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“Calhoun Institutional Archive of the Naval Postgraduate School Calhoun: The NPS Institutional Archive DSpace Repository Theses and Dissertations 1. Thesis and Dissertation Collection, all items 1969 The phase equation in potential scattering. Dettmann, Terry Robert Monterey, California. U.S. Naval Postgraduate School http://ndl.handle.net/10945/12820 This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States. Downloaded from NPS Archive: Calhoun : Calhoun is the Naval Postgraduate School's public access digital repository for / (8 D U DLEY research materials and institutional publications created by the NPS community. «ist : Calhoun is named for Professor of Mathematics Guy K. Calhoun, NPS's first NY KNOX appointed — and published — scholarly author. LIBRARY Dudley Knox Library / Naval Postgraduate School 411 Dyer Road / 1 University Circle Monterey, California USA 93943 http://www.nps.edu/library NPS ARCHIVE 1969 DETTMANN, T. THE PHASE EQUATION IN POTENTIAL SCATTERING by Terry Robert Dettmann DUDLEY KNOX LIBRARY NAVAL POSTGRADUAT E SCHOO MONTEREY, CA 93943-5104 ; United States Naval Postgraduate School THESIS THE PHASE BQUATION IN POTENTIAL SCATTERING by —— Terry Robert Dettmann June 1969 Ths docwment lias been approved gor puvlic re- Lease and sale; cts distribution 16 unkuncted, LIBRARY NAVAL POSTGRADUATE SCHOOL, MONTEREY, CALIF. 93940 The Phase Equation in Potential Scattering by Terry Robert Dettmann Lieutenant (junior gradd) ,United States Navy B.S., Marquette University, 1968 MASTER OF SCIENCE IN PHYSICS from the NAVAL POSTGRADUATE SCHOOL June 1969 ABSTRACT The solutions of the phase equation for a potential with the asymptotic properties of an atomic polarization potential are studied with the intention of developing a camputer program for direct integration of the phase equation for the s-wave phase shift of an arbitrary potential. Such a program is developed and its limitations are discussed. In addition, an appendix is devoted to the variational principles of Kohn and Schwinger for calculating phase shifts. LIBRARY NAVAL POSTGRADUATE SCHOOL DUDLEY KNOX LIBRARY MONTEREY, CALIF, 93940 NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5104 TABLE OF CONTENTS I. INTRODUCTION ---------------------------------------------- Q TI. THEORY ------------------------------------—--~---~---~-+-+--~— bm A. THE PHASE EQUATION ---—--———-——-——-=--___-=___________ a, B. THE POTENTIAL -----------------------------------~----~ Ws TII. CALCULATIONS ---------------------------------------------- 16 (SY, PR CUDERS ee eae 22 V. CONCLUSIONS --------~~-----~-----------------------—- + 28 Perron te ThE ronnie .ehOoo —=—$<—<—$—$=—$ 29 APPENDIX A.2 — THE SCHWINGER METHOD —-—-—--—-——-——————————-—-—-—_____ 33 SOE DAOC ONS ee ee eee 38 Pete Ce ———<——— 44 INITIAL DISTRIBUTION LIST --------------------------------------~ 46 POON, DID Daly s2sasesesese SSS SSS SS S55555 So 47 LIST OF ILLUSTRATIONS Figure Page 1. Plot of the Potential Function -------------------------- TS 2. Plot of Phase Function for K = 0.1, Yukawa Potential ---- 18 3, Plot of Phase Function for K = 1.0, Yukawa Potential ---- 19 4, Plot of Phase Function for K = 0.1, Yukawa Potential ---- 20 5. Plot of Phase Function for K = 0.1, Polarization BCE EIN a a 23 6. Plot of Phase Function for K = 1.0, Polarization Cen i eee a Ge a 24 7. Plot of Phase Shift vs. K for Polarization a — Za ig K a. (i, K) Ds (K) u(r) Wi) wor f(r) SYMBOLS THE RADIAL COORDINATE IN BOHR RADII THE ENERGY IN RYDBERGS THE PHASE-FUNCTION IN RADIANS THE PHASE SHIFT, No (rk), IN RADIANS THE SCHRODINGER RADIAL WAVE FUNCTION THE POTENTIAL FUNCTION IN RYDBERGS ACKNOWLEDGEMENTS The writer wishes to express his appreciation to Professor R. A. Armstead for his understanding, encouragement, and aid during the course of this investigation and also to my wife Pauline for her unquestioned support, loyality, and gentle goading as well as her skill with the typewriter. I. INTRODUCTION The Quantum Theory of scattering processes is of basic import- ance in studies of atomic, nuclear, and elementary particle physics. Analysis of Atomic collision processes is required in such diverse applications as gas dynamics, chemical reaction theory, and stellar dynamics. Nuclear Physics relies on scattering of various particles fram nuclei for a great deal of the quantitative information about nuclear processes while in Elementary Particle physics, very little other information is available except for the various scattering events that are observed. There are many approaches to the scattering problem, but one of the most useful (especially in low energy applications) is the method of partial wave analysis. The method was first introduced by Faxen and Holtzmark based on a method developed by Lord Rayleigh. Rather then delve into the basis of the theory, it will be assumed that the reader is familiar with the treatment of Schiff [Ref. 1] or Merzbacher (REE. 2). A particularly interesting procedure for calculating scattering phase shifts is the "Phase Method." Originally introduced by Courant and Hilbert [Ref. 3] and applied to scattering by Morse and Allis [Ref. 4], the method has recently been revived and expanded in a series of papers by Calogero [Ref. 5, 6] and his new book [Ref. 7]. This paper will investigate the application of the phase equation to a potential which has the asymptotic properties of an atomic polarization potential (1/x*) . The method employed is a direct integration of phase equation (11). In the appendix, the variational principles of Kohn and Schwinger are treated and are suggested as alternate methods for computation of phase shifts. 10 It... THEORY A. THE PHASE FUNCTION METHOD The phase function method is a method of reducing the second order linear Schrodinger Equation to a first order, non-linear Riccati equa- tion. For s-wave scattering, the solution for phase shifts reduces. to the solution of the differential equation (1) n(n) ee [very Je \ sant (xe +) To show how this equation. comes about, it is convenient to: approach the problem from the standpoint of A. Ronveaux [Ref. 8]. He starts by’ considering a second-order homogeneous differential equation with: two perturbative terms, Vy (X) and V. (X) ; (2) me + woul + Oida= Vdsu & KOO we’ with u(0) =p and u'(0) = v. This can be written in the form of the following system (3) Xie) = AGO Xx) + V(x) Xx) where ne On. Te X(x) (“| X(o) ‘? A(x) ?f O 1\ MVxjzf/O O Te -- ON, The unperturbed solution, xX, is, (4) X (x) = WON C aE where C = m C2 is a constant vector, and uae a on Us ie and Uy and u, are two linearly independent solutions. The solution of the complete system may then be written (5) K(x) = Wx) C(x) Using the Lagrange Method of Variation of Constants, then OC, we } WOO) VO) Wix) CCx+} Cx) = M(x)C(x) CCo) = j wean fa) If we now define the phase function (6) Ses mac, Ge) / CK and differentiate, we find that Cn S Cadence + [Ma (x) 7 Hy (0) S SOx) = Mi(x)SO) where Evaluating M gives the phase equation S (x) - } J uc) | + | u, (x) so] } Jet VW/(x) x } Mia] us + Uy (x) S (x) | t V, (x) | u, (x) -ufcasealt where det(W)# 0 and {[-we-l Oy [+ [ucor]$ } | u ol u | + [ucry]f 6) 2 Given the general form one now approaches the problem of s-wave phase shifts. The Schrodinger Equation (in natural units, oe | ) is (9) meee Ko eer) = Vor) adr choose Uo) ZT Wz O v.02). Then the general solution is Wir) 2 C, Sm ke ~- Co cos Kr ~ Cae) seine — Sag) coskr | OF Wee Men) ces Wier Fy te sim (Ket) where S€q)= ¢, 13 4, ¢r) = ~- tan 4, (rpx) Then, letting V, = Bio eG Chup sinks, U, = cosk ‘ga clef W: K the general phase equation becomes 7a (io) =(d/dr) tan Yo == [ very /k] (sim kr + tan yocoskr | with tan), (0,k)=0, Noting that (d/dr)tann, = sec ye(dyo/dr} it becomes apparent that (11) Cd/dr) No = I vor) /k | sin’ (Kr + "0 ) B. THE POTENTIAL FUNCTION The object of this paper is to investigate the properties of the numerical solutions for the Phase Equation (11) when applied to a potential which has the asymptotic properties at infinity of an Atomic Polarization potential, i.e., Ce) a (const. ) x ifr d [<7 oe Je In order to simplify the analysis, an additional condition was imposed on the potential in the form of a requirement that as [-? 0 , the potential should approach zero as ( , l.e., Clee > (const xr T——9 © The function chosen to satisfy these conditions was ¥ r—27 oo 4 o The asymptotic properties of this form are obvious, however, it is f ay UO ae a dee 6), ae eek difficult to calculate with. In order to simplify the problem further, the substitution pe xe was chosen which leaves { | : i rx (12) Cine a F lorena o It is no longer readily apparent that U(r) has the correct asymptotic properties, but it may be easily verified by integration and power series expansion. The success of this form is shown in Fig. l. 14 V(r) 0.08 0.04 Figure 1 Plot of the Potential Function 15 III. CALCULATIONS Having derived the phase equation (11), one now wishes to devise a method to put it to practical use. Calogero develops a variational scheme for the phase equation [Ref. 9] while other authors deal with the equation in other ways [Ref. 10, 11]. An excellent review has been prepared by Babikov [Ref. 12] which provides a good general introduction to actual calculations and an equally good bibliography. Owing to the fact that the equation is non-linear, the various ap- proaches have dealt with same form of approximate solution. This paper will deal with the relatively simple problem of constructing a program to directly integrate the phase equation using numerical approximation only. The particular method chosen was a fourth order Runge-Kutta Integration scheme to be implemented on the IBM-360 time sharing system. General discussions of the numerical method may be found in numerous places [Ref. 13, 14], and so only the necessary portion of that theory is presented here. The Runge-Kutta method proposes to solve the equation i) v a f ( x, Vv ) subject to the initial conditions X = Xo when Y = Yo° The iterative solution process proceeds via the following equation; (14) vi - vy, + nel 4+ 2k, + 2KRaee Ky | © Kas re eee) Ke? h {Cx nt 2 Ky2 hf 0X4 2h) Wt a &) where 16 Nig Mf Cknt hy Yn + Ks) and where X,,,, = X,¢h . The programming needed to provide a solution of the equation (11) which gives the exact phase shift, uy BMis ee Ny ©; k) , then reduces to programming a simple iterative procedure. The program developed is printed at the back and is in such a form that it may be run either on the time-sharing system for quick approximations or directly on the computer system in the regular batch processing. A second version of the program used to obtain a print out of the values of the phase function at various r values is also included. The use of the second form will become obvious as the discussion unfolds. It now becomes necessary to consider the actual running of the program and its particular limitations. In order to properly investi- gate the program, the Yukawa potential, ~mr VOr)# 4 c / mr is convenient to consider since Calogero treats the case with strength parameter, g = -10, m=1. The output produced by the second program is graphed in Figs. 2 and 3. The most obvious feature of the graphs is the presence of "levels" which arise from the presence of possible bound states in the potential. Calogero discusses the connection of this with Levinson's theorem and shows that this structure in the phase function can be used to locate bound states. This very feature though poses a distinct problem to the evaluation of the exact phase shift. Owing to the presence of this structure, a simple test for the end of the integration that depends on the phase function tending to a constant value at infinity is useless. 17 rKXX xX X x AXAA x Figure 2 Plot of Phase Function for K = 0.1, Yukawa Potential 18 4 Pi X orn 1% 2 x - 4 —— 0.4 Figure 3 PO ee Plot of Phase Function for K = 1.0, Yukawa Potential 19 For this reason, the integration must be carried out past any possible structure before a cutoff of the above type is allowed. In order that the program will do just that, an integer counter is established and required to be larger than a certain lower limit which is designed to carry the integration over the structure. The procedure for doing this is as follows; the second form of the program is first run with the intention of finding the structure and determining the value of DR which gives the best results. Having found the structure, the counter may be set appropriately. By use of this artifice, the program is enabled to reproduce the results given in Calogero's book to within a few percent. It should be noted that another way of treating this problem is to establish a lower limit on the basis of the value of r reached. This approach would be superior if there were a method for predicting the position of the last flat region in No" The counter method was chosen only on the basis of a desire to know the number of iterations used, for computer timing purposes. A further pecularity with the direct integration routine is the sensitivity to the s MRED, DR. In investigations with different 40 ra values of DR, it was found that when DR is chosen too large the structure at small r tends to destroy the ac- 20 x curacy of the routine by adding ¥ large terms over some of the flat x regions in the phase function. An | 3 pA example is for K = 0.1 and dr = 0.1. Figure 4 The result is es 44.1 in 110 steps as shown in the diagram (Fig. 4). 20 The true result (in agreement with Calogero's book, p. 22) is De 6.24 in 38 steps found by taking dr = 0.01 or less. Now the utility of the second program for preliminary tests become apparent since it allows us to establish the number of bound states for the potential and the proper increment and the lower counter limit in just a few test runs which may be quickly run on the computer time sharing system. The program also has a very important limitation in that the accuracy decreases sharply for a small k due to numerical round off errors arising from the necessity of dividing by a small number. The Situation may be improved by choosing a smaller increment, DR, but after a certain point this procedure tends to become self defeating because decreasing DR increases the program running time. A test run on a Yukawa potential with g = -2, K = 0.1, and dr = 0.1 gives a close answer to the zero energy phase shift, but if the value of k is reduced to K = ion) the accuracy in the calculated value of the phase shift actually decreases. Ja IV. RESULTS The final problem is the application of the programs to the polarization potential discussed earlier, eqn. (12). Figures 5 and 6 detail the results of the explicit calculation with the second program. They indicate that a value of DR of 0.1 will carry the program nicely through to the asymptotic region if the lower counter limit is set at 200. The lack of structure in the graphs tends to make this requirement useless though and it becames pos- Sible to set the limit at 10 iterations just to let the program get started before the asymptotic cutoff procedure is allowed to function. The lack of structure just commented on tends to indicate a lack of bound states in the potential as might be suspected from the very weak nature of the function. Tests could be done by imposing a strength parameter on the potential, i.e., { Vir) = qe |dxx%e O which could then be adjusted as suited the investigator. By means t X of such an artifice, bound states might begin to appear when the peak of the potential is high enough. In addition, increasing the strength of the potential would increase the magnitude of the phase shift. One might expect that such structure would appear for values of r less than 1.0 and that the way to locate such structure would be to decrease the value of the increment DR. Though interesting, such an investigation does not have any apparent physical use since the only physical feature of the potential is the asymptotic dependence at large r. Thus further discussion of this point has been amitted here. 22 Figure 5 Plot of Phase Function for K = 0.1, Polarization Potential 23 Figure 6 Plot of Phase Function for K = 1.0, Polarization Potential 24 The most interesting feature of the calculation fram the stand- point of possible application of the program to real, physical potentials is the necessity of carrying out the calculation to larger values of the radial coordinate than was the case for the Yukawa potential. The reason for this behavior is obvious, the polarization potential only drops off as fast as Ve" at infinity. This leads to the observed difference in the convergence of the scheme and probably to the reason why the only places that the Phase method is applied in the literature are in papers dealing with nuclear type potentials which may be assumed to fall off very rapidly. To get some idea of the magni- tude of this effect, the graph in Fig. 2 may be examined in comparison to the graph in Fig. 5. The plot for the Yukawan potential has sta- bilized by r = 1 while with the polarization potential, the plot is just stabilizing at r = 25. These observations hold great import for any application of the method to atomic like potentials since the matter of time available for computation is often very critical. Even on the IBM-360 time sharing system, timing for the program had become critical when increments less than 0.1 were chosen. In addition, output disk space became a problem. The problem of timing could be expected to be considerable worse on smaller computing systems. Having discussed the phase function and its properties, use may now be made of program 1 to obtain a plot of ny VS- energy as amieeeyateye. 7 « The most apparent feature of Fig. 7 is the 0.78 radian peak at k = 0.36. It is a real peak as opposed to an apparent peak that could be produced by a poor choice of the increment at such a low value of k. An inter- pretation of this peak will not be attempted at this point but a com ment is appropriate on the behavior of Ny (kK) on the low energy portion Z> of the peak. It seems to drop off drastically, possibly to zero. Below k = 0.1, no data taking was attempted due the necessity of integrating too far with too small an increment. 26 Figure 7 Plot of Phase Shift vs. K for Polarization Potential 27 V. CONCLUSIONS The program developed for the phase equation integration has been tried on a test case and applied to a second, semi-physical case with somewhat the expected results. This program has in these two cases performed well as long as the precautions mentioned were taken. It should be noted that no error analysis of the routine was attempted due to lack of time, however, the comparison of the results on the Yukawa potential for g = -10 with the results published by Calogero in his book [Ref. 7], tends to point to an accuracy that is within a few percent. Whether this result is valid for any potential other than the Yukawa potential and in particular whether it is equally true for atomic level potentials is a question that remains unan- swered at this point. It is expected that simple numerical errors arising from the Runge-Kutta integration scheme and the ordinary computer errors of round off, etc. will decrease the accuracy of the calculation. 28 APPENDIX (A.1) The Kohn-Hulthen Method The most frequently used variational procedure is the Kohn- Hulthen method [Ref. 15, 16] which is based on the stationary property of the functional ~ \ where (Al) se am |YCE-H) bdr Le and A is same function of the phase shift determined fram the normal- ization of b as —-700° . This paper will employ tangent normalization throughout, i.e., (A2) p 7 Ga ° | je (Kr) + tang ne Cer) | r—> co where Ne is the phase shift for the ee partial wave and fe and vy are the spherical Bessel and Neuman functions respectively. Consider- ing the special case of s-wave scattering, Q = 0, then (A2) becames Y a, (da)? [ sinke/ir A. Pay Yo MM: /cx) J ; < (uz) * | _ sin (Kirt 7) COSY, RG With this form for p , ohne now considers the explicit variation of (Al) «(AT jsucv 1K Ud AT + vests ce. U)Sedr Since for s-wave, p has no angular dependence, be BEC H)Y so that the second integral in (A3) becomes \+\ 3 ("Ged + Kr - v| od ar raday\ dr - fan _— aba" Ate 4 \6 dco) ode yr Zo Integrating by parts, this gives “an [as fd {oe age} - 4 _ dd *d (AY) dr 1 (shee -vo) de da \or} Weed (Sv) - 5¥r? at) 8d (gy 4 a VAT é ts (a a rac oe srs | be jshenwita , at Jr wn SI - 2 (sboviexs aybar 4 Ja[e(vash. adds The last term, upon application of the boundary conditions, reduces a 5 Yan yo /k } so that sy (I+ tan Ye /k) BZ 150s =) ee Thus (A3) becomes | aa But the Schrodinger equation is just (Vi4e-v)P=0 so that (A5) dCi + ton No /k ) : The stationary functional, [A] = Tan Yio /K , is given by (A6) (oe We ee a 2 \d(e- wd bdr An 30 The trials function, p , 1s chosen to have the form (A7) W = a” 4 lige X; (0) where uy = (di) “aes K i | and the X; are same convenient basis functions, that rapidly approach zero as r-”?@ . The asymptotic part of the wave function a , 1s chosen so as to shield the = singularity in cos kr/r as r~7 0. It is evident that p has the correct asymptotic form as | Y = (ui \? [sinkr/er ~ \k Csinkr /p2p? - coskr /er)| aa > (uz) ns [— (sin br /ker) + | (cos Kr/r) | 7 eee The Kohn-Hulthen principle then states that the variation of {\] with respect to all arbitrary constants, l.e., (a8) SI\l = S(A+I)=- 0 produces a value of [\] that is second order accurate ind) el he explicit variation is done by writing the functional in the form; 0 A] i at; City + 226, @ + B+ zy a) where 2 MM; = 2m \ x. CE- WX; a Mii \ Xj (e-wiP A sP B= 2m \ oe-m¢ AS Sil Variation of the n constants, C, then gives the system of linear equations (A10) z Mi Ci = = Ry J Since the trial function is linear in \ A (0) C %; = ic; + \ I? ; and since the /1, | are independent of \ COF Cc! Co ce + \ C; This leaves the equations: Co) 2. Mi le = ie; Ci) = IZ; (All) ve 2 My P C/ Solving for the ir,.8 : TM = wy + Aw, + dw, 4 Bo + \8, 4) Bat) where a Hiss (wo) Co) (0) W, : Vs oF ‘ M;; C +t 2 2 C; Ie, e | d Cr) = Ci) =e — - ¢ Co) : WW = py Ch S45 C. a + 2 2 G Ky 16 Z é C, Kk, 3) é Ci iy yy Ci) € 26° My Gf? + 2 26fPR J yA Then a final variation with respect to \ gives, to first order \- = Guinea. ae 2S; + Gs) and the second order accurate, stationary functional is PAS ae ech [hy = fw + B +1 a Bit!) ee SU, + By) 32 (A.2) The Schwinger Method A less frequently used but more elegant variational method is that due to Schwinger [Ref. 17, 18, 19, 20). It gives the stationary value of the functional [Kcoty | directly. The problem is to solve the radial Schrodinger Equation. (A13) Au + a = Bc2a) | ug() = Or) ur) oh i“ To do so, one solves for the Green's Function Giger it ) given by, ene i which gives me) gh Garin + [at aenaan | Galen Stree Ge (rir, )= ~K iro paler) my kre) GS la (A1L5) Se O(c Cig emai aire The integral equation for ?(r) is then a> (Al16) Din lee ) = ae ep = fdr. GGr tn, ) OG arene) For large r, the asymptotic forms are CIC ORS (rr- 2 (0+1)7) > sin(yr - 4 (es1)q) Kr yoCicr ) K¢ Ne (Kr) therefore, the asymptotic form of JU,(r) may be written as ? yous ee err O> + sin eps : oa] | de Vy fe) uCry / © But, this is to match to the asymptotic form U pg ( op (A17) UgCr) —-> Cos Ikr - “(e+1)i | + sinlkr- 3 (esi ] tanne 50 and so one sets (eos ee] SPR “{e i \ae. te yelX to) OCr,) Ug 6 ‘a0 Inserting the integral form for yecer) in terms of Ue(r) fram (Al6) ae (i> 7 IS therefore, Ktanv\g 5 \3: DARE: \ p Vbey Or) ugte) GOI) Ul) Ug) © © Oo dividing now by to ns "Ne oa ur) \1 Uor)ug dr) + {o |e Ur ue ce y Gale, JOOD Uely) ev (Als) K cot 2 5 Bos A “cee Oe re ee te ee 8 ee me ee a ow ee Oe en 2 | \a- if car yunerdjecer | .@) For s-waves this becames Vale GUE je fr Or duce) Grin) Ulry) olre) (ALS) Keedyy = vie | ee yee Te ee : i \sr Vic er yeaa cr | where G(rir,) is now: | Cigars ) ea aay Cos Kr, Ge Oe = u Sn KT pcos le res £ K In order to show that (Al19) is in fact stationary to second order for the exact wave function, consider (A20) S\ kot v0 | = | keed 7 (ot du)| = [kcot (uf 34 Expanding and collecting terms, we have S(kcot no | > (ar UCr)o(r) | eed - (ar Gr V¥y) Uy) ote | Gt + 2 Ar Gy Wow fucry- (cert. ye WL. 2 | le dy Uiervirsntr | +2] [i [ar oer Sosinte [xf Jor ver UCrolr) si mat | 6 lO 5 yD Oo© (a. Ur) vcr) Juco - |r. Gr try) Uerruens | gO eines ° : me \: dy Ul (r)olr) sin ky | Cry combining the terms the numerator for the expression will be i 2 So CK? at |. (ar UL sors | Nt UG Ge Jen jas Gr UGH ¥0%)| “ ), & v + 2 jar Ola duce) Je = js Fy Gr | rye) Or, ) UCf,) \\ oO mace al o oo ox) 2 ] a \a OCs yuced| eerd- jor Glrin)JOcy vcr, Life Uriearainnt @ + |: \4r Ur duces sin us| x # [3 Uce)ocrdsines ]f + XS?) However oF ave) ~@ (A21) {de Or) o¢r) | ein Joi. 6¢rh UCI oer kcoty, i {err aernnte| be trem (All7) this is Be = [as Ur) vers sin ler After factoring out Ie (a (< ry UCriudr) sinks = the numerator becomes es 6a tou! > (ar Uirivmsinkr| ‘}(or rO(r)vdr)sinkr = \s1 dr )udr) sinks > . O WN 21 BZ | Or) Svtr) [eee - sinkr (1. Gerig)ocsuealf OS) Sp) However, combining terms, this is, ie: \. Or) ocr) sin kr | 32 \o. Uc) Ser] ven =sinken lo r, Gtr In) Ula) u% 2 ‘ + &(S*) But from (Al6), for u(x) the correct wave function, Os © (it Ot(r) -smky - i ry GOr Voy Ot )ucrg) Thus, neglecting terms of order Be , the numerator is zero and hence 5\k Cot 1 | S4O Therefore, we see that (neglecting second order variations) the expres- sion is stationary when u is the correct solution of the Schrodinger Equation. Now going back to equation (Al19) and inverting 2 7 | la Uladucrysin Kr | (A24) klong, |= ee ee aay a rr Udryutir) - Jas fac, Uervo«rr Grin) Utadoer G Letting HOME he x; then 2 (A25) [Kton,| : (2c; V, ) - 26 Ci Mj in ° 2 Cty (Uj os Gy) VE Cyc; Nii where Oo V5 “ \ax Ur ) Xi Cr) sin [rp Uy) 2 \s0 Ohl AS) Kj Cr) = Ui, CH = {2 (a Or) X; (VY GONG) UG IX ()- Gi and : Ni Vi ae Ni Mij * Vii ~ Gis 36 / The problem remaining is to solve for ik 4 y\ | and tlie Cs . do so, define (A26) Q = vy. C70; (Lkiony, [My ~- Nii ) =O but Q is obviously stationary and so, (A27) AD® = 2 2¢;((ktany, |Mij- My )2 0 OC; In order to have a solution for the CS , the determinant of the matrix of coefficients must be zero, l1.e., (A28) e+ (Ci tan Mol Mij ~My) ) = O However, it is a simple matter to show that the matrix N is of rank one, i.e., there are no non-zero sub-determinants of the matrix N of order 2, and further, it can be shown that such a system has only one nontrivial solution for ii tan Y) oa ; Nj, My “ae 4 Mu My--. (A29) ary Wp ae ea i Mii N42 sare TF * ue 4 | Ke a 8] i} | a re el A PAE PEI TY POY TOE NEA LNT PE LI RA TES ve OM Li PELE BF TLE TOMS EY NT MT I LUE I EE We bn, PAO) ey eames Pb Saat CH FR fo\* M i Thus one finds the solution for | K tan n | without recourse toa ] solution for the C$. 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N OO Oe NOMS W OO O00 © od and ae = = om 41 OF DTORNM SIAM DOOUPREEERERE COOOVVCOOCOUO COODCOOVOCOGO ADODCOVOUDGDOOO UUNUUNUYUUUUe BEPC OE DG Mo 2 ie Gs ep Gi Gp aaadkaaadadad EGRATION ROUTINE UNCTICN PROVIDES THE MAINLINE INT Lis ~, x N ~ # AN + . d ~~ -_ # < & -_ = e «<Y Li Ww - + o WW Oo” <{ > ¥ _ ‘a 4 LJ LJ oe. SC ad - Oo ~~ < (a 4 —_— rol — —_ — (Ta) Ww WO WM O. # a * = = > =~ = a) uu qu - Ww aA alia W - ow = = uUrfE = tC ia, x We” Ww WW bly bd Or bas md and pa) -= WWUWUWOUY OCOOOOOQVOOVDOWVDOVGVOVWVODOOCVWOVO00O KHOA SN OP OKOMNO YS AOM DUO OHANMS UO OM QOUDKVOO VON XM St He et Stet HN AIQINUNICUAIN QUA ODDO ODQMQQMDO OVO QO OVOVGVIOOVOODWMDIOIOOOD00092 OOOODOOOV OOO VDOOADIQOTVOMDVOQVVUBOMOO0ORM000 LET DT SE oe eee oe ee ee eee WU WOO UOUU OU OO UUOUUOU OOO U UO UUO MMN NNN NNANNANNHNNNNNANANNNNNNNNNNM P(R) QUADRATURE INTEGRATION OF FUNCTION WWIOW RE INT (X*®*®48EXP(-R*X)DX) a, Cc ~ uN Om + KxTTNDODOS TOC SHO SHIOO BWOOEMOMNA OS SOLS DO mDODMMOO ODD ONM SEAIOM DWAUAHO DO OFM HAIN OMS MOHMO®B @ AIGMPF=§ OCOMHONO OWUM TOD N ZHAO CQ\NUASIMSHORNAMNOO ON HO BOBANOAM VN ONGH O ZHNAMOMO eQied © el emir Oeeee eM © 0O0O eO 0 0 oe @ Cn aaee a O°O 1 I a | ooo0o e ' oziuuuwu nnunnawn we 00 LLY came came, com, tte, ee, OE ee oe am, ey, Ay EE A, a, em, a, om ZT HNMG PTO DaNOALSMOOFPO Seek ee ee Se a ae? Ce a ee ee ee ee ee a Ca NOdd dd dl qm OK OK EK KK 42 e) )*(U**4*EXP(-R*¥U)) l I + —_ mJ CO m= @uw > od 2K wh = lt 35> WNW) — © ££ COlaew =~ nxunRK—o Oo > w2 ODNaAaw = OOOQWVOVOO0O KANO HOMO © elelelelelel ee] er. DOOQGDQO0O000C0O DOOODOOCVOU0O0O me eb CO0CCOOdOOdoO aga00aga0gaag0c0 4006. od C ac a *% = © = 5 ~ = a ow om © Sd a e a “OC | a OF ~— «0 Oa. ZO eOmM UW QOwoowWw DO — eet of Ce 0a. Oar Oe fa =e D ee | ee | oe iL wo We wee (DEO OOK UW =~ 43 i. ee 1p 13 14. iS: Hr REPERENCES Schiff, L. I., Quantum Mechanics, p. 92-121, McGraw Hill, 1955. Merzbacher, E., Quantum Mechanics, p. 213-247, John Wiley & Sons, 1961. R. Courant and D. Hilbert, Methods of Mathematical Physics, p. 303, Interscience, New York, 1953. P. M. Morse and W. P. Allis, Physical Review, vol. 44, p. 269, 1933. F. Calogero, "A Novel Approach to Elementary Scattering Theory," Nuovo Cimento, vol. 27, p. 261, 1 January 1963. F. Calogero, "Maximum and Minimum Principle in Potential Scat- tering," Nuovo Cimento, vol. 28, p. 320, 16 April 1963. F. Calogero, A Variable Phase Approach to Potential Scattering, Academic Press, 1967. A. Ronveaux, "Phase Equation in Quantum Mechanics," American Journal of Physics, vol. 37, no. 2, p. 135, Feb. 1969. F. Calogero, "A Variational Principle for Scattering Phase Shifts," Nuovo Cimento, vol. 27, p. 947, 16 February 1963. S. Franchetti, "Simple Treatment of Central Force Collisions with Particular Reference to Phase Shift Calculation," Nuovo Cimento, vol. 6, p. 601, 1 September 1957. B. Levy and J. Keller, "Low-Energy Expansion of Scattering Phase Shifts for Long-Range Potentials," Journal of Mathematical Physics, vol. 4, p. 54, January, 1963. V. Babikov, "The Phase-Function Method in Quantum Mechanics," Soviet Physics Uspekhi, vol. 92, p. 271, November-December, 1967. R. G. Stanton, Numerical Methods for Science and Engineering, p. 151-154, Prentice Hall, 1961. Digital Computer User's Handbook, p. 2-144-162, McGraw-Hill, 1967. Kohn W., "Variational Methods in Nuclear Collision Problems," Physical Review, vol. 74, no. 12, p. 1763-1772, 15 December 1948. Schwartz, C., "Variational Calculations of Scattering," Annals of Physics, vol. 16, p. 36-50, 1961. 44 Lae 18. i. ZO. Schwinger, J., Unpublished Lecture Notes, Harvard University, 1947. Altshuler, S., "Applications of Variational Principles to Scattering Problems," Physical Review, vol. 89, no. 6, p. 1278-1283, 15 March 1953. T. Wu and T. Ohmura, Quantum Theory of Scattering, p. 64-65, Prentice Hall, 1962. Schwartz, C., "Application of the Schwinger Variational Principle for Scattering," Physical Review, vol. 141, no. 4, p. 1468- 1470, January, 1966. 45 INITIAL DISTRIBUTION LIST Defense Documentation Center Cameron Station Alexandria, Virginia 22314 Library, Code 0212 Naval Postgraduate School Monterey, California 93940 Professor R. L. Armstead, Code 61 Ar Department of Physics Naval Postgraduate School Monterey, California 93940 LTjg Terry R. Dettmann, USN 10826 W. Grant Ave. #2 West Allis, Wisconsin 53227 Commander, Naval Ordnance Systems Command Department of the Navy Washington, D. C. 20360 46 No. Copies 20 Unclass} Security Classification DOCUMENT CONTROL DATA-R &D and Indexing annotation muat be entered when the overall report Is classified 2a. REPORT SECURITY CLASSIFICATION Unclassified (Security classification of title, body of abstract 1. ORIGINATING ACTIVITY (Corporate author) Naval Postgraduate School Monterey, California 93940 3. REPORT TITLE The Phase Equation in Potential Scattering 4. OESCRIPTIVE NOTES (Type of report and inclusive dates) Master's Thesis, June 1969 $. AUTHOR(S) (First name, middle initial, last name) Terry R. Dettmann June 1969 46 20 Ba. CONTRACT OR GRANT NO. 92a. ORIGINATOR’'S REPORT NUMBER(S) b. PROJECT NO. 0b. OTHER REPORT NO(S) (Any other numbere that may be aesigned this report) (10. DISTRIBUTION STATEMENT Distribution of this document is unlimited. tt. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Postgraduate School Monterey, California 93940 13. ABSTRACT The solutions of the phase equation for a potential with the asymptotic properties of an atamic polarization potential are studied with the intention of developing a computer program for direct integration 6 the phase equation for the s-wave phase shift of an arbitrary potential. Such a program is developed and its limitations are discussed. In addition, an appendix is devoted to the Variational principles of Kohn and Schwinger for calculating phase shifts. DD °""..1473 (Pace 1) 53 “ Say OTUT-807=a08T7 aS ee -eipacilicadedic alee A- 31408 Unclassified ~ Security Classification KEY WORDS a Phase Equation Potential Scattering DD 12h"..1473 (eack) Unc S/N 0101-807-6821 48 Security Classification A-31409 hela he phase equation in potential scatteri ATA 3 2768 002 10859 7 DUDLEY KNOX LIBRARY