Numerov-Cooley Method : 1-D Schr. Eq. Last time: Rydberg, Klein, Rees Method and Long-Range Model G(v), B(v) rotation-vibration constants.

Transcription

1 Numrov-Cooly Mthod : 1-D Schr. Eq. Last tim: Rydbrg, Kli, Rs Mthod ad Log-Rag Modl G(v), B(v) rotatio-vibratio costats 9-1 V J (x) pottial rgy curv x = R R Ev,J, v,j, all cocivabl xprimts wp( x, t) = ai i i ie t / h i iitial prparatio of wp: a i = * i [ wp(x,0) ]dx Mthod: A(E,J) = ara of V(x) blow E: dtrmid by V J (x) fr volutio of wp usd WKB QC obtaid x ± (E,J) E V(x) Today: What do w do wh w hav V J (x) (spcially wh V(x) is ot suitd for WKB)? Solv Schr. Eq. umrically! No modls 15 digit rproducibility chap This is th fial tool w will dvlop for us i th Schrödigr rprstatio. To summariz th classs of 1 D problm w hav solvd: * picwis costat pottials (matrix approach for joiig at boudaris) * Airy fuctios (liar pottial ad joiig JWKB across turig poit) * JWKB (quatizatio coditio ad smi-classical wavfuctios) * umrical itgratio (today)

2 9-2 Numrical Itgratio of th 1-D Schrödigr Equatio widly usd icrdibly accurat o rstrictios o V(x) or o E V(x) [.g. oclassical rgio, ar turig poits, doubl miimum pottial, kiks i V(x).] For most 1-D problms, whr all o cars about is a st of {E i, i }, whr i is dfid o a grid of poits x i, o uss Numrov-Cooly S 1. Cooly, Math. Comput. 15, 363 (1961). 2. Prss t. al., Numrical Rcips, Chaptrs 16 ad 17 Hadouts 1. Classic upublishd papr by Zar ad Cashio with listig of Fortra program (ow s LRoy wb sit) 2. Tsts of N-C vs. othr mthods by Tllighuis Basic Ida: grid mthod * solv diffrtial quatio by startig at som x i ad propagatig trial solutio from o grid poit to th xt * apply (x) = 0 BCs at x = 0 ad by two diffrt tricks ad th forc agrmt at som itrmdiat poit by adjustig E.

3 Eulr s Mthod wat (x) at a sris of grid poits x 0, x 1, x =x 0 +h call ths i = (x i ) 9-3 h h h x 0 x 1 x 2 Nd a gratig fuctio f(x, ) ( ) +1 = + hf x, ( is a umbr, ot th tir wavfuctio.) For th Eulr mthod, th gratig fuctio is simply: fx (, )= d +1 dx x x +1 x For th Schrödigr Eq. d 2 dx prscriptio for goig + 1 must dpd o both x ad. x sampls pottial, sampls prvious valu of. icrmt i x x +1 x = h [NOT Plack s costat] = +1 h Th valu of this drivativ actually coms from th diffrtial quatio that must satisfy, ot from prior kowldg of (x) (which w do ot yt hav!) 2µ = ( E U( x)) 2 2 h d 2 dx 2 = V(x)(x) d i+1 i dx xi h All costats absorbd i V(x). V(x) Must b i uits of Å 2. [ ] V( x) C U( x) E U( x) is pottial C= 10 8π cµ h h is icrmt of distac, ( ) = µ 12 ( amu, C ) A i Å. E ad U(x) ar i -1 cm uits (E / hc) d 2 dx 2 x i = i+1 i h i i 1 h h = h 2 [ i+1 2 i + i 1 ] µ A = m 1m 2 m 1 + m 2

4 9-4 Schr. Eq. tlls us th rul for propagatig. Employig Eulr s mthod (h is ot Plack s costat): h 2 [ i+1 2 i + i 1 ]= V i i i+1 2 i + i 1 = h 2 V i i i+1 = 2 i i 1 + h 2 V i i a rcursio rlatioship. Nd both i ad i 1 to gt i+1. i ordr to gt thigs startd w d two valus of startig at ithr dg of th rgio whr is dfid ad starts out vry small. S Prss t. al. hadout for discussio of th-ordr Rug-Kutta mthod. Th grator is chos mor clvrly tha i th Eulr mthod so that stppig rrors ar miimizd by takig mor drivativs at itrmdiat poits i th x i, x i+1 itrval. Cooly spcifis 2 y = 2y y + h V i+ 1 i i 1 i i [ ] 2 y 1 h 12 V = ( ) i i i * us i to gt y i (ad vic vrsa) * us i ad y i (ad y i 1 ) to gt y i+1 * us y i+1 to gt i+1 Th rsult is that th rror i y i+1 is o th ordr of h i V i smallr rror if h is smallr (much bttr tha Eulr)

5 9-5 So what do w do? 0 D 0 R (E) R R+(E) R x R R β R R.g., V ( R) = D 1 ( ) D MORSE V( ) = 0, V( R ) = D 2 [ ] at R = 0 x = R ( R ) = 0 R = ( ) = 0 boudary coditios * 2 boudary coditios hadld diffrtly bcaus w wat to dfi a fiit # of qually spacd grid poits (ot actually cssary s Prss: variabl grid spacig which is dd to sampl ifiit rag of x with a fiit umbr of grid poits) at R = (rquird) 1 = (arbitrarily chos small umbr to b corrctd latr upo ormalizatio) us this to start th itgratio outward. If w hav mad a wrog choic for 1, this ca b corrctd mrly by dividig all i i 1 by a i-idpdt corrctio factor.

6 9-6 At larg R (th classically forbidd rgio), choos at th last grid poit, x, to b small ad us WKB oly oc to comput th xt to last grid poit. W do this bcaus w hav o raso to go to x. = (th fial grid poit) = rcall JWKB R ( V ) 1 1 R ( V ) = p 12 / 12 / p 12 / 1 h ~ x R+ ( E) V pdx 12 / Th xt to fial grid poit [This is th oly plac WKB trs ito this problm!] 12 / 1 x 1 umrator p 1 xp p h R + ( E) 12 / 1 x domiator p xp p h R + ( E) 1 dx dx pr-xpotial factors ar approximatly qual itgrals i xpotial factors ar valuatd as summatios i 1 /, th commo trms i th summatios i th xpotial factors cacl Oc -1 is gratd from by JWKB, rtur to Cooly s mthod of umrical itgratio for all succssiv grid poits. So ow w propagat o from i = 0 out toward right ad th othr o from i = i toward th lft. Th shootig mthod. i = 0 m

7 9-7 Stop th iward propagatio of wh a poit is rachd whr, for th first tim, m m+1. Sic i is xpotially icrasig from at i= util it rachs its first maximum isid th classically allowd rgio, this outr lob of is also th most importat fatur of (bcaus most of th probability rsids i it). outrmost lob of maximum of at ( x m+1 ) x m+1 V(x) asymptotic approach to 0 Us outrmost lob bcaus this is th global maximum of (x), this miimizs th problm of prcisio big limitd by fiit umbr of sigificat figurs i th computr. St valu of m = 1.0 by rormalizig both fuctios * from, 1, m M rplac ach iby i (from th right) for all i dow to m. * from i = 0, 1, m M rplac ach iby i (from th lft) for all i up to m. m m i = i m = 1 m must b cotiuous, v at th joiig grid poit, m. Th rormalizd s ar dotd by.

8 9-8 This surs that (x) is cotiuous vrywhr ad that it satisfis grid form of Schr. Eq. vrywhr xcpt i = m 0 = ( y i+1 + 2y i y i 1 )+ h 2 V i i I ordr to satisfy Schr. Eq. for i = m, it is cssary to adjust E. Th abov quatio ca b viwd as a oliar rquirmt o E. At th crucial grid poit i = m, dfi a rror fuctio, F(E). E E FE ( )= y m+1 + 2y m E ( ym 1 )+ h 2 E E V mm whr w wat to sarch for zros of F(E). Assum that F(E) ca b xpadd about E 1 (E 1 is th iitial, radomly chos valu of E.) FE ( )= FE ( 1 )+ df ( E E 1 ) + discard highr trms de E1 ad solv for th valu of E whr F(E) = 0. Call this E 2 df 0 = FE ( 1)+ E2 E1 de E 1 ( ) E 2 = FE ( 1 ) + E ( df de) 1 E1 Corrctio to E 1 This givs a stimat of whr th zro of F(E) arst E 1 is locatd.

9 9-9 Usual approach: comput df = FE ( 1 +δ) FE 1 de E1 δ ( ) Oc th drivativ is kow, us it to comput corrctio to E 1 (assumig liarity). Nwto-Raphso mthod for solvig oliar quatio E 2 = E 1 + FE ( 1 ) ( df de) E1 Itrat util th corrctio,, to E is smallr tha a pr-st covrgc critrio ε. Now w hav a igfuctio of H ad igvalu, E. box ormalizd: 1/2 Normaliz E by dividig by * dx = NE E ( x i )= * dx = j i 2 j h i=0 1/2 i 2 h itgral valuatd by summatio ovr grid poits. ral for boud 1-D fuctio This procdur has b usd ad tstd by may workrs. A good vrsio, Lvl 7.1 (schrq. f), is obtaiabl at Robrt LRoy s wb sit: I will assig som problms basd o Numrov-Cooly mthod for itgratig th 1-D Schr. Eq.