DISCRETE VARIABLE REPRESENTATION OF THE ANGULAR VARIABLES IN QUANTUM THREE-BODY SCATTERING DAVID CABALLERO

Transcription

1 DISCRETE VARIABLE REPRESETATIO OF THE AGULAR VARIABLES I QUATUM THREE-BODY SCATTERIG BY DAVID CABALLERO A Dssertaton submtted to the Facut of Caremont Graduate Unverst and Caforna State Unverst, Long Beach n parta fufment of the requrements for the degree of Doctor Of Phosoph n the Graduate Facut of Engneerng and Industra Apped Mathematcs Caremont Graduate Unverst Caforna State Unverst, Long Beach 0 Coprght b Davd Cabaero 0 A rghts Reserved

2 APPROVAL OF THE REVIEW COMMITTEE Ths thess has been du read, revewed, and crtqued b the Commttee sted beow, whch hereb approves the manuscrpt of Davd Cabaero as fufng the scope and quat requrements for mertng the degree of Doctor Of Phosoph n the Graduate Facut of Engneerng and Industra Apped Mathematcs Afonso Rueda, Ph.D., Char Caforna State Unverst, Long Beach C.Y. Hu, Ph.D. Caforna State Unverst, Long Beach Es Cumberbatch, Ph.D. Caremont Graduate Unverst A adm, Ph.D. Caremont Graduate Unverst

3 Abstract Dscrete Varabe Representaton Of The Anguar Varabes In Quantum Three-Bod Scatterng b Davd Cabaero Caremont Graduate Unverst: 0 There are man numerca methods to stud the quantum mechanca three-bod scatterng sstem usng the Schrodnger equaton. Tradtona, a parta-wave decomposton of the tota wave functon s carred out frst, aowng the scatterng sstem to be soved one parta wave at a tme. Ths s convenent when the nteracton s centra, causng the tota anguar momentum to be conserved durng the coson process. Ths s not possbe n the presence of a non-centra nteracton such as a aser fed, where the tota anguar momentum s not conserved durng the coson process. The Dscrete Varabe Representaton s a new method for sovng the quantum-mechanca three-bod scatterng probem to obtan the tota cross secton. The mpementaton of ths new method for the two-bod probem has been successfu apped to rea

4 sstems. The etenson to the three-bod probem s the net ogca step. For ths thess bpoar spherca harmoncs are used n the mpementaton of the three-bod Dscrete Varabe Representaton. Ths Dscrete Varabe Representaton s capabe of worng wth an combnaton of nteractons, ncudng non-centra nteractons. The tota cross secton computaton for a three-partce eastcscatterng numerca eampe s used to ustrate the potenta of ths Dscrete Varabe Representaton method. The three-partce sstem conssts of a postron scatterng aganst a ground state hdrogen atom an eectron bound to a proton.

5 ACKOWLEDGMETS The author woud e to than the members of the commttee for ther hep and advce n the preparaton of ths thess. The author s partcuar gratefu to Dr. Hu for her nvauabe advce, mmense hep and drecton n mang ths thess possbe. The author s aso gratefu for the generous aocaton of computer tme and for the use of computer resources made possbe b the SF, Teas Advanced Computng Center TACC and the Teragrd aocaton commttee. v

6 TABLE OF COTETS ACKOWLEDGEMET... TABLE OF COTETS... LIST OF TABLES... LIST OF FIGURES... Page v v CHAPTER ITRODUCTIO... COMPUTATIOAL METHODOLOGY Three-bod Eastc-scatterng Coordnates. 8. Dscrete Varabe Representaton Rada Spnes And Spne Bass umerca Schrodnger Equaton Asmptotc Boundar Condtons ear Interacton Regon Boundar Condtons Scatterng Cross Secton RESULTS Seecton Of Knots And Anguar Grd Resuts Usng Y-As Azmutha Shft Resuts And Anass Of Other Y-As Azmutha Shfts COCLUSIO v

7 APPEDIX A ATOMIC UITS AD MASS-SCALED JACOBI COORDIATES A. Introducton A. Eectron Atomc Unts A.3 Jacob Coordnates... 9 A.4 Mass-scaed Jacob Coordnates A.5 The Couomb Potenta In Mass Scaed Jacob Coordnates A.6 The Two-Partce Schrodnger Equaton In Mass-scaed Jacob Coordnates B DISCRETE VARIABLE REPRESETATIO DVR B. Introducton B. Two-Dmensona FBR Bass Functons B.3 Anguar Grd Gaussan Quadrature Ponts And Weghts... B.4 Anguar Momentum Quantum umbers... 3 B.5 Gram-Schmdt Orthogona And Competeness... 6 B.6 Two-Dmensona DVR... B.7 Four-Dmensona FBR Bass Functons... 4 C SPLIE ITERPOLATIO C. Introducton C. Hermte Ponoma Spnes C.3 Interpoaton Of A Functon Usng Hermte Ponoma Spnes C.4 Generc Equaton Usng Hermte Ponoma Spnes C.5 Hermte Spnes In Two Dmensons C.6 Interpoaton Of A Functon Usng Bass Functons v

8 APPEDIX D SOLVIG THE SCHRODIGER EQUATIO USIG DVR BASIS FUCTIOS AD SPLIES D. Two-Partce Schrodnger Equaton D. Fu Two-Partce Boundar Condton D.3 Appromatng Two-Partce Boundar Condton... 6 D.4 Three-Partce Schrodnger Equaton D.5 Three-Partce ear Zero Boundar Condton Usng Bass Functons And Spnes D.6 Three-Partce on-appromate Asmptotc Boundar Condton D.7 Three-Partce Appromated Asmptotc Boundar Condton... 7 REFERECES v

9 LIST OF TABLES TABLE. Orthogonat And Competeness Mamum Tota Anguar Momentum Inde Eastc Cross Secton Vs Incomng Momentum Phsca Constants... 89

10 LIST OF FIGURES FIGURE. Jacob Coordnates Seected Jacob Coordnates Pont DVR Anguar Grd Jacob Coordnates At th Grd Pont Cumuatve Probabt Of The Ground State Hdrogen Eectron At Rada Poston Coordnate Dagram For The Incomng Vector And The th DVR Grd Ange Rotated Coordnate Sstem Ground State Hdrogen Wave Functon And Its Frst And Second Dervatves Eastc Cross Secton Vs Incomng Momentum Usng Quntc Spnes On Eastc Cross Secton Vs Incomng Momentum Usng as Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.3 And 0 0 Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.4 And 0 0 Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.5 And 0 0 Y-As Azmutha Shft... 60

11 FIGURE d. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.6 And 0 0 Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.3 And Y-As Azmutha Shft... 6 b. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.4 And Y-As Azmutha Shft... 6 c. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.5 And Y-As Azmutha Shft... 6 d. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.4 And Y-As Azmutha Shft... 66

12 FIGURE 4c. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.5, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.3 And 0 0 Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.4 And 0 0 Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.5 And 0 0 Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.6 And 0 0 Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.3 And Y-As Azmutha Shft... 74

13 FIGURE 7b. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.6 And Y-As Azmutha Shft a. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.6 And Y-As Azmutha Shft... 79

14 FIGURE 0a. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.3 And Y-As Azmutha Shft b. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.4 And Y-As Azmutha Shft c. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.5 And Y-As Azmutha Shft d. Rada Wave Functon 0 For Frst Y Knot 0.8, 0.6 And Y-As Azmutha Shft Eastc Cross Secton Vs Incomng Momentum For Frst Knot Locaton Eastc Cross Secton Vs Incomng Momentum For Frst Knot Locaton Eastc Cross Secton Vs Incomng Momentum Usng as Azmutha Shft Jacob Coordnates For A Three Partce Sstem Jacob Coordnates Two-partce Coordnate Reducton Tradtona Anguar Grd Eampe on-tradtona Quantum umbers Eampe Tradtona Quantum umbers Eampe Pots of Spherca Harmonc Functons for 4, L 3, and M... 0 v

15 FIGURE 3. Imagnar Part of Spherca Harmonc Functons on Even umber Grd Four-Dmensona Anguar Momentum Grd For 9, Four-Dmensona Anguar Momentum Grd For 9, Jacob Coordnates At The Four-Dmensona DVR Grd Pont Indeed B Jacob As Orgns Agn Eampe Of φ Shft For Cubc B-spne Bass Quntc B-spne Bass Cubc Spne Coocaton Ponts Between Knots And Boundar Condtons Quntc Spne Coocaton Ponts Between Knots And Boundar Condtons Eampe Of Two-Dmensona Knots And Coocaton Ponts v

16 CHAPTER ITRODUCTIO Quantum mechancs came about durng the ear twenteth centur through the wors of a number of phscsts e Bohr, Schrodnger, Drac, Hesenberg and man others. It provdes a mathematca frame wor to cacuate the dnamc propertes of atomc and nucear nteracton sstems. Amost everthng nown about partces at the quantum scae has been found b scatterng eperments. Thus a mathematca frame wor descrbng the scatterng process can be used to stmuate a quantum scatterng eperment b epanng the underng detas of the resuts of the scatterng process. However, computatona dffcut has severe mted the cacuaton of the three-bod probem. On recent, wth the advent of massve parae computers, has progress been made towards sovng the three-bod probem. Even usng toda's computer sstems, new computatona methods are needed to app the quantum mechanca equatons to obtan soutons to the precson necessar for current scatterng eperments.

17 Current researches are deveopng new numerca methods for the appcaton of quantum mechancs to the most genera ow energ quantum three-bod scatterng sstem. The method n ths thess provdes a step towards that goa. The method descrbed n ths thess produces the tota and dfferenta scatterng cross sectons that can be compared drect wth epermenta data. In ths research the epermenta data conssts of a postron scatterng aganst a ground state hdrogen atom whch conssts of a proton and an eectron. Ths scatterng process s denoted as e H. As ths method becomes more refned t coud smuate ant-hdrogen producton eperments n eterna feds snce t woud ndcate how to obtan the argest cross sectons correspondng to ant-hdrogen producton. The ong range Couomb nteracton s a maor dffcut n obtanng soutons to the Schrodnger equaton governng the e H scatterng process. There are man methods to stud ths probem usng the Schrodnger equaton,,3,4,5,6,7,8,9. However, the method that provdes the most compete pcture for ow-energ scatterng s the Modfed Faddeev Method MFE,

18 especa when there are rearrangement channes, and a three partces are dfferent. The modfed Faddeev method nvoves the sovng of a set of couped equatons that s we suted to anaze mut-channe scatterng processes wth a ong range potenta 0,. The effcenc of the numerca souton to the MFE and other mpementatons of the Schrodnger equaton are a functon of the choce of bass functons used n the epanson of the wave functon. The epanson and proecton of the bass functons reduces the dfferenta probem to a arge set of couped near equatons. The souton to these probems requres the manpuaton of arge matrces and a poor choce of bass functons can mae the probem unwed n both matr denst and computer tme requred for a souton. One effectve method to sove the equatons s to epand the wave functon n terms of goba bass functons,3. Once the anguar momentum states have been proected out, dagonazng the netc energ operator, a Gaussan quadrature procedure s used for the numerca ntegraton of the potenta operator. For three-bod scatterng an effcent method for evauatng the MFE equatons s to use a bpoar spherca harmoncs 3

19 epanson to represent a the anguar momentum states 4,5. Once the anguar momentum states have been proected out, a Quntc-Hermte ponoma spne and coocaton procedure s used for the and coordnate numerca cacuatons 6,7,8,9,0,,,3. The three-bod probem n ths form ma be soved usng parta-wave decomposton. For scatterng probems ths s convenent when the nteracton s centra so that the tota anguar momentum s conserved. Wth ths method the souton s obtaned one parta wave at a tme. Ths approach becomes mpractca for scatterng n the presence of a non-centra nteracton such as the presence of a aser fed or when the coson energ ncreases so that man rearrangement channes are possbe. The Dscrete Varabe Representaton DVR s a bass-set representaton n whch the matr eements of a mutpcatve potenta energ operator V are dagona 4. DVRs can be n man tpes of functons, such as ponoma, trgonometrc or anguar spherca harmoncs 5. In 965 Harrs chose a ponoma bass to dagonaze the poston operator, or a monotonca ncreasng or decreasng functon of. He then stated 4

20 that ths bass aso dagonazes V 6. In 968, Dcnson showed that f the bass were ponomas mutped b a weght W then the matr eement computaton usng DVR woud be equvaent to usng an - pont Gaussan quadrature 7,8. In 99, Lght used the DVR method to sove the nucear Schrodnger equaton 9. The use of Dscrete Varabe Representaton n the two-bod probem has been successfu apped to other phsca sstems 30,3,3,33,34. The etenson to the threebod probem has aso been nvestgated b defnng a three-varabe DVR as the drect product of three dstnct one-varabe rada DVRs, one for each dmenson,, and z 35. The advantage of the DVR s that the matr eements of the potenta energ are dagona when proected out at the DVR grd ponts. The couped netc energ matr eements are eft to be evauated anatca or numerca. B dagnozng the potenta, and numerca ntegratng the netc energ, computatona effcenc s obtaned wthout the use of parta waves. The method proposed for ths thess uses the Dscrete Varabe Representaton as the anguar bass functons used n the epanson of the wave functon to sove the Schrodnger equaton. Once proven, ths method ma be 5

21 mpemented nto the Modfed Faddeev Method. Foowng the sprt of Hu and Kvtsns, who use the bpoar spherca harmoncs consstent as a convenent set of bass functons to sove the Modfed Faddeev Equatons, the DVR bass functons used to sove the Schrodnger equaton are aso based on the bpoar spherca harmoncs rather than a drect product n the mpementaton of the three-bod four-varabe DVR. The DVR cacuaton s parta wave ndependent and w end tsef we to cacuatons of non-centra nteractons. Quntc spnes and three-bod eh rada bass functons are mpemented n the and coordnate numerca cacuatons after the DVR anguar bass functons have been proected out. Ths thess computes the scatterng cross secton for a three-partce eastc-scatterng probem numerca. The numerca eampe s used to ustrate the potenta of the Dscrete Varabe Representaton method to sove the genera three-partce scatterng probem. 6

22 CHAPTER COMPUTATIOAL METHODOLOGY The descrpton of the dscretzaton of the Schrodnger equaton s gven n ths chapter. The frst secton dscusses a convenent coordnate sstem used to smpf the mathematcs. The second secton descrbes the DVR bass functons that are used to smpf the Schrodnger equaton when these functons are proected out of the wave functon. The thrd secton descrbes the rada bass functons that are used to nterpoate the resutng rada wave functon. The fourth secton gves the dscretzed Schrodnger equaton usng the resuts of the prevous 3 sectons. The ffth secton descrbes the asmptotc boundar condtons and the amptude functon. The sth secton descrbes the boundar condtons near the nteracton regon. The fna secton descrbes the scatterng cross secton obtaned from the amptude functon. A detaed descrpton of each of these sectons s gven n the append. 7

23 . Three-bod Eastc-scatterng Coordnates Append A contans more detas of the mathematcs behnd ths overvew of the Mass-scaed Jacob coordnates 36,37,38. The mass-scaed eectron-atomc unts have η e m e K so that the unt of ength s the eectron Bohr radus a correspondng to the ength n MKS MKS unts of a η /Km e M. Smar, the e unt of energ s E, and the eectron Bohr energ E B -½ correspondng to E MKS B J ev. Three-bod nematcs s convenent descrbed usng Jacob vectors. There are three domans of confguraton space that gve rse to three sets of Jacob coordnates. A set of Jacob coordnates s chosen that best descrbes the asmptotc state of the probem to be soved. For postron-hdrogen scatterng the three sets of Jacob coordnates whch represent the three possbe asmptotc scatterng states are shown n Fgure. 8

24 e p e - 3 θ p e - e Channe Channe Channe 3 Fgure. Jacob Coordnates θ e - p 3 θ 3 e The convenent set of ndces used n the phscs communt s used to descrbe the mass-scaed Jacob equatons. The reatonshp between the mass-scaed Jacob coordnates are summarzed b defnng the ndces, β, γ as ccc,, 3 where,, 3 represents the Jacob channe and,, 3 aso represents the partces e, e -, p respectve. Aso defned are the pars β, γ, β γ, and γ, β. Let m and r be the mass and poston vector for the partce ndeed b. Then the mass-scaed Jacob vectors are defned b t r r Eq. β γ mβrβ mγrγ µ r Eq. mβ mγ where 9

25 t m m β β m γ m γ Eq. 3 and u mmβ mγ Eq. 4 M and M s the tota atomc mass m m m 3. ote that t / s the reduced mass of the par, and u / s the reduced mass of the partce and the par. The Jacob vectors of dfferent channes are reated b orthogona transformatons s Eq. 5 β cβ β c Eq. 6 β sβ β where the mass dependent coeffcents are c β mmβ Eq. 7 m m m m γ β γ s β β sgn β c β Eq. 8 For scatterng probems, a more convenent set of oca coordnates are the engths of the mass-scaed Jacob vectors and the ange between them, Eq. 9 Eq. 0 0

26 z cos θ Eq. and the coordnate transformaton for the engths are s c z Eq. β cβ sβ β β s c s c z Eq. 3 β β β β β z β c β s β z β s β β c β Eq. 4 In mass-scaed coordnates the Couomb potenta, V, for the partce par s gven b V t e β e γ / Eq. 5 Defnng q t e β e γ as the mass-scaed charge for the par, the Couomb potenta for the par β,γ becomes V q / Eq. 6 In terms of the rotatona anguar momentum of the bound par gven b the operator, ˆ, and of the orbta anguar momentum of the free partce about the bound par, ˆ, the quantum-mechanca form of the mass-scaed Schrodnger equaton, wth par wse Couomb potentas, s ˆ ˆ q q β β q γ γ Ψ EΨ Eq. 7

27 Ths thess computes the cross secton resuts for postron-hdrogen eastc-scatterng. For convenence of cacuatons, the Jacob coordnates ndeed b s mpemented nto the Schrodnger equaton snce t s best reated to the asmptotc phscs. e n p out e e e - θ p e - Channe Fgure. Seected Jacob Coordnates. Hereafter and the subscrpt s dropped from the equatons. β and γ 3 reman n the equatons as needed. The prevous equaton s rewrtten as ˆ ˆ q q q 3 Ψ EΨ Eq. 8 3 b Ψ, where For convenence et the wave functon be represented Ψ ~,,Ω,Ω Ψ,,Ω,Ω Eq. 9

28 Ω ange representaton of the par of anges θ,φ that ndcate the poar and azmutha anges of the coordnate as Ω ange representaton of the par of anges θ,φ that ndcate the poar and azmutha anges of the coordnate as Substtutng Eq. 9 nto Eq. 8, the wave functon dfferenta equaton becomes ~ ~ ~ ~ Ψ Ψ ˆ ˆ Ψ Ψ ~ q q q 3 ~ EΨ 0 Ψ Eq Dscrete Varabe Representaton Append B contans more detas of the mathematcs behnd ths overvew of the DVR bass representaton. Sovng the tme ndependent two-bod Schrodnger equaton usng an anguar two-varabe DVR approach has been successfu mpemented. The DVR s a untar transformaton of a Fnte Bass Representaton FBR defned for some quadrature scheme assocated wth the FBR. The two-varabe anguar bass functons are a near combnaton of spherca harmoncs. The rada functons are not epanded but are appromated usng spnes or some other appromatng technque. 3

29 Snce the three-bod anguar four-varabe DVR s anaogous to the two-bod anguar two-varabe DVR, a descrpton of the two-bod DVR s descrbed frst and then etended to the three-bod DVR. The dervaton of a set of two-varabe anguar DVR bass functons begns b defnng Gauss ponts on a two-dmensona unt sphere Ω θ θ,φ φ where θ s the spherca coordnate poar ange and φ s the spherca coordnate azmutha ange. The nde represents the anguar grd pont ndcated b the par of ndces θ, φ. z Ω 4 θ, φ, Ω θ,φ Ω 7 θ θ Ω 5 Ω 8 Ω 6 φ φ Ω Ω 9 3 θ, φ 3, Ω 3 θ 3,φ Fgure 3. 9 Pont DVR Anguar Grd. 4

30 Aso defned wth ths grd are a set of assocated Gauss-quadrature weghts w, and a set of DVR bass functons φ Ω. B constructon, these functons satsf the propert φ Ω δ / w Eq. where δ s the Kronecer deta functon. On ths grd the two-dmensona Gauss quadrature appromaton for a functon gω s 39,40 g Ω dω g Ω Eq. w The DVR functons are orthogona * φ Ω φ Ω dω φ Ω φ Ω w δ Eq. 3 * The advantage of the DVR bass functons s that the potenta matr of a mutpcatve potenta s dagona n anguar space * φ Ω V Ω, r φ Ω dω φ Ω V Ω, r φ Ω w V Ω, r δ * Eq. 4 The anguar DVR bass functons are constructed from the FBR bass functons as descrbed n append B. For the case of spherca harmonc FBR bass functons, whch are orthonorma * Ω J ' Ω dω δ ' J Eq. 5 5

31 the DVR bass functons are gven b * φ Ω w S S Ω Eq. 6 wth ' v ' J' Ω S Ω C Eq. 7 beng the dgta Gram-Schmdt orthogonazaton of the FBR spherca harmonc bass functons, J v Ω. The Gram- Schmdt orthogonazaton s requred so that the bass functons are orthogona n the Gauss ntegraton * S Ω S Ω dω S Ω S Ω w δ Eq. 8 * Let the nde represents the par of anguar momentum quantum numbers,m wth 0<< ma and -<m< and the nde represents the anguar grd pont ndcated b the par θ, φ. w are the approprate seected Gaussan weghts. There s not a set of unque quantum numbers, grd ponts and weghts for the DVR bass 4. The seecton of the convenent set of quantum numbers, anguar grd ponts and weghts used for ths thess s descrbed n append B. For ths thess the tota number of anguar momentum quantum numbers,, s equa to the tota number of anguar 6

32 quadrature ponts,. The tota number s aso an odd number. Ths s not a strct requrement on the use of DVR functons 43. However, for the seecton of anguar bass functons, the seecton of anguar quantum grd ponts, and the seecton of grd anges, the Gram-Schmdt epanson fas when the tota number of DVR bass functons s an even number snce the bass functons are not near ndependent on the grd. The FBR transformaton to DVR s a untar transformaton thus competeness s aso obtaned S Ω w S Ω w δ Eq. 9 * Ths guarantees the DVR bass functons obe the propert gven b Eq.. In ths anguar two-varabe DVR approach the twobod wave functon s epanded, usng spherca coordnates, approprate seected weghts and Gram- Schmdt coeffcents, as ~ Ψ r, Ω * ψ r φ Ω w S S Ω ψ r Eq. 30 v When evauated at the th anguar grd pont, the wave functon reduces to the correspondng rada epanson 7

33 coeffcent, ψ r, dvded b the square root of the correspondng weght. ~ Ψ r, Ω ψ r φ Ω ψ r w δ ψ r w Eq. 3 When sovng the tme ndependent three-bod Schrodnger equaton Eq. 0, the four-varabe anguar fnte bass s chosen to be the bpoar spherca harmoncs. Y L,M, where ˆ, ŷ Y m m M [Y m m ˆ Y θ φ Y m ŷ] m L,M θ φ C L, M, m,, m Eq. 3 L Tota anguar momentum quantum number, L, -,..., - M Tota anguar momentum proecton Rotatona anguar momentum of the bound par The orbta anguar momentum of the free partce about the bound par m Bound par anguar momentum proecton aong the reatve coordnate -as m Orbta anguar momentum proecton aong the reatve coordnate -as C Cebsch-Gordon vector coupng coeffcent, L,M,m,,m <,,m,m,,l,m> 8

34 ˆ ange representaton of the par of anges Ω θ,φ ŷ ange representaton of the par of anges Ω θ,φ To gve the bpoar bass a convenent one-dmensona oo the foowng notaton s mpemented nde representng the quartet of quantum numbers L, M,, Ω ange representaton of the quartet of anges ˆ, ŷ θ, φ, θ, φ z a θ θ a φ a φ Fgure 4. Jacob Coordnates At th Grd Pont. Usng ths notaton, the bpoar bass functon s rewrtten as m L, M Y Ω Y ˆY ˆ C, m,, m Eq. 33 m m M m 9

35 Let nde representng the quartet of ndces θ, φ, θ, φ Ω anguar Gaussan quadrature pont ndeed b, θ, θ φ φ, θ θ, φ φ umber of Ω Gaussan quadrature ponts reatve to the -as, θ * φ umber of Ω Gaussan quadrature ponts reatve to the -as, θ * φ Tota number of four-dmensona Gaussan quadrature ponts, * The bpoar bass functon evauated at the th Gaussan quadrature grd pont s denoted as Y Ω and s gven b m L M Ω C, Y Y, Y, θ φ θ φ m m Eq. 34 m m M θ φ m θ φ The bpoar spherca harmoncs are orthogona * ' ' Y Ω Y Ω dω δ Eq. 35 Anaogous to the two-bod case, the three-bod anguar DVR bass functons are constructed from the FBR,,, 0

36 bass functons, as descrbed n append B. The fourvarabe anguar DVR bass functons are gven b * φ Ω W S S Ω Eq. 36 wth ' v S Ω C Y Ω Eq. 37 ' ' beng the dgta Gram-Schmdt orthogonazaton of the bpoar spherca harmoncs bass functons, Y v Ω, to ensure the Gauss appromaton s orthogona * * Ω S ' Ω dω S Ω S ' Ω W δ ' S Eq. 38 Agan the tota number of anguar grd ponts s an odd number so that the Gram-Schmdt orthogonazaton process s successfu. Anaogous to the two-dmensona anguar case we have the DVR propert φ Ω δ / W Eq. 39 and the DVR functons are orthogona * φ Ω φ Ω dω φ Ω φ Ω W δ Eq. 40 and compete * φ Ω W φ Ω W δ * ' ' ' Eq. 4

37 The weghts are approprate chosen Gaussan quadrature weghts: w Eq. 4 θ sn ' cos θ P θ θ θ θ w π / Eq. 43 φ φ W w * w w θ * w φ * w θ * w φ Eq. 44 ote that the Gram-Schmdt modfed bpoar spherca harmonc functons have the propertes: * ' > S ' W δ ' < S Eq. 45 * ' > S S' WW ' δ' < Eq. 46 where S v represents S Ω v. As an eampe of the mportance of usng the Gram- Schmdt orthogonazaton, Tabe sts the resuts of the orthogonat and competeness propertes of the off dagona matr eements of the Gauss quadrature representatons. The top row n the tabe sts the DVR grd sze for the tabe coumn beneath the headng. The numerca vaues of Y Ω and S Ω are evauated at the DVR anguar grd ponts usng doube precson computer representaton.

38 The second and thrd rows show that the orthogonat and competeness propertes fa when usng the Y Ω representaton. The fourth and ffth rows show that the orthogonat and competeness propertes are obtaned when usng the Gram-Schmdt S Ω representaton. Tabe. Orthogonat And Competeness. DVRsze-> * Y Y W ' Y Y * ' * S S ' S S * ' W W W ' W W ' As the DVR grd sze ncreases, the accurac of the orthogonat and competeness propertes of the Gram- Schmdt representaton decreases when usng doube precson numerca representaton. As the sze of the DVR grd sze ncreases, hgher precson representaton s requred to mantan hgh accurac resuts. ote that the competeness and orthogonat of the S v Ω functons ensures that the matr eements of the mutpcatve potenta are dagona. For the fnte bass, the tota anguar momentum quantum number, L, s between 0 and L ma where L ma s the product of ma of each Jacob as: L ma ma * ma. 3

39 Tabe sts the vaues of L ma for each three-bod anguar DVR bass functon sze. Tabe. Mamum Tota Anguar Momentum Inde. DVRsze-> L ma A s varabe wave functon Ψ,θ,φ,,θ,φ s appromated usng the anguar four-varabe DVR functons as foows Ψ ~,, Ω * ψ, φ Ω W S S Ω ψ, Eq. 47 where ψ, are the two-varabe rada epanson coeffcents. v When evauated at the th anguar grd pont the wave functon reduces to the correspondng rada epanson coeffcent, ψ,, dvded b the square root of the correspondng weght. ~ Ψ,, Ω ψ, φ Ω ψ, W δ ψ, W Eq Rada Spnes And Spne Bass Append C descrbes the detaed mathematcs for the Hermte Quntc Spnes that are used to numerca evauate the rada coeffcent of the wave functon that 4

40 resuts after the anguar DVR bass functons have been proected out. Snce the three-bod spne bass s a drect product of the two-bod spne bass, a descrpton of the twobod spne bass s descrbed frst and then etended to the three-bod spne bass. In the two-dmensona anguar DVR case we are eft to evauate the rada coeffcents, ψ r. Interpoaton s the process of estmatng the ntermedate vaues of a contnuous functon from dscrete sampes. The rada coordnate s dscretzed over a rada grd. These dscrete ponts are caed nots. The set of ponts that mae up the grd s represented b { } { 0 r mn,,,..., r ma } Eq. 49 where s the number of segments between the nots: number of nots. In between each set of nots s a set of ponts caed coocaton ponts. A fundamenta propert of nterpoaton functons s that the must concde wth the samped data at the coocaton ponts. Each set of coocaton ponts conssts of f ponts. For eampe f f s the samped functon and g s the correspondng 5

41 nterpoaton functon then fr gr where r s a coocaton pont. Hermte Quntc spnes are a set of three nterpoaton functons defned on two adacent ntervas descrbe b three contnuous nots [, ]U[, ]. These functons are zero outsde ther ntervas. The quntc spne bass conssts of * f pecewse ponomas of ffth degree. Usng Hermte Quntc spnes the nterpoatng functon s wrtten as f rs 0 g r A rs φ r Eq. 50 rs where φ rs r s the rs th ndeed Quntc spne functon A rs s the correspondng scang coeffcent, s the number of nots and f s the number of coocaton ponts between nots. Pots of the quntc spne functons are gven n Append C. Over an one nterva, [, ], there are s nonautomatc zero bass functons and s scang coeffcents to be evauated. Gauss-Legendre coocaton ponts are found between each adacent par of nots. For Quntc spnes f 3 coocaton ponts are requred for the Gauss quadrature ntegraton appromaton to be a reasonabe appromaton 6

42 to the functon. Ths assumes that the functon beng appromated s smooth enough over the nterva that a ffth order ponoma can appromate t. For the three-dmensona two-bod wave functon, after the DVR bass are proected out, the rada wave functon s appromated usng the quntc spne bass functons f A, rsφrs r rs 0 ψ r Eq. 5 In ths DVR quntc spne approach the two-bod wave functon s epanded as ~ Ψ r, Ω w S v f * S Ω A, rsφ rs r Eq. 5 rs 0 When evauated at the th anguar grd pont, the foowng equat s obtaned f A, rs rs r rs 0 Ψ ~ r, Ω φ Eq. 53 w For the three-bod case, the two-dmensona nots are the drect product of the one-dmensona nots. The two-dmensona coocaton ponts are aso the drect product of the one-dmensona coocaton ponts. The two-dmensona quntc spne bass functons are a drect product of two one-dmensona quntc spne bass 7

43 functons. The two-dmensona rada wave functon s wrtten as f rs 0 f ψ, A φ φ Eq. 54 rs 0, rs, rs rs rs where are the number of nots on the Jacob -as, are the number of nots on the Jacob -as and t s assumed that both as use the same spne bass so that f s the same for both as. Smar, rs s the spne nde for the -as and rs s the spne nde for the - as. In ths DVR quntc spne approach the three-bod wave functon s epanded as ~ Ψ,, Ω f * W S v S Ω f rs 0 rs 0 A, rs, rs φ φ Eq. 55 rs rs When evauated at the th anguar grd pont ~ Ψ r, Ω W f f A, rs, rs rs φ rs rs 0 rs 0 φ Eq. 56 As descrbed n Append C, f the quntc spne approach cannot meet the Gauss quadrature ntegraton appromaton requrements over an nterva between two nots, then another set of bass functons must be chosen to appromate the functon over that nterva. 8

44 It was found emprca that the postron-proton scatterng bass functons provde a better wave functon appromaton near the nteracton regon on the -as than Hermte spnes aone. These bass functons are gven n Append C and denoted b Frs. When these bass functons are mpemented aong wth the quntc spnes then the two-dmensona rada wave functon appromaton s gven b ψ, f,ma rs, rs rs 0 rs 0 f rs 0 f A rs 0 A F rs, rs rs rs φ rs φ φ rs otherwse t Eq. 57 where t s the frst coocaton pont after the second not. ote that the ponoma order of rs s reated to the number of DVR bass functons va the anguar momentum number,ma. Dependng on ths number, the number of coocaton ponts between the frst and second nots ma be dfferent than that for quntc spnes n order for the Gauss quadrature ntegraton to be vad for these bass functons on ths nterva. For notaton, n the net sectons whenever the rada epanson s denoted, t s gven as F 9

45 f rs 0 f ψ, A φ φ Eq. 58 rs 0, rs, rs rs rs wth the understandng that the over the frst nterva, the frst bass functons are rs. On when the rs bass functons are specfca descrbed w the be denoted proper.e. when the boundar condtons are descrbed. Append C aso descrbes how to dscretze a near dfferenta equaton nvovng the wave functon b evauatng at the seected coocaton ponts. The resut s a near set of agebrac equatons that are soved edng the spne scang coeffcents. F F.4 umerca Schrodnger Equaton Append D descrbes the mathematcs for the dscretzaton of the Schrodnger equaton usng the massscaed Jacob vectors gven n channe, mpementng the DVR bass functons and mpementng the rada spnes. Append D descrbes both the two-bod case and the threebod case so that the reader can see the anaog between the cases. On the three-bod resuts are dscussed n ths secton. 30

46 Substtutng the DVR epanded wave functon ~ Ψ,, Ω W S v * S Ω ψ, Eq. 59 nto the reatve moton Schrodnger equaton ~ ~ ~ ~ Ψ Ψ ˆ ˆ Ψ Ψ ~ q ~ q ~ q3 ~ EΨ Ψ Ψ Ψ 0 Eq. 60 and evauatng a DVR anguar grd pont, denoted wth nde, ange Ω, we get the foowng dfferenta equaton for the rada wave functon 3 q q 0 SS, SS, ψ, Ω q 3 3 Ω E δ, Eq. 6 where *, W W C '' ' Y 'S ' S S Eq. 6 *, W W C '' ' Y 'S ' S S Eq. 63 β s the transformaton of the β-jacob coordnate to Ω the -Jacob coordnates usng the coordnate transformaton for the engths as descrbed prevous wth z beng the cosne of the ange between the β-jacob 3

47 coordnates defned b anges Ω. In ths descrpton and β and 3. otce that the potenta has been dagonazed. On the S S, and S S, terms are off dagona and most of these terms are zero due to the propertes of the DVR the Gram-Schmdt orthogonazed bpoar spherca harmoncs. Snce there are DVR ndces, to, a set of couped dfferenta equatons s obtaned for the set of rada coeffcents, ψ,. Substtutng the rada spne epanson nto ths set of dfferenta equatons eds the foowng set of couped near dfferenta equatons for the unnown coeffcents, A,rs, rs 0 f f SS φrs rs 0, rs 0 A SS,,rs,rs φrs φ rs q q Ω q φ rs φrs Eq. 64 φ rs δ, There are * * f * * f unnown coeffcents f on quntc spnes are mpemented. Otherwse the number of coeffcents s gven b *[ * f * Ω E δ,

48 * f ' f ], where ' f s the number of postron-proton scatterng bass functons mpemented between the frst and second -as nots. Usng the spne souton technque descrbed n Append C to obtan an agebrac equaton for the coeffcents, each equaton s evauated at a the coocaton ponts. There are DVR anges, -* f coocaton ponts aong the -as and -* f coocaton ponts aong the -as, gvng a tota of * -* f * -* f equatons. Boundar condtons supp the remanng equatons to provde a unque souton for the coeffcents. As descrbed n Append B, snce the mass of the proton s much greater than that of the postron or eectron, the and Jacob aes have ther orgns near agned. If the aes coocaton ponts are chosen so that the some coocaton ponts are the same for each as, then there are cases that correspond wth the eectron concdng wth the postron causng numerca nstabt due to the potenta energ between the twopartces. 33

49 The DVR -as azmutha grd ponts are shfted so that the -as s not parae to the -as. Ths mtgates the nterference between the two-partces..5 Asmptotc Boundar Condtons Append D aso defnes the boundar condtons that descrbe the phscs of the eastc-scatterng process. When the scattered partce s far awa from the scatterng center so that the center's nfuence on the partce s neggbe, the scattered partce s a free partce. Ths regon s caed the asmptotc regon. In the asmptotc, regon the scattered partce wave functon s the sum of an ncomng pane wave and an outgong spherca scattered wave wth an anguar dependent amptude factor 43. The hdrogen wave functon s unchanged. Thus the three partce asmptotc wave functon s Ψ, e A φ e f θ Eq. 65 where A s the normazaton constant, s the ncomng partce momentum vector, s the outgong vector drecton, fθs the scatterng amptude factor and s the shft n the outgong wave functon amptude and θ s 34

50 the ange between the ncomng drecton vector and the Jacob -as vector. φ s the bound partce hdrogen wave functon whch for convenence s denoted as h φ φ /. For convenence et Ψ, Ψ ~, /, and the asmptotc wave functon s wrtten as Ψ ~, h h φ e φ e f θ A Eq. 66 where h φ φ h Y m θ,φ Eq. 67 and φ h s the product of and the hdrogen rada wave functon, R n. The scatterng amptude s found b matchng the outgong wave above and the epanded nteror wave functon at the asmptotc dstance at a DVR ange. The foowng technque s mpemented to obtan the boundar condtons as we as the scatterng amptude. B evauatng Ψ ~, at the th DVR grd ange and at two adacent asmptotc coocaton ponts, and ', fθ s emnated edng the equat ψ, e ' cosθ 'cosθ ' h ψ, ' A φ W e 'e Eq

51 where θ represents the ange between the ncomng drecton vector and the Jacob -as vector correspondng to the grd ange. The ncomng wave vector ˆ has orentatons gven b the anges on the - as anguar grd θ,φ. The goa of ths thess s to fnd an appromatng wave functon that aows the dervaton of the amptude functon to determne the scatterng cross secton. Snce an appromated wave functon s beng mpemented usng a fnte number of parta anguar momenta an appromated boundar condton for that wave functon must aso be mpemented to obtan a reasonabe match at the asmptotc boundar. The epanded form of functons s gven b e e cos θ usng anguar bass e cos θ e r P cos θ / Eq. 69 where θ s the ange between the ncomng vector and the outgong vector. Ψ ~, The epanded asmptotc wave functon s gven b h h Aφ e r P cos θ Aφ e f θ Eq. 70 / The amptude functon, fθ, s aso epanded 36

52 f θ s P cos θ Eq. 7 where s represents the th dagona eement of the scatterng or S operator. The actua vaue of s s not computed nor requred for asmptotc matchng. In each of these epansons the sum over goes from zero to nfnt. If ths sum s truncated then these epanson equatons become appromatons. In the appromated asmptotc wave functon the sum over goes from 0 to the mamum gven n the DVR quartet of quantum numbers L, M,, used for the wave functon appromaton. That s, the sum goes from 0 to,ma. Usng ths notaton the asmptotc wave functon s represented b Ψ ~ asm h, Aφ h Aφ e e π / s P cos θ P cos θ Eq. 7 whch s rewrtten as Ψ ~ h h asm, Aφ S Aφ e G Eq. 73 where S P cos θ Eq

53 f θ G P cos θ s Eq. 75 Evauaton at the DVR ange for the outgong -as fnds h ψ, A φ S e G Eq. 76 where the hdrogen wave functon s evauated on at the two approprate -as anges out of the four anges the nde represents. S P cos θ Eq. 77 f θ G P cos θ s Eq. 78 Usng the appromated asmptotc wave functon, evauatng at two dfferent scatterng dstances and ' n the asmptotc regon, and emnatng G eds the foowng asmptotc boundar condton -' h -' ψ, e ψ, ' A W φ S e S ' Eq. 79 Usng the dervatve at two dfferent scatterng dstances eds boundar condtons for the dervatves -' h -' ψ ', e ψ ', ' A W φ S' e S' ' Eq. 80 Contnung, the second dervatve eds another boundar condton 38

54 -' h -' ψ ", e ψ ", ' A W φ S" e S" ' Eq. 8 wth S ', P cos θ ' Eq. 8 and S ", P cos θ ' " Eq. 83 S, and ts dervatves are a functon on of the - as components, θ,φ, of the 4 anges the nde represents. ote that the amptude functon, evauated at the th DVR grd ange, s wrtten as - e ψ, - f θ e S 0 Eq. 84 h AY ˆ φ W 0 where S s a functon on of the -as anguar components that represents, and the hdrogen anguar component s on a functon of the -as anguar components that represents. The amptude functon represents the fu scatterng amptude matr f, where and each run over a the possbe grd ponts ndeed from to. That s, the matr represents a ncomng drectons as we as a outgong drectons. 39

55 .6 ear Interacton Regon Boundar Condtons In addton to the asmptotc regon boundar condtons, boundar condtons are requred n the nteracton regon. At 0 or 0 the boundar condton ψ 0, 0 ψ,0 0 Eq. 85 s mpemented. Ths mantans that for the mt that goes to zero and/or goes to zero the fu wave functon, ~ Ψ, Ψ,, Ω, Ω /, remans fnte. Snce ths thess focuses on eastc-scatterng t s epected that the wave functon for the scattered partce near the nteracton regon at ocatons that do not nterfere wth the hdrogen partce's eectron shoud have a smar functona form of the prevous derved wave functon for a partce scatterng off a proton. That s, the quntc spne bass functons are repaced b the postron-proton scatterng bass functons. Ths quntc spne repacement s done snce the proton-proton scatterng bass functons are of hgher order than quntc spnes as,ma ncreases. 40

56 The ow order postron-proton scatterng bass functons are gven b F π e C [...] Eq. 86 wth C C C 0! / 3! π / π / e / π / π / e /... / π / π / e Eq. 87 However, these constants are not used snce on the form of the bass functons s requred n the spne epanson F e Eq. 88 Usng these bass functons for sma aows a better ft of the spnes to the scatterng wave functon. That s, the Gauss quadrature ntegraton appromaton s vad n ths regon. Of course, f more coocaton ponts are requred, due to the order of the bass functons, the must aso be added between the frst two nots. The cumuatve probabt of the ground state hdrogen wave functon near zero s gven n Fgure 5. 4

57 Cumuatve Probabt atomc unts Fgure 5. Cumuatve Probabt Of The Ground State Hdrogen Eectron At Rada Poston. As ong as the probabt of the hdrogen eectron nteracton wth the ncomng partce s sma, then repacng the quntc bass functons b the postronproton scatterng bass functons shoud ed a good representaton of the tota wave functon. Snce the wave functon s appromated b ψ,,ma rs, rs rs 0 rs 0 f rs 0 f f rs 0 A A rs, rs F rs rs φ rs φ φ rs otherwse t Eq. 89 where t s the frst coocaton pont after the second not and s seected n the regon where the cumuatve probabt of the hdrogen eectron at ts rada 4

58 poston s sma, the boundar condtons for the wave functon and ts frst and second dervatve must be f f,ma f f,ma A,rs,rs rs t rs, rs, t rs rs rs rs 0 rs,ma rs 0 rs 0 φ φ A F φ 0 Eq. 90 f f,ma f f,ma A,rs,rs rs t rs, rs, t rs rs rs rs 0 rs,ma rs 0 rs 0 φ ' φ A F' φ 0 Eq. 9 f f,ma f f,ma A,rs,rs rs t rs, rs, t rs rs rs rs 0 rs,ma rs 0 rs 0 φ " φ A F" φ 0 Eq. 9 In other words, the wave functon usng the postronproton scatterng bass functons must ed the same vaue at t as the wave functon usng the quntc spne bass functons. ote that F' F [ / ] Eq. 93 F" F [ ]/ Eq Scatterng Cross Secton The scatterng cross secton s obtaned from the amptude functon b assumng that the asmptotc partce s a free partce and has a free partce wave functon modfed b the amptude functon. 43

59 For a gven nput drecton, the amptude functon for the outgong th DVR grd ange s reated to the wave functon b h φ f θ, - e ψ, h - φ e S 0 Eq. 95 AY ˆ W 0 Snce on the -as s requred to compute the scatterng cross secton, the rada hdrogen wave functon s proected out of ths equaton. f p h h θ < φ φ f θ > Eq. 96 e ψ, - h h h - < φ > < φ φ e S 0 AY ˆ 0 W > where < f * f f g > g w Eq. 97 s the Gauss quadrature ntegraton over the coordnates. The resutng amptude functon, f p θ, no onger has an dependenc on. Fgure 6 ustrates the genera coordnate dagram for the ncomng partce vector and the outgong th DVR grd ange as confguraton. 44

60 z a θ, θ θ θ a φ φ a φ Fgure 6. Coordnate Dagram For The Incomng Vector And The th DVR Grd Ange. The ange between the ncomng vector and the outgong Jacob -as s found b cosθ cosθ cosθ snθ snθ cosφ -φ Eq. 98 The cross secton s gven b dσ p σ dω Ω f θ sn θ dθdφ d Eq. 99 Snce the potenta s spherca smmetrc the seecton of the ncomng drecton s arbtrar. Aso, the ntegraton over the arbtrar azmutha ange, φ, s trva snce the amptude functon s ndependent of ths ange. To mae the cross secton cacuatons more convenent, the coordnate sstem s rotated so that the vector s aong the z-as, θ 0. The ange between the 45

61 46 and aes n the orgna coordnates, θ, s eact the same as the poar ange of the th -as n the rotated coordnate sstem θ so that and cosθ cosθ. The -as azmutha ange, φ remans arbtrar wth respect to the amptude functon. The amptude functon remans unchanged: f p θ f p θ. θ z θ φ n φ θ z n φ θ θ φ Fgure 7. Rotated Coordnate Sstem. Thus the tota cross secton s now gven b p p p p p W f d f d d f d f d d f Ω Ω Ω Ω sn θ π θ π θ π θ φ θ θ θ σ Eq. 00 usng the DVR quadrature ntegraton.

62 CHAPTER 3 RESULTS The scatterng cross secton s computed for a postron scatterng wth a ground state hdrogen atom whch conssts of a proton and a bound eectron. The resuts of numerous ndependent scatterng cacuatons obtaned b researchers are quantfed n reference 44. In partcuar reference 44 gves the eastc scatterng cross sectons for the postron- Hdrogen probem that has been obtaned b the research communt. Verfcaton of the resuts of the DVR technque descrbed n ths paper s obtaned b comparng the resutng converged cross sectons to the eastc scatterng resuts gven n reference 44. The DVR grd that generates the smaest set of couped equatons s the 99 grd edng 8 DVR bass functons. The ncomng partce's vector anguar orentaton s chosen as one of the DVR -as grd anges. The potenta functon smmetr mpes that resuts for equa cross sectons are ndependent of the ncomng ange chosen. 47

63 3. Seecton Of Knots And Anguar Grd For the numerca ground state rada hdrogen wave functon, t has been found emprca that good convergence usng quntc spnes are obtaned usng on eght strategca paced -as nots wth vaues between 0 and 0. Specfca pacng the nots at 0, 0.5,.0,.5, 5.0, 8.5,, and the cut off pont at 0 aows the quntc spne mpementaton to cose appromate the rada hdrogen wave functon and ts dervatves. Fgure 8 shows a pot of the wave functon and the spne ft vaues. R, R', R" are the ground state wave functon and ts frst and second dervatves respectve. Rs, R's and R"s are the spne fts. 48

64 Ground State Hdrogen Wave Functon And Its Frst And Second Dervatves Rs R R's R' R"s R" atomc unts Fgure 8. Ground State Hdrogen Wave Functon And Its Frst And Second Dervatves. The mamum error between the spne fts and the wave functon s on the order of 0-4. The -as spata not ocatons had to be chosen wth suffcent denst to proper ft the asmptotc wave functon as we as the wave functon n the nteracton regon. The asmptotc wave functon for eastc-scatterng has a De Broge waveength λ π/. The number of nots and -as cut off pont had to be chosen so that the asmptotc regon woud be obtaned. The number of nots aso had to be traded aganst the resutng matr sze and computer resources. As a 49

65 resut, a suffcent cut off pont was found to be at 80 wth 35 equa spaced segments between 0 and 80. The smaest DVR set that can be mpemented to obtan a reasonabe cross secton resut s a 8 set of DVR bass functons, consstng of 9 and 9. Usng 8 -nots and 36 -nots for the spata coordnates, ths eds a matr sze of ~87,500 87,500 compe eements or ~375, ,000 rea eements. The net smaest DVR set of bass functons conssts of 5 eements, consstng of 9 and 5. If ths set s mpemented the resutant matr sze s ~500, ,000 compe eements or,000,000,000,000 rea eements. The net sze DVR bass functon set s 65 whch resuts n a matr sze of 7,700,000 square. On the Unverst Of Teas Sun Consteaton Lnu Custer Ranger parae computer sstem, the 8 DVR generated matr equaton requres ~5,600 core hours to obtan a souton usng the SCALAPAK software. Sovng the 5 DVR generated matr equaton requres ~45,635 core hours. Due to mts on the computer resources, on the 8 DVR set of equatons can 50

66 current be soved to convergence. The resuts are shown n ths secton. Other computer resources and software are current beng nvestgated ncudng parae and sparse matr computng technques such as TAUCS, BSCLIB-EXT, SuperLU, PARDISO, UMFPACK, Sparse Matr SCALAPAK, and MUMPS 46. Once these technques have advanced to the pont where the can sove probems of the sze generated b ths DVR technque, addtona anass can be performed edng hgher fdet cross secton resuts. 3. Resuts Usng Y-As Azmutha Shft Usng quntc spnes on, convergence of the souton s found usng the -as and -as nots descrbed above. Fgure 9 shows the DVR resuts, usng a -as shft of 6.8 0, potted aongsde that of reference 44 for 0., 0.3, 0.4, 0.5, 0.6 and

67 4.5 σ eastc Reference 99 DVR Quntc spnes on a.u. Fgure 9. Eastc Cross Secton Vs Incomng Momentum Usng Quntc Spnes On. In genera, the resuts at 0. are not epected to be we converged snce the De Broge waveength at ths energ s far ong. As the waveength ncreases, the asmptotc regon aso ncreases. In ths case the asmptotc regon s beond the -as cut off pont. Impementng a onger cut off pont requres addtona - as spata coordnates. Addtona spata coordnates ncreases the sze of the matr whch ncreases the requred computer resources. On the other hand, at 0.7 the sstem s cose to the bound e e - postronum formaton whch has a profound poarzaton effect on the scatterng process. The current DVR formuzaton does not have the structure 5

68 for the e e - vrtua postronum formaton, especa n the -drecton whch current uses the hdrogen wave functon spne appromaton and grd ponts. Sovng for ths case aso requres more spata coordnates on both the and aes, ncreasng the sze of the matr and the requred computer resources. Consderng the sma DVR bass sze used n ths cacuaton, the agreement s ver good at the 0.3 to 0.6 energes. Fgure 9 demonstrates the proof of prncpe of the DVR technque. As the sze of the set of DVR bass ncreases, the -as anguar momentum aso ncreases and the use of the postron-proton bass becomes mportant. Usng quntc spnes and postron-proton bass functons, convergence of the souton s recomputed usng the same spata and anguar grd for the prevous case, ecept that the frst non zero -as not was paced at 0.5, 0.5, and 0.8. The frst coocaton pont after the frst not s denoted as st. The postron-proton scatterng bass functons are mpemented n the regon between 0 and st. The quntc spnes are mpemented n the st to the asmptotc regon. The near zero matchng boundar condton occurs at st. Fgure 0 shows the 99 53