Integral expressions for the difference between contiguous Dirac phase shifts

Nuclear Physics 75 (1966) 470--474; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permission from the publisher

INTEGRAL EXPRESSIONS FOR THE DIFFERENCE BETWEEN CONTIGUOUS

DIRAC PHASE SHIFTS

F. CALOGERO Istituto di Fisica dell' Universit~, Roma and Istituto Nazionale di Fisica Nucleare, Sezione di Roma and D. M. FRADKIN t Istituto di Fisica dell'Universit~, Roma Received 6 July 1965 Abstract: The extensions to the Dirac case of a formula by Tietz are presented. This formula relates

phase shifts differing by one unit of angular momentum to an appropriate average of the radial rate of change of the potential.

Recently Tietz 1) has derived the following r e l a t i o n s h i p between two scattering p h a s e shifts a n d the c o r r e s p o n d i n g r a d i a l wave functions: sin ( 6 , + 1 - 6 , ) = - ( p 2 / 2 m ) -1

E

d r d V ( r ) yt+,(r)y,(r). dr

(1)

I n this e q u a t i o n 6, is the scattering phase shift for energy p2/2m a n d a n g u l a r m o m e n t u m l, V(r) is the p o t e n t i a l which p r o d u c e s it, a n d the r a d i a l wave functions y , ( r ) are n o r m a l i z e d so as to reduce to circular functions at infinity: yl(r) ~ sin ( p r - ½hz + 51).

(2)

r -.~ g o

A nice feature o f the Tietz f o r m u l a , eq. (1), is its simple physical i n t e r p r e t a t i o n ; it relates the difference between two c o n t i g u o u s p h a s e shifts (which p r o v i d e s a m e a s ure o f the change in the effect o f the p o t e n t i a l as the a n g u l a r m o m e n t u m varies) to an a p p r o p r i a t e average o f the r a d i a l rate o f change o f the potential. Hence, this form u l a c o r r e s p o n d s to the semiclassical picture, which associates different a n g u l a r m o m e n t a with different i m p a c t p a r a m e t e r s . It is the p u r p o s e o f the present note to present the n a t u r a l extensions o f the Tietz f o r m u l a to the D i r a c case tt. T h e f o r m u l a e we o b t a i n relate again the difference bet North Atlantic Treaty Organization Postdoctoral Fellow in Science, 1964-65. Present address: Physics Department, Wayne State University, Detroit, Michigan. tt Tietz's claim that his formula holds also in the Dirac case is not valid except in limiting situations, as we discuss below. 470

DIRAC

PHASE

471

SHIFTS

tween contiguous phase shifts to an appropriate average of the radial rate of change of the potential, and they reduce in the non-relativistic limit to Tietz's formula. We also mention that an extension of the Tietz formula in another direction is also possible, namely to the case when the two angular momenta differ by an arbitrary amount, not just by one unit. Such generalizations are discussed in the following paper 2). The solution of the Dirac equation V+#m-

=

(3)

has the form 3) t ~b~ = 1 [ y~l)(r)x~

r ~ - iy~2)(r)x~] "

(4)

Here, X~ are the orthonormal spin-angular functions, # is the quantum number of the z component of the total angular momentum, and x = _ 1, +_2, ___3 . . . is the quantum number associated with the Dirac operator K~

= fl(~r. L + 1 ) ~ = - I¢~u.

(5)

We have adopted units such that h = c = 1. The quantum number K defines the quantum number j, characterizing the total angular momentum, and the quantum number I, characterizing the "orbital" angular momentum, by the relations J = 1~1-½ =

l-½~/l~l.

(6)

The radial wave functions y~l)(r) and y~2)(r) satisfy the system of coupled firstorder differential equations dy(~')(r) dr dy~2)(r) dr

K + - y~l)(r) + (E + m - V)y~2)(r) = 0, r

tc yt2)(r)_(E - m - V)y~l)(r) = 0. r

(7)

(8)

As is well-known, in the non-relativistic limit (E << 2m, V(r) << m) the "small component" y~2)(r) is of order [ ( E - m ) / ( E + m)]½ relative to the "large component" y~t)(r). The scattering phase shifts 6jr are related to the asymptotic behaviour of the radial wave functions by the relations

y~')(r) ~ sin [pr-- ½1n + cSj,],

(9)

r--~ o3

y(K2)(F)

"+

-- [(E

--

m)/(E + m)] ~ cos [ p r - ½hz + t~jt],

(10)

r.--~ oo

t Note however that we have changed the relative normalization of the radial wave functions.

472

F. CALOGERO AND D. M. FRADKIN

wherep = (E 2 - m 2 ) ~, a n d j and l are related to x by eq. (6). We recall that for each l (except for l = 0) there are two phase shifts, corresponding t o j = l + ½, which become identical in the non-relativistic limit; in the extreme relativistic limit (or, equivalently, in the zero-mass case), it is instead the two phase shifts with equal j which coincide. Also note that with these asymptotic relations we have fixed the normalization of the radial wave functions y(~l)(r) and y(2)(r). As is well known, the system of coupled first-order equations can be manipulated to give a single "Schr6dinger" equation, which turns out to be d

E

dr2 +p2_2EV(r)+V2(r)

x(t¢+l) /.2

+ -a(r)--½

--¼a2(r [E+m-V(r)]-)y(~°(r) = O, (11)

r

where a(r) = - d l n ( E + m - V ) / d r . The effective potential in this Schr~Sdinger equation depends on the index x, through the term xr-la(r), a fact which was ignored by Tietz 1) in discussing the applicability of his approach to the Dirac equation. Moreover it depends on the original potential in a fashion complicated enough to obscure all physical significance. It is more convenient to consider directly the inhomogeneous second-order equations derivable from the first-order equations by straightforward differentiation, namely: d2y(l__~)

+ Fp2_2EV(r)+V2(r)

dr 2

L

d2yT___~ ) [ dr 2 + p 2 _ 2 E V ( r ) +

V 2 ( 7 ")

x(t¢+l)l y(1) = d___VVy(2), r 2 .J dr

(12)

x ( x - 1 ) q .(2) dV (1) r 2 J Y~ = -- d~ y~ "

(13)

We now apply directly to these equations the procedure proposed by Tietz 1), i.e. we consider the Wronskian between the functions Y~+x, (1). (r ) and yt~2)(r), and we thus obtain d r,~,,(1) I ~YJ¢+I L dr

--

dr

, , ( 2 ) _ _ , (1) •Y~ )'~ +

1

dyt 2) 1 --~-r J

( -----

) dV . (1) +.(2) ,/2)1 ~ r LYe+lye1) ~ + x ~ j,

(x ~ - 1 ) .

(14)

Integrating this equation from the origin to infinity and using the fact that the Wronskian vanishes in the origin and that it is connected asymptotically to the scattering phase shifts as implied by eqs. (9) and (10), we find sin(6j+l,,+l-6~,,) =

-(E-m)-XfO°dr dVr,,(1) .(t)_..(2) y(2)], Jo

dr LJ'Ic+lYa¢ "~-YK+I

(x ~ --1).

(15)

DIRAC

473

PHASE SHIFTS

We emphasize that here l and j are defined by tc through eq. (6). In a completely similar way, starting from the Wronskian between y~l)(r) and y(2)(r), we also obtain sin(6j,~-~j,~_~) = - ( E - m ) -1

fo d r ~dV r [y ~ y_~

y~ y_~j

. (1). (1) ..F,,(2)_(2)"1,

(16)

where now l = [xl a n d j = l - ½ . These two equations are the desired generalizations of the Tietz formula. The first one, eq. (15), relates phase shifts differing by one unit of orbital angular momentum and both referring to "spin up" or to "spin down" states, so that also the total angular momentum differs by one unit; the second one, eq. (16), relates spin up phase shifts to spin down phase shifts with the same total angular momentum, the orbital angular momentum always differing by one unit. Both equations reduce to the Tietz formula in the non-relativistic limit, because in that limit the small component y~2)(r) vanishes, the large component y~l)(r) becomes the Schr/Sdinger radial wave function, and the phase shifts depend only on the orbital angular momentum. In the zero-mass case, (or, equivalently, the extreme relativistic limit) the first equation remains significant, whereas the r.h.s, of the second one vanishes identically . ( 1 ) ( r ) -- y~2)(r), y~2)(r) = -y~°(r). because for m = 0 we have the formal symmetry y~ It is thus demonstrated that in this case the phase shifts corresponding to equal j and different I are identical. In fact the r.h.s, both of eq. (15) and of eq. (16) vanishes also in the high-energy limit pR >> l, where R indicates some measure of the range of the potential, because in this limit we may substitute for the radial wave functions their asymptotic expressions eqs. (9) and (10). This finding is consistent with Parzen's theorem 4), which states that all phase shifts tend in this limit to the same value, and therefore implies that the left-hand sides of eqs. (15) and (16) vanish. Finally, we mention one elementary application of the Tietz formula that may have some pedagogical interest. Consider the special case of the "square well" potential

V(r) = VoO(?-r),

(17)

where O(x) is the usual step function. The Tietz formula becomes simply sin (61+ 1 - gil) = (2mVo/p2)yz(r)Yt+ l(r).

(18)

But on the other hand, in this case yl(~) = (cos 6z)3z(p~ ) - (sin 6,)~l(p~).

(19)

The Riccati-Bessel functions)z(x) and fit(x) are the usual spherical Bessel functions multiplied by the argument; they reduce asymptotically to circular functions: Jr(x) --} sin (x-½1n), x-. oo -, x ..-~.oo

- cos (x-

½1 ).

(20)

474

F. C A L O G E R O A N D D . M . F R A D K I N

F r o m eqs. (18) and (19) we easily obtain (tg ~,+,) = [-(tg

6,)(I-A~,)t+,)+A),),+,]/[I+A~t+,(),-(tg

c5,)~,)],

(21)

where A = (2mVo/p2) and the a r g u m e n t of the Riccati-Bessel functions is p~. This equation allows the calculation o f all phase shifts once one is known. Its validity m a y on the other h a n d be verified using the k n o w n explicit expressions for the phase shifts and the special properties o f the Riccati-Bessel functions. This same procedure can be applied to the scattering o f a Dirac particle on a square well potential.

References 1) 2) 3) 4)

T. Tietz, Nuclear Physics 44 (1963) 633 D. M. Fradkin and F. Calogero, Nuclear Physics 75 (1966) 475 M. E. Rose, Relativistic electron theory (John Wiley and Sons, New York, 1961) p. 159. G. Parzen, Phys. Rev. $0 (1950) 261