INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE ALGEBRAIC APPROACH TO THE SCATTERING PROBLEM ABSTRACT

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1 IC/69/7 INTERNAL REPORT (Limited distribution) INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE ALGEBRAIC APPROACH TO THE SCATTERING PROBLEM Lot. IXARQ * Institute of Atomic Physics, P.O. Box 35, Buchrest, Romni. ABSTRACT In this pper the uthor pplies the lgebric pproch in order to evlute the prtil scttering mplitude S.. A recurrence formul leding to S. is obtined. Jnury 1969 MIRAMARE - TRIESTE At Interntionl Centre for Theoreticl Physics, Trieste, Itly, Jnury-Februry 1969.

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3 THE ALGEBRAIC APPROACH TO THE SCATTERING PROBLEM I. INTRODUCTION In previous pper we hve discussed the solution of the penetrbility problem for centrl brrier potentils by the lgebric method, i. e., by trnsforming the Schrddinger eqution corresponding to this problem into n lgebric system of liner equtions. This trnsformtion cme s result of the pproximtion of the potentil by sequence of step functions. We now use this method to study the scttering problem of two spinless prticles. For the studied cse, where the interction of the two prticles is described by centrl potentil, simple recurrence formul leding to phse shifts is derived. II. THE ALGEBRAIC SYSTEM The scttering of two spinless prticles rm. nd rru hving the position vectors r, nd lr, their interction being described by the potentil function V( ]r, -?* ) ) cn be studied in c, m. s. by mens of the Schrodinger eqution 2 2 * ±J&El + v(r) 0(r) = e^{r) (1) 2m dr 2 with the boundry condition = 0. (2) Writing this eqution we hve hd in mind only the Schrodinger eqution corresponding to the jg'th prtil wve* The following nottions hve been used: -2-

4 r r ^ - ^, r s r], (then re[0,+ oo)), (3) V( r ) = V( r ) + A U + l)/(2mr 2 ), (4) nd m = m 1 m 2 /(m 1 +m 2 ). (5) The function ^(r) is r times the rdil prt of the wve function of the system. We now derive the system of liner equtions ttched to the problem (1) nd (2). For doingsowe replce the potentil V(r) by the potentil V (r) (Fig. 1) nd lbel by A the intervl [ r x, r x + J ], (X= 0, 1, 2,... ). The Schrbdinger eqution in this intervl is nd hs the generl solution + V(r)0(r)= 0(P) (6) ^( r ) = p exp(u^r) + Q^ exp(-w^r) (7) A A A A where w. = [2m (V. - e)] l/2 /* (8) A A (we note in pssing tht if VI - e is positive (negtive) the frequency UK is rel (imginry)). By inspecting (7) for A = 0, it is immeditely estblished tht the condition (2) gets the form 2 2 "^T*- dr -3-

5 Now we impose on the wve function ^(r) the fmilir condition of being continuous nd of dmitting the first order derivtive which is continuous nywhere on the rel xis. These requests must be prticulrly vlid t the joining point r-. between the intervls X nd X-l. At r^ the continuity condition of 0(r) gives X-1 Q X-1 = P (10) nd tht of the continuity of its first derivtive ppers W X-1 P X-1 "Vl Q X-1 If we llow X = 2, 3,...,, there re 2{-l) such reltions. In ddition to these, we tke into ccount eq. (9) in order to estblish the conditions imposed by the bove requirements t the point r,. The derivtion is strightforwrd: -Qj expfu^) + QQ exp(-w () r 1 ) '= P^ + (12) -Q* expfu)^) -Q^ The 2(-l) reltionships (10) nd (11) together with (12) nd (13) form homogeneous lgebric system of 2 equtions for unknowns, P^, P" P P,..., P^, Q, Now, by denoting (13) B 2X (14» -4-

6 nd this system cn be written s 2 V ly. if = m", i = 1, 2 2 (15) j = 1 where ex (Obviously, for the scttering problem only on the spot in this pper, w is n imginry quntity.} b n = b 21 = ( <r> ' b l ex b 2+l 2cr ~ exp^n- r +l ' 2or+l 2CT+1 2 % ex^<y Vl 1 ' b 2cr l,,, = -exp(w r r ) 2-l 2 b_ = -u exp(u r ) 2 2 ff ^ or' In the bove formuls, cr = 1, 2,... (r-1) -5-

7 All the remining terms of column mtrix n (with 2 elements) s well s 2 x 2 squre mtrix b re equl to zero. Obviously, ll the informtion concerning phse shifts is embodied in the coefficient ' 9 of the system (15). B. which is one of the unknowns 2 III. CALCULATION OF THE UNKNOWN B lot We perform this clcultion by using the rule of Crmer 2) B 2 " ' b 2* (18) Here lb I nd [b 2 ct re the determinnts of the mtrices b nd b. The lst is obtined from b by replcing the lst column {j = 2) by the. column m. The whole problem is then reduced to the clcultion of these determinnts. Generlly the clcultion is very difficult, but for the problem t hnd we obtin importnt fcilities if we use the simple form of the mtrices n nd b. For resons which will become cler lter we consider the determinnts > b 21 b 22 (19) nd b" 1!, 13 exp[-(u 21 K b 23 (20) -6-

8 s bse for constructing two other determinnts. More -2, precisely, we construct new pir of determinnts ( b nd j b" J) by the following rule: The mtrix b (hving the determinnt is obtined by dding to the mtrix b the next two rows s well s the next two columns, i. e., b 2 l) 11 5 D 22 D 23 D > J 43 J 44 (21) while, to obtin the mtrix b -2 we dd to the mtrix b 1-1 (this, not b 1 [ ) the next two columns but not the next two rows, i. e., we dd the first nd the third. To be clerer we write this mtrix explicitly hi -2 D 42 D 43 45/ (22) In order to clculte jb j we use the Lplcen expnsion of 2) determinnt ccording to minors obtined from the lst two rows nd their cofctors from the first two rows. By noting tht in our cse only two minors re different from zero, -7-

9 3 33 D 34 ) -Ho ) exp[(u -u )rj Z 1 Z 1 i (23) b 43 b 44,-2, 34 (24) A 9 A. A one immeditely gets * * *) + (25) _ n If we pply the sme Lplcen tretment to the determinnt the result is n " ] b - _ 1 (26) 2 3 Strting from the mtrix b we cn construct the mtrices b nd b by the sme rule s the one used for obtining b nd b from b. The next step will be the derivtion of b nd b from 3 b, etc. Clculting the" corresponding determinnts by mens of the sme Lplcen method leds to the result tht similr dependences re involved between two successive steps:

10 + (w +u ) exp[i(u> +u )r (27) This whole construction hs mening if we observe tht t the -th step we obtin precisely the determinnts involved in the formul (18); i.e., b derived by mens of the recurrence formul (27) is just Jb") jbfj of (18). of (18), while b" ff ] of (26) is exctly If we now note tht ] b j nd b [ cn gin be derived by using the formul (26) for ju = 1 nd b = b =1 we cn conclude the pper s follows: The weight S of the I'th prtil wve of the scttering theory for centrl potentils is given in the frme of the lgebric method by the rtio where ] b j re obtined by pplying the recurrence formul (2 7) for (J. = I, 2, 3,...,, with ] b j = ) b J = 1 s strting quntities. Obviously the finer the division of the rel xis r, the more ccurte the vlue of S fl. A CKNOWLEDGMENTS I should like to thnk Professors Abdus Slm nd P. Budini nd the Interntionl Atomic Energy Agency for hospitlity t the Interntionl Centre for Theoreticl Physics, Trieste. -9- *

11 REFERENCES 1) L. Gr. Ixru, ICTP, Trieste, Internl Report IC/69/6, (to be submitted to J. Mth. Phys.) 2) AC. Aitken Determinnts nd Mtrices, Oliver nd Boys, Ltd., Edinburgh nd London, FOOTNOTES The connection of B with the S coefficient of the usul scttering theory cn esily be derived. Using the nottions of this theory the symptotic form of the rdil wve function corresponding to fixed S. multiplied by r is Mr) = e ' S i where S is connected to the phse shift by the formul In our nottions ik = w nd the intervl is the symptotic one. Consequently ^ Rigorously speking, only ]b is used for the construction. The determinnt jb J will intervene s complementry quntity. * *' Now it is cler why the introduction of b ws lso necessry. TIO-

12 V(r) A V X+2 r ->- The true potentil V(r) nd the pproximte potentil V w (r) used by.the lgebric method. -11-