integrals we choose ~ as the direction of the polar axis and specify with the angles (0, ~o) and (0', ~o') the directions of the vectors r and k', respectively. Then, calling 0" the angle between the directions ? and K' we have • ~ = cos 0, k' • P = cos 0" = cos 0 cos 0'+sin 0 sin 0' cos(q3-q¢), and the following relations holds: f + tp,(cos O)Vt,(cos 0")d(cos 0) = P,(cos 0'). -l 2/+l

(8)

Using this result we can easily evaluate the integrals appearing in eq. (7). For instance, consider

It =

rdr

rdre -'k'"

df2f*(K" P)e"''-ik'" = 2n a

sin OdOf*(cos O")e ik.... 0.(9) ,dO

As by hypothesis (i.e., by the choice of a) r is large, we have e,k.... 0 ~ E (2/+ 1)e½'"' sin (kr+½ht) P,(cos 0). i kr

(10)

Further, the partial wave expansion of the scattering amplitude is t

f*(O") = E (2l'+ l)a*(k')P,,(cos 0"),

(11)

1"

where the partial transition amplitude at(k) is defined by 1 at(k) = ~lk (St(k)- 1).

(12)

Introducing now eqs. (10) and (11) into eq. (9) and using eq. (8) we obtain ll = 2 = f °°dr E (2/+ l)a*(k')Pt(cos 0'){et'k-k')'--e i(k+k')'+i"'} ik3a t

2~ , ik Jk W J"

_ {

1 + rotS(k-k')} +F(k,k'), i(k-k')

(13)

where F(k, k') does not diverge at the limit k' = k. (We have assumed of course that the partial wave expansion of the scattering amplitude converges in the cases of physical interest.) In the same way we find

I2

= f ;rdr f dfU~(ti " P)e-'k'''+'~" 1 2~L(O,)e~k-k')° / i-£' " " t-i(k-k')

/

+rta(k-k')j +V*(k',k),

t We write for simplicity fk' (0") instead of f k, (cos 0").

(14)

270

13

i. SAAVEDRA

= f~dre~tk-k'~" f dt2A(k " Of~,(k' " O = &ze '(k-k')°

1

1

+ nb(k-k')

~ (21+ l)at(k)a~(k')P,(cos 0').

(15)

Finally, introducing eqs. (6), (7) and ( 1 3 ) - 0 5)into eq. (5), multiplying throughout by ( k - k ' ) and taking the limit k ' = k we obtain

0 = -- 1. imfk(O)+ ~ (2•+ l)]al(k)12pt(cos 0), k

(16)

i=o

which is the unitarity condition for the scattering amplitude written for an arbitrary angle *. The optical theorem is obtained as a special case for forward scattering, 0 = 0. We have thus proved that the orthogonality relation is, in this sense, a statement of unitarity. Let us consider now the completeness relation, eq. (4). We take Irl and ]r'l sufficiently large so that eq. (2) holds. As in the previous case, the 6-function is cancelled and we are left with three integrals which we now evaluate choosing a system of coordinates in which ~ defines the direction of the polar axis. Let the direction of and ~' be specified by the angles (0, ~o) and (0', ~0'), respectively, and let 0" be the angle between ~ and ~'. Then we have, i~.~ = c o s 0 , • ~' = cos 0" = cos 0 cos 0 ' + s i n 0 sin 0 ' c o s ( , p - 9 ' ) . The integration problem is essentially the same as above and a straightforward calculation yields ,t f

eik • r - ikr'

i,(~, ¢) = d 3 k f ? ( l ~ . r ) - - - p t"

=

kdkf*(O')e ik''-r'' trr d o

irr d o

k d k Z f f f ( k , O')e -ikt'+'''+i"t, l

f

(17)

e - ik • r" + ikr

. . . . ~) . 12(r, r') . . d~kA(t~

l,(r, * ' r),

(18)

i•

eik(r_ r, )

Is(r, r') =

dSkfk(~ • ~)f*(,~. t~'1 -

rr'

_

4re rr

+ The connection

(21+ l)la,(k)l~e,(cos O3,

k2dkeik"-r"~ ~

with the usual statement

(19)

l of unitarity

Sl(k)St(k)*

=

1 is i m m e d i a t e l y

obtained

by

remarking that this implies the relation al(k)at(k)* = ( I / k ) lm at(k) for the partial transition amplitude. ~r For the moment we writeft(k , O) -- (2l+ l)at(k)Pt(cos 0).

POTENTIAL SCATTERING

271

these integrals satisfying, according to eq. (4), the relation (20)

11 + ]2 -'~ 13 = O.

Introducing eqs. ( 1 7 ) - ( 1 9 ) into eq. (20) and setting r' = r we obtain

0 = -4rr

fo

kdk I m f k ( 0 ) + 4 ~

%

k2dk

~ (2t+ l)la,(k)12e,(coso) 1

_ 2_~f~kd k 2f.(k,O)e_2ik,+i~, + 2_rr I ~ k d k 2fl(k,O)e2ikr_int, i dO t i jo t

(21)

and we see that the first two terms cancel out due to the unitarity condition (16). The remaining relation can be transformed into the equation

f

~ k d k Y" ( - ) r ( 2 / ' +

1)a,,(k)P,,(cos O)e2'k' = O,

(22)

if the usual symmetry property of the S-matrix (for real k)

S*(k) = S , ( - k ) ,

(23)

a*(-k)

(24)

which implies that = a,(k),

is introduced as a new assumption. It is well known that eq. (23) follows from the symmetry properties of the Schr6dinger equation, but in our discussion it must be considered as a postulate. Let us now consider eq. (22). T w o possibilities arise: either eq. (2) holds in the limit r ~ ~ , in which case the exponential factor in cq. (22) oscillates violently and thus transforms this equation into an identity, or the interaction is of finite range, in which case it is always possible to find a sufficiently large, but finite, R such that eq. (22) holds for any r > R. In this latter case we obtain, after multiplying eq. (22) by P~(cos 0) with l fixed and integrating over 0

(_)t

kdkat(k)e2~k% 'kt = 0,

(25)

where t = 2 , ' - 2 R , 0 _< t _< ce. We remark that if we had considered the presence of bound states the right-hand side of this equation would have been different from zero, and therefore the sign appearing in the left-hand side would have been important; in our discussion it is of course irrelevant. Let us multiply eq. (25) by e -i''~', c~ > 0 and integrate over all t. Using the notation ~p,(k) - ka,(k)e 2'k",

(26)

we obtain

( ~ dtO(t).ff[,pt(i,+m)] = O, -r,o

). = k-o~,

(27)

272

i. SAAVEDRA

where O(t) is a step function and ~" a Fourier transform. Parseval's theorem for generalized functions then yields t

that is

f

o~ d2 tp,(). + to)6 +(2) = 0,

(28)

--o0

o~a,(to)e21'°a:liPf~ooka'(k)e2'k'dk, -

(29)

k-to

which is a dispersion relation for the partial transition amplitude and arbitrary 1. This dispersion relation can be given a more familiar form using eq. (12) and the result 1 p

ni

f oo e2ikR

dk

e 21c°R.

-® k-o)

We obtain thus P

Sl(c°)eEi'°R = ~i

-~o

S'(k)e2ikRdk,

(30)

k-e)

a dispersion relation for the S-matrix and each value of the angular momentum. It is worth remarking that it is for the S-matrix, and not for the total scattering amplitude as one could have expected (as we always worked with eq. (2)), that dispersion relations follow from the completeness relation. I am grateful to Professor S. Okubo, who asked the relevant question, and to Professors Abdus Salam and P. Budini as well as the IAEA for the hospitality extended to me at the International Centre for Theoretical Physics, Trieste. t Clearly, the procedure is purely formal. For a rigorous p r o o f for the case 1 = 0 see ref. a)

References 1) 1. Saavedra, Nuclear Physics 29 (1962) 137