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Funaional Integration for Solving the Schroedinger Equation

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Funaional Integration for Solving the Schroedinger Equation Comitato Nazionale Energia Nucleare Funaional Integration for Solving the Schroedinger Equation A. BOVE. G. FANO. A.G. TEOLIS RT/FIMÀ(7$ Comitato Nazionale Energia Nucleare Funaional Integration for Solving the Schroedinger Equation A. BOVE, G.FANO, A.G.TEOLIS 3 1. INTRODUCTION Vesto pervenuto il 2 agosto 197) It ia well-known (aee e.g. tei. [lj - |3| ) that the infinite-diaeneional r«nctional integration ia a powerful tool in all the evolution processes toth of the type of the heat -tanefer or Scbroediager equation. Since the 'ieid of applications of functional integration ia rapidly increasing, it *«*ae dcairahle to study the «stood from the nuaerical point of view, at leaat in siaple caaea. Such a orograa baa been already carried out by Cria* and Scorer (ace ref. |«| - |7| ), via Monte-Carlo techniques. We use extensively an iteration aethod which coneiecs in successively squa£ ing < denaicy matrix (aee Eq.(9)), since in principle ic allows a great accuracy. This aetbod was used only tarely before (aee ref. |4| ). Fur- chersjore we give an catiaate of che error (e«e Eqs. (U),(26)) boch in Che caaa of Wiener and unleabeck-Ometein integral. Contrary to the cur­ rent opinion (aae ref. |13| ), we find that the last integral ia too sen­ sitive to the choice of the parameter appearing in the haraonic oscilla­ tor can, in order to be useful except in particular caaea. In this paper we study the one-particle spherically sysssetric case; an application to the three nucleon probUa is in progress (sea ref. |U| ). St»wp«wirìf^m<'oUN)p^woMG>miutoN«i»OMtop«fl'Ht>p>i«Wi*faif»,Piijl(Mi^fcri \ptt,mt><r\A, i uwh P.ei#*mià, Uffcio Eanfeai SaMfiM», " ~ Hot*, Vi.1. lUfitM MM|MM 125 (Mi. 199) 5 2. FUNCTIONAL INTEGRATION AM) TROTTER'S FORMULA I*t «.(tXq^O^U) ,«.,((> <*) denote a vector-valued continuous func- tioo on the interval [Otfl|. He iapose tLs condition q(0)->x , q(B)-y. This is called a path connecting x and y. Suppose F(.) is a functional of the path q; e.g. ve vili consid­ er the particular functional 0 (1) F(q) - exp (-- I| V(q(t» dt) , 0 where V(q) is a function bounded frost belov and sufficiently regular so that the integral exists. Wiener's original idea was to sua a functional F over all paths connect­ ing x and y, weighting each path q by the differential probability that • particle undergoing brovnian notion will follow thr path itself. Indeed he was able to give a rigorous sxtaning to this idea, defining: (2) f F(Ì) dP m - f F(i) dPw (q) - - li- (vf ^ * J ... J r^.,^^1 -£ CVt*y Idq-^.dq,, * ' q denotes a point of Che physical space Jr. Sonatinas ve shall writ* •iayly q in place of q. * 7 where q -x , ^."y and F(q.,...,« ) is the functional F computed on a Trotter's theorsnt. Let T,» and T • V be linear operators defined on a broken curve q(t) which coincides with q(t) on the points t-j— , j>l,..,s. f*C lilbert space * ' H, all self-adjoint anJ bounded froa below. Then for all * c H and for any positive nuaber 0: Let T - - •=• A b* the quantua-nechanical kinetic energy operator, V(q) he the potential operator and H-T*V the haailtoaian of a particle of aass M. -t\ -sj - -B(T*¥) (6) lin (e * e " )" * - e * . Let now * be the g:ound-state and •. be the first excited state of the n • • particle and let E denote the energy difference between the two states. t In oar case T • - js & and V is the potential. Hence, putting Obviously we have -•I -£ a - e " e B the corresponding kernel is (3) lin e ~fiH|*> - lie •"«• P |* >, f*> c L2(tf> a - - a -• - 2 , ki - i - UHl * - £ V(i, 2e b B where E denotes the ground state energy and P is the projection oaerat- a) < ;u, > - iSiiV) • « or on the ground-state. Our nuaerical nethod consists in ccaputing e for 0 sufficiently large 'e.g. such that 0AE*1O), and then using the ap­ Froa (6), (7), the Feynaaa-Kac f omul J (5) follows easily; indeed: proximation suggested by the L.H.S. of (3) (8) < x, •"*" J > - lin < i,a0y > - n •*• ~ (*) E„--TU» - * m * m +~ -• >2 - lin J ...J <i^j> <ij.M2> .-< ^ >d^1...dqn 4 >oC the where < •» *>2 " '* "">** product in the Hilbert space iV). Fuaccioiul integration enable* us to compute the kernel < x,e y > , aad asking use of (7) one has unly to notice that I e J ~ • 3 'SB j-l *,ye /? corresponding, to the operator e by nwans of the Tiji—n Kac formula: COM verges alaost everywhere to F(q) because of Leaesgue's bounded con- vergeace tbeorea. la order to canjiiira a "*B for large 0 we are faced with - W = (5) < x, e y>- J F(q) «P„fT<«) Che aaaarical pro*lea of coapatiag large powers of the operator A. where F is the functional considered in (1). An alternative way of computing the kernel (5) is offered by Trotter's f omnia; ** Tmttr't forswla noMfoader aer* general condition* (see ref. [3] , [•) 9. my. • • Choosing for instance outers a of the type i, -earn caa iterate the cedute of squaring the operator A: "~ a """2a 3 (ID ^ "" a " e [-J(T*^ (lr[v.[T.vll.yTp[T,v]iK..] 2 2 2 2 (9) A* "* - (... (A ) ) the L.I.S. ia a eelf-adjoint operator which differs from A only by • times higher order tema. Let na coaaider the "error1* For instance with • • 8, tbe equivalent faactioaal iategratioa ia faraala (2) is over a space with 256 dimensions, which has beea scovasi to ha vary accurate in anst cases. U2) --Jr[MM.]» n[*-r».»l] We have considered only the case of a spherically symmetric potential: v(q) - V<|qf ). Let P be tb* projection operator oa the ayaaatric e-eta T r r—UI te. We can replace the original Hilbert space L (JO by the svbepece * *l • »r " r 7 h " *** ««•««•»••• 2 3 P L (R ), and the operator A by P AP . Of coarse the gravai state of o 2 o o the system belongs to f L (/?"). Therefore it is safficiest to know the Straightforward coapntations show that kernel corresponding to the operator (11 (*)2 P e " P - e " P «£>'»-4 r dr* dr* dr o o o One can obtain such a kernel integrating the expression «ap I - **~3r This "error" aaat he with T • V (saa Ea.(U)). Since we are in­ 1 Peff 24 a2 terescad ia the state a of the systaa, wa take the expectation (where 6 " „ ~g> °**T cn* angles between the vectors a and y. Mara g{{ value of hath a precisely, one can average over the Bear measure of the rotatioa grana. ia the atate a ; aa obtain the condition The result is: o cu) (J )2 < *_.!• (i-t r - r<*>- ( 1 )2 | * > « i2E < *,* > 2 (10) K(x,y) - < x.PAPv >- \- Crjrff) -L " - • la ardar ta abtain a estiaate of the L.I.S. of I*.(14) it is pos­ -ar where, of course, x • |xj and y • I yi , sible to aamrnai—ra a aith a faactiaa of the farà r e with a* ^. Oar calculation is parforaai according to the facaala (f) where ids» her» nel ia given by (10). Siacc • ia finita the rasale cannot ha exact. Ha lb atfaciala, aaa caa alaaya a • 2* aafficiaatly large ia each a disci'ss now briefly the Magnitude of tat error. way «let (1*) ia netieffed. , aa aa anali aaa, • ia ••••ill' by fallowing ref. [5] a* have, osiag the iahsr aaeaserif fonala: /*&tjH.i>,:- rJ«»i. '•r.r'M'0~',Ì^.Ì'é*Ì r. ?,,**• W'i"f .%*.'• «-» I- 11 3. THE NUMERICAL PROCEDURE The first problem faced is the choice of the integration interval [o,LJ. which must contain the region where the radial wave-function is appre­ ciably different from zero. We solve the radial Schroedinger equation for a zero energy virtual state: (15) (xy(x))" • \<x)y(x) - 0 with initial conditions y(0) - 0, y'(0) » 1 for the solution y. If there are many bound states a good estimate of the range of the wave function is given by the last zero x of y(x), since the wave-function falls very rapidly to zero short after this point; if these zeroes are not found, we use the effective range theory assuming that the range of the wave-function is of the order of the scattering length a (see e.g. [9], p.1088). The integration interval [O,L] has been chosen as being twice the pre- ceeding estimate. A guess for the number of bound states is given by (see e.g. [io] p.89) (16) ^ f /- MM dx •m CD rounded to the nearest integer. The upper bound M for the integration of equation (15) and for the eva­ luation of the integral (16) must be chosen in such • way that for x > M, V (x) » 0. r 1.0 «lItt ZIA— — m torn si iu 11.25 H 14 116 12 13 The number N of base points which decompose the integration interval a way that both inequalities (19 and (15) are roughly satisfied. Hence, must be gre/.t enough so that the halt-width /B .. of gaussian (10) con in the case m • 8, we have tains about six points, making the numerical integration sufficiently accurate; therefore (20) - —*£! (17) '6ef f > 6 L/N where » • — .
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