A simplified expression for the Talmi coefficients

1.D.1

I I

Nuclear Physics A96 (1967) 115--120; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A SIMPLIFIED EXPRESSION FOR THE TALMI COEFFICIENTS M. M. BAKRI

Applied Mathematics Department, Faculty of Science, Cairo University, Giza, UAR Received 3 October 1966 Abstract: An expression is derived for the Talmi coefficients for two particles with unequal masses, which is much simpler than that given previously by Smirnov. This is achieved by using a theorem proved recently by the author, stating that the expansion coefficients of the two-particle function r 2nl +lt r2n2 +12 y(rD y(r2) in terms of functions of the c.m. and relative coordinates -1

-2

ltmt

12m2

are, apart from a normalization factor, the Talmi coefficients.

1. Introduction

The Talmi coefficients 1) of the expansion of the two-particle harmonic oscillator wave function in terms of the wave functions of the c.m. and relative coordinates are of particular interest in the nuclear shell model. These coefficients are the matrix elements of a unitary transformation 2) connecting the Hilbert spaces of the two particles and the c.m. and relative coordinates; the two-particle wave functions are (1.1)

gnlhml,n212rn2(rl, 1"2) = ~n~l~(rl)~n2h(r2)Yl:n,(rl)Yt2mz(r2),

where Y~') is a surface spherical harmonic function of the unit vector

~n,(r) = x/2"n t/r(n + l +~)e-½"2r'I~+~(r2),

r/r, (1.2)

and Lln+½(r2) is an associated Laguerre polynomial. This Talmi matrix is reducible under the three-dimensional rotation group. The irreducible representations are matrix elements with respect to the functions ~n,ll.n2'2, Ira(r1,

r2) =

~n,l,(F1)~n212(r2)~lralll2(rl,

Y2),

(1.3)

where

=

5".

C(1,,

l;

m,_,

(1.4)

m l -t- ll~2 ~ m

and C(ll, 12, l; ml, m2, m) is a Clebsch-Gordan coefficient. Moshinsky 3) obtained these irreducible coefficients for n 1 = n 2 --- 0 and equal masses of the two particles, by expanding the wave function (1.3) in powers of the c.m. and relative position vectors of the two particles. Recurrence formulae were obtained for these coefficients 3 - 7), and the Moshinsky coefficients were evaluated numerically s). Smirnov 9,10) obtained expressions for the irreducible coefficients for unequal masses of the two particles, by expanding the wave function for arbitrary values of nl n21112. However, 115

116

M.M. BAKRI

these expressions are very complicated, containing summations over thirteen parameters. The purpose of the present paper is to give a much simpler expression for the Smirnov coefficients, using the fact that these coefficients are also the expansion coefficients of the function 2) A

~2nl+It~2n2+12~ (r -2 ,mh,2, 1, r2)

A

q).,h,.2h, ,m(r~, r2) = a.lh-'~.2,2"1

(1.5)

in terms of the corresponding functions of the c.m. and relative position vectors, where

A,, = ( - 1)"F(n~/2 2-+')n !r(n + l+ ½)3+.

(1.6)

A representation of the function (1.3) in an abstract space of creation and annihilation operators, similar to (1.5), was given by Bargmann and Moshinsky 11), and was utilized to study the Moshinsky coefficients s, t2). The expansion of the function (1.5) is obviously much simpler than that of the function (1.3), and is carried out here. Smirnov 9) expanded (1.5) as an intermediate step, and then he substituted this expression into the expansion of the associated Laguerre polynomial in powers of r 2"+1. Now we can dispense with the last step, and we may evaluate the Smirnov coefficients from his expansion of the function (1.5). However, even this expansion can be simplified further, using Sack's expansion 13) for r = r 1 + r 2, r2"+tYt.,(r) = ~ Z ( - 1)½('+k+'O~[k'][f] ~i

0

--kj~ 1, r2)~tm~j(rl, r2),

(1.7)

where

Ik-jJ <- l <_ k +j <=2n+l,

[l] = 2l+1,

(1.8)

kO00l jr ~ is a 3j-coefficient 14), which vanishes if k + j + l is odd, and

R ~ ( r l , r 2 ) = ( - 1)"- NnlV(n+ , ~- ~)rlkr22"+z-k n

(rl/r2)2"

,

(1.9)

x Z=ov!(N_OW(n+L+l_OF(k+l+v ) where

N = n-½(k+j-l),

0 < N < n,

L = ½(l+j-k),

0 < L < l.

(1.10)

Sack's expansion is simpler than that given by Smimov, who considered the product of the expansions of r 2" and rtY)m(r). In what follows we use dimensionless vectors

rs=V~2~Ps,

s--- 1,2,

(1.11)

where Ps is the usual position vector, gt and g2 are the masses of the two particles,

117

TALMI COEFFICIENTS

and co is the angular frequency o f the harmonic oscillator hamiltonian H = - 1 132+ - -IP 2.,2.~_~2 .(.D2/~#1p2+/~2~2) • 2/~1 2#2

(1.12)

The wave function (1.3), expressed in terms o f the dimensionless vectors r I and r 2, is the eigenfunction o f H. The c.m. and relative position vectors o f the two particles

are

1

g = - (mrl+.2r2), //

~ = Pl-P2,

(1.13)

]/~'lizc° P = V /th '

(1.14)

where # = / l t + / t 2. Introducing the dimensional vectors

R=

k,

r

the wave function o f the c.m. and relative coordinates is ,/,(s,,) Wnllt, n212,Ira" The vectors (1.11) and (1.14) are related to each other as follows: R = rl cos ½fl+r2 sin ½#, r = r 1 sin ½ f l - - r 2 COS ½fl,

(1.15)

where cos ½fl = ~//t;,

sin ½fl = ~//t2,

0 < fl < 7~.

The required expansion may be written in the form

(P.:x..2h,,.(R, r) = ~, (ns 13, n i l , , llfiln111, n212, l>~p.,h,.,,,.,m(rl, r2),

(1.16)

n3n4

1314

where

(n313, n414, llfllnx ll , n212, l)

(1.17)

is the Smirnov coefficient, which is independent o f m, as is well known o). F o r equal masses the Moshinsky coefficients are obtained by substituting fl = ½~. The Talmi coefficients o f the expansion ){nll:a,,n212m2(R'r) =

E

113114

/n3

~n,

13 ma rl nl 14 m4 n2

11 m l ~ . . tr 12 m2/~n313m3'n414ra4k 1'r2)

m3-I-m4hl, mml -l-m2

(1.18)

are given by 7) n4

14 rn4

n2

12 m 2 /

= ~ C(ll, 12, l; m l, m2, mt+m2)C(13,14, I; m3, m4, m 3 + m 4 ) l

x (n313, n414, IIfl]ni11, n212, l>.

(1.19)

118

M.M. BAKRI

2. The Smirnov coefficients N o w from (1.5), (1.7) and (1.15) we have

(~nl|ii,212,1m(R, r) = ~2~Ii1111~l-li212Z (--) ~(11+|2+klWk2+Jt-J2)4[kl][~2][Jl][J2] klk2 JlJ2 rlntll (~ n212 x ~k,s,t'l cos ½fl, r2 sin ½P)Rk~j~(rl sin ½p, r2 cos ½fl)

X (~01 Jl

/l'~{k2 J2 ~)~l(rl,le2;lel,il'2)

0

0

/\0

(2.1)

"~lml1(kl.jl)12(k2J2),

0

where ~/lmli(kljl)12(k2j2)(P1, r2 ; r l , r2)

=

~,

C ( l , , 12, l; m , , m2, m)~It,~,kxj,(rl, r2)~h.~k~S~(rx, r2)

Z

4[l][11][12]( ll

ml -I-In2~ m

:

12

)( kl Jl

/

m1 m2

#1+21=ml #2 + ~,2ram2

m

~tI

li l ( k 2

~1

--ml/\]A2

J2

/2 )

~2 --m2

ml "bm2=m

(2.2)

X Ykl/il(rl) ]/k2li2(rl)YjlJ.1(ili2)Yj2~.2(r2). In the last step we have put

(--i)/l+~:l-jl = (--I)/l+k2-j2 = I, since

l 1 -]-k 1 -Jl

and l 2 -I-k 2 --J2 are both even. N o w substituting 1#) in (2.2)

Yk,.,(r~) Yk~.~(r~) =

Z

( - 1 ) m3

/3, m3= # I +112

X

V~~ [k~][k2][/a]

13](kl

(k01 k2 0

k2 0 ] \ # 1 /X2

13 ) Y/3m3(rl) ~ --m 3

and using

~1 , , ~ Z( --~1 k, --21 ml/\--/x2

~2 ,2)(~,

--,;L2 m2

mlm2 P1112

--~1

~

,~)

--/~2 m3

2.122

X

--'~1

{k, li)

=(/3 l, , ) k 2 . m3

m4

m

--12

13

14

m4/\ml

!"112,

TALMI COEFFICIENTS

119

where the last bracket on the right-hand side is a 9j-coefficient 14), we obtain °2¢,,,z,O,,m,~¢k~j~)(rx, r2 ;r~, r2) = ( - ) t,-,, 1 ~/[lx]El2]Ek~][k2][j~q[j2]

×

~x/~ (~ k2 la~(jx

J2

131,

0

0

0/ \ 0

~){klJll~} k2

J2

alJlml,12(rl, r2)"

1+

I3

(2.3)

Further, from (1.6) and (1.9) we get A Dnlll[ .212 A,,~t, "-',uh "'k,jl~rl COS ½fl, r 2 sin ½fl)Rkzj2(r I Sill ½fl, r 2 COS ½fl)

= ( - 1)N~ + S~n~X/2- pnl ! n2 !F(nl + ll + ½)F(n2 + 12 + 3). × rk~+k2~--k~-k~(cos ½fl)2n~++~+k~- k~(sin ½fl)2,~+Z,+k~-kt NI N2

X )--, Z .,v~v2fN'N2L'L2(cot ~ , , ~, ½fl)2va - 2.2(r1/r2)2., + 2v2

(2.4)

VireO v 2 = O

where I/:NxN2L1L2 --~ Vl ! v2!(NI_Vl)!(N2_v2)! / J viv2

x F(nl + LI + ½- Vl)F(n2 +L2 + ½- v2)F(kl + ½+ vl)F(k2 + ½+ v2), (2.5) p = 2nl + ll + 2n2 + 12,

(2.6)

and

N i ---- nz--½(ki.-l-ji--li) ,

Li = ½ ( l i w j i - k i ) ,

i = 1, 2.

(2.7)

Now since kx + k 2 q- 13 and 11+ l 2 -- l 3 - 14 are both even, we may write

kl+k2+2vl+2v2 = 2na-F13,

p - 2 n 3 - I a = 2n++l+.

(2.8)

The latter relation corresponds to the energy conservation. Using (2.1), (2.3), (2.4) and (2.8), and comparing with (1.16) we obtain finally the Smirnov coefficient ( n 3 1 3 , h a l 4 , llfllnl/1,

n2/2, I)

4

= ¼~ 1-~ (-1)"'~/[IJni! F(ni+/i-]-~)(cos ½fl)2"2+':(sin ½fl)2,, +t,. i=1

x

z

kl +k2 = k v I +V2~V

k2 l~) (cotg½fl)kl-k2+2va-2v2 0

Ek,lEk21(

k+2v=2n3+13

14(k2 J2 jljz

0

o:

o

o

J2 o:

o

o

kl Jl J2

l+

•

(2.9) The summation extends over six parameters, between which we have the relations

120

M.M.

BAKRI

= k , V1 + v 2 = V, k + 2v = 2na + la. This is a remarkable simplification over Smirnov's expression 9), which extends over thirteen parameters, between which there are six relations. The M o s h i n s k y coefficients for equal masses are obtained by substituting fl = ½re. kl +k2

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

I. Talmi, Hdv. Phys. Acta 25 (1952) 185 M. M. Bakri, to be published M. Moshinsky, Nuclear Physics 13 (1959) 104 T. A. Brody, G. Jacob and M. Moshinsky, Nuclear Physics 17 (1960) 16 M. Moshinsky and T. A. Brody, Rev. Mex. Fisica 9 (1960) 181 A. Arima and T. Terasawa, Prog. Theor. Phys. (Kyoto) 23 (1960) 115 R. D. Lawson and M. Goeppert-Mayer, Phys. Rev. 117 (1960) 174 T. A. Brody and M. Moshinsky, Tablas de par6ntesis de transformaci6n (Monografias del Instituto de Fisica, Universidad de M6xico, M~xico, D.F., 1960) Yu. F. Smirnov, Nuclear Physics 27 (1961) 177 Yu. F. Smirnov, Nuclear Physics 39 (1962) 346 V. Bargmann and M. Moshinsky, Nuclear Physics 18 (1960) 697 M. Moshinsky, Group theory and collective motions (La Escuela Latino Americana de Fisica, Julio 2-Agosto 10, 1962, M6xico) R. A. Sack, J. Math. Phys. 5 (1964) 252 cf. A. Messiah, Quantum mechanics, vol. II (North-Holland Publ. Co., Amsterdam, 1962) appendix C, p. 1053