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> 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 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))" • \ 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 It«lt 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 » • — . Experience has already been made so that N is the maximum N „m-2 storage dimension. To determine 0 "6 2™"* we proceed as follows. -BH ff The calculation of < x, e y > is carried out, according to (9), by from che uncertainty principle it follows that the kinetic energy (and performing m successive integrations, practically reduced to matrix tnulcj_ hence the total energy in normal bound state situations) is of the order plications (see (9)). The base points are chosed using the Causs techni que with 4 points,translated a sufficient number of times. —T , where R is the length of the subinterval of [O,L] where the wave- It is eisily seen, from formula (2), that the matrix representing the rune ci on is appreciably non-vanishing. kernel (10) is very small outside a band of width e along the main dia Therefore E * —a- . Taking into account the condition tot p gonal, where 0 initially depends on B ff and enlarges at the rate /l at each successive step. In order to take advantage of this fact, the (18) BE Ì 5 <£, ground-state energy), matrix elements outside the band are automatically zeroed. Ve can also note that, due to the gaussian form of the kernel (2), ini — 6H which assures that the operator e "projects" onto the ground-state, tially -he band is narrow and sharp becoming more and more smooth during one has the repeated integrations. Therefore we must perform the first integrat 8 E • i ì 5 ions with a rather large amount of bas« points, which become unnecessary ° R* in the last calculations. To save computer time we halve the number of points at a suitably chosen so that step; so the old gaussian coefficients for the integration become obso lete. We are facing the problem of integrating a function known at cer (19) 'eff tain points different from the usual Gauss base-points. The following well known formula for finding the new weights has been used: which is approximately equal to 1/4 for • • 8. The inequalities (1 3 and (15) give lower bound* for 0 . Since Che rtU- (21) H. *— If "2zis1L dx j-1 n. tioh e 'n e " e M « « is precise only lot small 0 y'Cxj) J x-y3 (see Eq. (14)), it is reasonable to take B as small as possible» in such 14 15 where (22) y(x) - n (x-x.) meters a new estimate of the length of che interval in which the wave- j-o J function differs appreciably from zero is given: then the entire process is repeated. With 2 or 3 iterations we obtain satisfactory resu1ts for and where the x.'s are affine images of n consecutive base points on the all tested potentials, using about 3D sec. of coopuCer cime for Che IM interval [~l,l] (see e.g. [ll] ) 360/75 of Cenerò di Calcolo dei CKEN di Bologna. integration range fig. 1 old band width (A) 2 new band width (A ) rror affected zone matrix A Due to the fact we have approximated an integration of an infinite inter- vai by a finite matrix multiplication the kernel K 2 , obtained after the first integration, is affected by an error in the low-diagonal elements as shown in fig.l. Hence we are forced to chore the integration range as large as possible, to avoid chat the error affects all aacrix elements after a few iterations of the "matrix squaring". This effect is obviously reduced by the presence of « strongly bounding potential: for this reason the free-particle case leads to less accurate results than the bound case. At the end of this process, once Che "matrix" < x,e y > is obtained, by means of formula (4) wc calculate the ground-stace energy E and Ch< wave function if/ . from * we compute < x > , < x 2 > ; using Chase para' 17 4. THE UHLPBECK-ORNSTEIN DISTRIBUTION A possible way of avoiding the difficulties due to the spreading of the error (see fig.l) is described in ref. [13 ]. This method consists in adding and subtracting to the Hamiltonian H • - A • V an harmonic osci 2 2 lator term u r , considering the new Green-function 2 2 e( A+L1 r V2 (23) < x, e " " > y > - (w/2ir sinh(2«B)) *) 9 — — exp { - j coth(2w6) (x +y ) +u cosech (2u6) x • y } 2 ano the new potential V* - • V- w r 2 ; Trotter's formula holds with this different splitting of the Hamiltor.ian. Kirthermore one makes a si milarity transformation on the op.rator (2 3) by means of the formula w 2 0/ . 2 2. w 2 - "rr -e(-At-) r ) 7 r (24) Sp-e"8 e * e e * which gives - f (*2-y2> . - e<-A'«2r2) w6 (25) < J.K^y > - e e < 18 19 The kernel (25) is well-known in statistical Mechanics as the "Uhle£ beck-Ornstein Cteen function". It gives rise to a randoa process, but with smaller mean and variance (see ref. [l3],fig.2). Therefore it seems suitable to solve our problem. As in section 2, we consider only spherically symmetric potentials; in K > order to obtain the kernel corresponding to the operator P FP' ** integrate over the angles between x and y. The result is 5. NL*ERICAL RESULTS 1) Wiener integral. ,-3,2 e„B K'x- VFPV> The subsequent results are obtained by squaring 8 times the original -4«fl 4xv 1- matrix, which corresponds to taking the 256 power in Trotter's fojr mula (6), and are compared with analogue results obtained by diffe rent methods available in the literature. - ,,*.-„ -_2wS\2 / ^ -2u8^2 u(x e W(X e (, p) 1 gXp y >_ - exp " V L (a) Saxon-Wood's potential V(x)—V (l+exp(~ " >" -4wfi 1 1 - e -A«e o a where V « 350 MeV , p « 1.25 • (ll)l/3 f , a - 0.53 f o However, the advantages of the Uhlenbeck-Ornstein over the Wiener inte (~) fi E our result E Lovitch result gral are bounded by the fact that the "error tens" (see Eq. (12)) is o o rapidly increasing with u> . Substituting V -• V and T •+ T' - -L*u 2r 2 9.164 MeV -273.309 MeV -273.302 MeV in Eq. (12) the error becomes: (b) sq are well potential , „ ,d2V 2, d4V (26) 12 D' ,dV.2 .42 E (—) - 4H r V e o dr r dr dr o -6.25 2.46 -1.7 Of course this term vanishes when V(r) • w2 r 2 . -11.10 2.41 -5.6 -W6.00 1.28 -977.3 Suitable values of u (i.e. values making < 4> ,D'+ > small) can improve o o the accuracy of the method. However in practice it is not easy to find such values, since * is not known a priori and D' is strongly dependent 0 Private communication on w due to the term 4w2 r 2 . 2i 20 -c x ve) Exponential potentialI V(xV(x) - - V e where t-l o E our result E o o f") 6 24.35 -0.2035 -0.2024 8 12.70 -0.5270 -0.5265 10 8.64 -0.9528 -0.9526 12 6.55 -1.458 -1.458 REFERENCES 14 5.30 -2.027 -2.027 [ 1 ] R.P. Feynman and A.R. Hibbs, Quantua Mechanics and Path Integrals, 16 3.87 -3.318 -3.318 1965. 20 3.41 -4.026 -4.026 2 ] I.M. Cel'fand and A.M. Yagloa. J. Math. 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