IL NUOVO CIMENTO VoT,. 93 A, N. 2 21 Maggio 1986

Quantum Probabilities in Open Dissipative Systems: a Path Integral Derivation (*).

T. S ~ and J. l~Icm~R~:

Centre de ~echerches .Ncecldaires et Universitd Loq~is t)asteur - Strasbourg Groupe de l~hysique Nueldaire Thdorique . B t ) 20, 67037 Strasbourg Cedex, Erance

(ricevuto il 10 Ottobre 1985)

Summary. - - We define and derive the probability of presence at a given point in space and time of a one-dimensional quantum system which interacts with an external system. The probability is expressed in terms of a double path integral which contains memory effects. We show that it is possible to get a simple compact algebraic expression of the prob- ability as the continuum limit of a discretized form of the path integrals.

PACS. 05.30. - Quantum statistical mechanics. PACS. 05.60. - Transport processes: theory. PACS. 24.60. - Statistical theory and fluctuations.

l . - I n t r o d u c t i o n .

During the last years the problem concerning the description of open

quan tum systems has been investigated in several fields of physics (1,~). I n

nuclear physics this problem came up with the introduct ion of t ranspor t con-

cepts used for the description of heavy-ion collisions and fission processes (8,4).

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) A.O. CALD~IRA and A. J. L~GO~TT: A~va. Phys. (SV. Y.), 149, 374 (1983); Physica (Utrecht) A, 121, 587 (1983). (2) D.M. B~I~X, M.C. N ~ s and D. VAIJT~I~ : Ann. Phys. (~V. Y.), 147, 171 (1983). (3) D.M. B ~ I ~ , J. NvTo and H.A. W~I~)]~miJL~: Phys. Lett. B, 80, 170 (1978). (4) K. M6m~ING and U. SmILA~S~V: 5Vncl. Phys. A, 338, 227 (1980).

159

160 T. S~*M~ and J. RICHE:RT

The t ranspor t approaches generally ground on phenomenological derivations (5) in the framework of classical statistical mechanics (~,7). There have been at- tempts made in order to introduce explicitly the microscopic features of the

external intrinsic subsystem into the derivation of the equations of evolution of the ~ctuM collective subsystem coupled to it. These a t tempts use Feynman ' s

influence functional formalism and semi-classical approximations (3,4,s.~o) of ~t.

Recent ly we introduced a microscopic description of the intrinsic degrees

of f reedom in order to derive a classical Langevin equation for a collective one-dimensionM system describing deep inelastic heavy-ion collision pro- cesses (~). The description is based on a random mat r ix model which describes the coupling to an external system and uses a pa th integral formulat ion which was first introduced by BRINK et al. (a). This approach revealed itself a ve ry promising way of t reat ing dissipative systems and it comes out tha t it might be, in some specific cases, ra ther easy to extend this description to the quan tum- mechanical framework. We should like to show here tha t i t is in fact possible

to derive compact analytical expressions of quan tum probabilities if the collec-

t ive potent ia l of the system is harmonic, even in the case in which the coupling to the external system is strong and induces memory effects characterized

by arb i t rary strengths and relaxat ion times. At tempts and technical suc-

cesses of this type have been ~chieved recently on simple pa th integrals in other fields of physics (~m3).

2. - Q u a n t u m probabi l i t i e s in a n o p e n s y s t e m .

Suppose tha t at the initiM t ime t~ the system is defined to be in a state of wave function ~(q~), where q~ is the co-ordinate which characterizes the one- dimensional system. Then the probabi l i ty tha t the system will be at a point

q, at t ime tf is given by the expression

(1) d d ~ * ' - ' H ( q , , t f ) = f q,f q, cf (q,)cf(qi)K(q,q,t,[qfq,$~)

(~) (o) (2) (8) K. (9) A. (10) ~X~.

Sendal (11) T . (1~) B. (13) D. (1983).

U. BnOSA: X I V Summer School on Nuclear Physics, .Kikolaji, Poland (1981). W. N(~RENBERG: Phys. Lett. B, 53, 298 (1974). H.A. W]~ID]~NMtiLL~R: Prog. Part, N~cl. Phys., 3, 49 (1980).

L. SV, BASTI),N: Phys. Let, t. A, 95, 131 (1983). B. BAL~NT~XIN and N. TAXIG~WA: Ann. Phys. (N. Y.), 160, 441 (1985). TAKIGAWA and K. II~DA: preprint Department Physics, Tohoku University, (1985). SA~I and J. :RICH]~RT: Z. Phys. A, 317, 101 (1984). K. CHANG: J. Phys. A, 17, 2475 (1984). C. KHANDEKAR, S. V. LAWA:NDE and K. V. BHAGWAT: J. Phys. A, 16, 4209

Q'UANTITM PROBABILITI]~S IN OPeN DISSIPATIV]~ SYST]~MS ~TC. 161

w h e r e / ~ is the quantum-mechanical propagat ion kernel for the system to go

from t ime t~ and any points q,, q~ to t ime tf and point q~. The q u a n t i t y / ~ can be expressed as a double pa th integral (in)

(2) q(tf)=qf ~(tf)=q~

We derived in a former work an explicit expression of the generalized action

~(q, ~) for the specific physical case of large-amplitude mot ion (heavy-ion

collision or nuclear fission) in which q stands for the radial relative distance

between the centres of mass of the ions. The collective motion associated with q ma y be weakly or strongly coupled to the intrinsic mot ion of the nucleons in the ions (11). I t comes out tha t the explicit expression of the exponent

in (2) can be worked out analytically if one introduces a random matr ix as- sumption on the mat r ix elements which couple the intrinsic to the collective motion (3). Then, to second order in the pa th difference ( q - ~)

(3)

~f ~m

ti

with

i 2 ~ (~ + ~) (q - - ~) +

tf tf

t i ti

(3a) Z(q, ~) = ~ M ~ - - U(q).

Here M is a collective mass, U a collective potential , $m and ? a mass correc- t ion and a friction coefficient which are generated by the coupling between the collective and the intrinsic systems and finally B(t, t') a nonlocal diffusion coefficient which depends on the t ime difference t - t' and which is charac- terized by a finite relaxat ion time. The explicit algebraic expressions of $m, y and B can be found in the paper by SA~I and RICHEST (11) (an explicit ex-

pression which is discrete in t ime is given by (B.7) in appendix B). In practice

they are slowly varying functions of q, ~. We suppose here tha t y and 8m are constant. First this is a physically sensible approximat ion and second we are

not so much interested in discussing here a fully realistic system as in deriving

an expression for the probabi l i ty II(q~, t) which might be compared to the corresponding classical ease under the same physical assumptions (see sect. 3).

As last remark we should like to ment ion tha t expression (1) reduces to well known-expressions for a free system if we pu t 3m = y ~ B = 0 (14). In

(14) R.P. F]~YNMAN and A. 1~. HIBBS : Q~anium .Mechanics and ?ath Integrals (AcadcH~ic Press, New York, N.Y., 1965).

162 T. SAMI and J. ntCr~nT

this case the probability amplitude /~ reduces to a product of propagators

/~ ! [ = * l l (q,q,t,]q,q,t,) K(q,t ,]q,t ,) K (q,t, Lq, t,) ,

where K is the propagator of the free system governed by the Lagrangian of (3a).

3. - Analytic derivation of the probability.

a) Diseret ized expression o] K . We want to obtain an algebraic expres- sion of II(q~, tf) as g function of At ~ t~ -- i~ and qf using (1), (2) and (3). W'e shall show now that this is possible in a rigorous way provided the potential U is a harmonic oscillator which we write

(4) U = 1M~o~q~ = ~[.t(9 2qZ ,

where /~ = M § ~m and whatever the explicit expression of the memory kernel B ( t - t') might be.

We first apply the saddle point method to ~ in (3) in order to derive the stationary trajectories of q and ~. One gets a system of coupled differential equations:

(5a) d 8 L f

dt 8~

t f

8 L f i f d 8q + ~m~ + y~ - - ~ t' [q(t') - - ~t ( t ' ) ]B( t - - t ') = O,

tl

t f

t~

(5b) d 8~,~f

dt e~

If we impose the boundary conditions which show up in expression (2) of/~

(5c) q(t,) ---- q, , q(tf) = q, ---- ~[(t~) , ~(t~) = q, ,

we straightforwardly get the unique solutions (q(t)} and <~(t)}:

(6a)

(6b)

C 1 (q( t )} = D~(t) exp [--2xt] + f f exp [2~t] + D2(t) cxp [--22t] +

+ 2- exp [2,~t],

C1 (~(t)> = Di(t) exp [ - - 2 ~ t ] - - -~ - exp [2~t] + D2(t ) exp [ - - 2 2 t ] - -

C~ - - f f exp [}~ t] ,

Q U A N T ' U M P R O B A B I L I T ] : E S I N OPEIh" DISSIPATIVJ:~ S Y S T E M S E T C . 163

where

~ - - ~ ~ V W - : ~ g ,

H e r e we consider the case in which ~ > COo which co r responds to t he clas-

sically o v e r d a m p e d reg ime of a d a m p e d osci l lator . One could also w o r k ou t

the case in which ~ < COo (osci l la tory reg ime) which we do no t g ive here (~5).

The coefficients in (6a) a n d (6b) r ead

( 6 e ) r = (q, - - ql) exp [ - - 2,t~]

1 - - exp [(2~-- 22)At] '

(6d)

and

wi th

C2 := (q~ - - q~) e x p [ - - 22t,] 1 - - exp [().2 - - 21)At]

D~(t) = b~ -k d~(t), D~(t) ---- b~ ~- d~(t)

~ exp [A2t,] - - 72 e x p [22t,] bl - - e x p [(22 - - 2~)t,1 - - exp [(22 - - 21)t,] '

Yl exp [21t~] - - ~ exp [21t~]

exp [(2~ - - 22)t,] - - exp [(2~ - - 2~) t , ] '

where

q ~ - ' q' 71 - 2 ' ~'2 = q , - [d,(t,) e x p [ - - 21t,] § g2(t,) exp[--) .~t , ] ]

and wi th t

1 dl(t) = ;t~ - - r

ti

where

1 f d t ' E ( t ' ) exp [).2t'], e x p [ ) . i t ' ] , g2(t) - ).1 - - 2~

ti

tf

i fdt ' l [(q(t ' )} - - ( ~ ( t ' ) } ] B ( t - - t ' ) .

t!

Solut ions (6a) a n d (6b) are complex . As m e n t i o n e d a b o v e t h e y are un ique

as one can convince oneself b y p e r f o r m i n g the expl ic i t de r iva t i on in t e r m s

of ~ : q - - ~ and Q : ( q - ~ - ~ ) / 2 .

One in t roduces now the new va r i ab les u and v which descr ibe the dev ia t ions

(15) L .D . LA~'DAV and E.M. LIFSHITZ: Mechanics (Pergamon Press, Oxford, 1959).

11 - I I N u o v o C i m e n t o A .

164

from the s ta t ionary trajectories

T . S A M I and Z. R I C H E 1 R T

(7) q = <q> ~- u , ~ = <q> + v .

By definition of (5c) the boundary conditions are fixed by

(7a) u(t,) = u(t ,) = v(t,) = v(t,) = o .

The propagat ion kernel ~ can be cast into the form

(8a)

where u(If) ' -o v(tf)--O

i ,

U(ti)~O V(r

is a double pa th integral over u and v with end points fixed by (7a) and is for.-

m~lly the same action as the one given in expression (3) except for the replace-

(~ classical ment of q by u and ~ by v. The ~) action , ~ reads

(so) tf

- - - 2 ( < q ( t ) > 2 - ( q ( t ) ) 2 ) - fi

<i(t)>)n} + | f i f

-~- ~ t' (<q(t)}--<q(t)>)(<q(t')>--<~(t')>)B(t--t') t i fl

which can be easily worked out by using the explicit expressions of <q> and <~>

given by (6a) and (6b). Hence there remains to calculate the normalizat ion factor C. If one dis-

cretizes expression (Sb) by introducing N t ime intervals e~ ~-At /N, then

-{-r

(9) C = lim C~ ---- lira JV~ 1-[ u~ exp -- Z u,~P,~,u~, 2752~_A f k ~ l --r L re ,n--1

where the constant ~C~ corresponds to the normalization which can be obtained in the same way as in ref. (1~) on a small t ime interval s~. The derivation is

shown in appendix A and given there by (A.7). The explicit form of the 2(37--1)-dimensional mat r ix P is given in ap-

Q I ' A N T I : 3 I P l t O B A B 1 L I T I ] - S ] N O P e N D I S S I P A T I V E S Y S T E M S E T C . 1 ~ 5

pcndix B. The constant C,. reduces trivially to

V[z + (9a) Cx = ~ ~2]/eN] 2~t ] "

As (;an be seen by examinat ion of the expressions given in the appendix, the matr ix P has 2 ( N - - 1) complex eigenvalues which are pairwise complex

conjugate and different from zero. Hence det P~(N_~) is a real positive quant i ty .

b) The continuum limit o] I~. If one uses the explicit expression of P ,

a series of formal manipulations on its de terminant reduces the upper left (N- -1 ) -d imens iona l quadran t to zero. With the help of Schur's formula (is)

it is possible to cast it in the form

(let P2(x_~)= (-- 1)N-~(det .~(x-1)) 2.

Int roducing ..~, ~ = (2i~./#)ex~,._~ and put t ing this expression into (9a), we

have finally

(10) C,~. = 1 =- e~ det~,,._~) -~ ,

~-@'N--I =

a(Y)

c(iV)

0 , o ,

. ~

0

c(~) - - b(~) + 0

c(N) a(N) --b(N) § 4 -

0 - - b ( N ) - - c,~(N~ 4

~ " ~ 0

" ~ " 0

o ~

0

c(~) - - b(h r) + ~ -

a(h r)

with

and

Introducing the notat ion

i 2 "~ 4 ] c (y ) = 2 7_ ~,, b ( N ) = 1-; ~- : 1 , /~

s,,, = A t l N .

x.~._l -- det ~.V_l,

(16) F .R . GA.~TMACnm~: Thgorie des Matrices, Tome 1 (Dunod, Paris, 1966).

1 6 6

one obtains the recurrence relation

with

Xo=l , xl=a(N).

F[ero

~ ~(~) - a(ar) = 2 1 - ~ ] ,

with

me At [ y2 ~ , = ~ , ~ , = - ~

T. SAMI and J. RICI:I]ERT

( fl(hr) --: ~ b2(~V) -:z - - 1 +

3~o ) o~ At" At 2 , f12 ~ 3 6

This recursion formula leads very easily to the finite series expression

"(1~-1'/2)(N 1 ) (11) a~_, = ~ - - - - q a(N)N-'-2qfl(N)q,

r q

where E( (N--1) /2 ) stands for the integer par t of (N- - l ) / 2 and ( ( N - - l - - q ) / q ) is the ordinary binomial coefficient.

Expression (11) can be expressed in the form

(12) ~_~ = 1 + + ~ (--1)~ q 2 ~-~-~-~.

bu t

where

Using the algebraic relations (17)

sinnyo = sinyo (-- 1)" 2"-H~(cos yo) "-l-~q ,

(17) I .S. GR.~DSI1TE'I'N and I.M. RYZH]K: Tables o] Integrals, Series and Products (Academic Press, New York, N.Y., 1980).

QUANTUM PI~OBABILITI~S IN OPEN DISSIPATIV]~ SYSTEMS :ETC. 167

and

sinh nyo = sinh y0~(~"~ ))~2 q ' 0

(--1)q(n -- ~ -- q)2,,-~-2q(cosh yo),,-~-2q , if y~ < 0 ,

taking the limit N = oo of xN_~ in (12) and going back to (9) through (9a) and (10) one ends up with

(13a) C ---- lim C~ -- # - - V r ~ exp [TAt/2#], ~ 2~h v/sin [At V~,, _ ~ ' / 4 / ]

if w~ -- ~2/4~u~ > 0

and

(13b) C - - lira Cs = N--~r

- exp [r5~/2~], 2z?~ sinh [At %/~14# ~ _ Wo 2]

i f o~0 ~ - ~ / 4 / < 0 .

The exponential torms are obtained from lim [1 + 7 Atl21~N]N. I t is easy to see tha t one retrieves the expression of C corresponding to the

free harmonic oscillator (~----~m = 0) and also the free particle (O~o = ~ = ---- ~m = 0) of Feynman and Hibbs (1~).

Finally

(14) I~(q,q~t, lq, t,) = 2 ~ sinh~/~)~-/~'--e)l] exp ~ A ~ exp ~Sfo,

in the case in which a~o < ~,/2#, which corresponds to the classically ovcrdamped regime introduced above (expressions (6a) and (6b)) and a similar expression in the damped oscillatory regime OJo > 7/2/~.

The double propagator _~ given by (1~) is a rigorous and properly normalized expression. I t is remarkable to see tha t this quantum-mechanical quant i ty possesses two different normalizations which are imposed by the classically different overdamped and oscillatory regimes.

e) Expression o/ the probability. I t is now easy to work out the final expression of the probubility II(qf, tf). We suppose the system to be, at t ime t~, in a state of wave function

~ ( q , ) - - e x p [ - - v(q, - - q ,o )2] .

I t corresponds to a S-state harmonic-oscillator wave function centred at q~o" The choice of this expression will be justified in a further work in connection with the derivation of classical probabilities.

168 T. SAM] and a. RICllERT

The double in tegra t ion over q~ and ql in (1) can be expl ic i t ly pe r fo rmed

and one gets the final exi)ression , in the o v e r d a m p e d regime, for the prob- ab i l i ty

(15) l I (q , , t,) = C ( n / E ) "2 exp [-- K ( q , - q,o)2],

whero C is g iven by (13b) and

v B" E - . - ~ -= D - 8~

with

B = i# ().2 cxp [(22 - - 2x)At] - - ).~) /t i-- exp [(2.~ -- 2~)At]

- - i B ' ,

t f t f

I ox, f f D ~ ~ exp L22At] [ i - - ( ~ L ~ ) A t ] (t~ dt '[C,o exp [2 , f ] + Qo exp [ 2 # ' ] ] .

t i t l

t f f f

e x P [ - -215f] ftfd �9 exp [;%t]B(t--~ t) - - 1 - - exp [(22 - - 2,)At] d t t [Cloexp 12~t'] + C~oexp [22~']] �9

t i It

f f ~f

} ?i -exp [2~ t] B ( t - - t ) - - ~.~ dt ' [C~o exp [).~ t] + C2o exp [2~ tJ]-

t l t i

�9 [C~o exp 1.2~t t] + C~o exp I2~tr]] B ( t - - t ' ) ,

i "v t "T C, : (q~-- q,)6~o, C2-- (q~--q~)6~o,

where C1 and C, are defined in (6c) and (6d) and o therwise obvious no ta t ions .

Fin~Uy

A = i A ' q, A ' # ()'~ - - 22) exp [22At] ' • ~ i ~ exp [(2_, - &)Ai ]

and

B r A t

q~0 ~--~--7 q i o , K : 4---E"

The express ion / / is normal ized to 1 and the no rm is conserved in t ime as should be. One can see t h a t the l imit At = 0 gives H = ~o2(qt) and der ive the equi l ib r ium d is t r ibu t ion a t At = c~:

, ~ ( ) . - ).-.)~ ~I 0 6 ) zz(q,, ~, = oo) - ~ ( ~ - - z~)~xp ~ q, j

QUANT~2M P]ROBABILITI)]B IN OPEN ] ) ISSIPATIV~ S~(BT~]MS ETC. 169

with

(16a) A', - - - -

where off~

2 , - - 2, exp [b~**o/4Jerf~

is the complementary complex error function

b, : 2~ ~- i/v~, 53 = L, d- i/vs.

Here 3~ z~ are two finite characteristic times connected to the memory effects contained in the diffusion function B(t, t') and C in (16a) is a coefficient which stands in front of the expression of this function (see (11) and appendix B).

Finally it might be interesting to get the semi-classiea.1 expression (h --> 0) of this probability since it leads to the classical limit.

One obtains

(17)

lira H(q,, t~)

_ ~ ' : ~A' ,

D I~SD,

-- lLo(q,, * , ) - 0 exp [ - R ( q , - q,o)-'],

/~ : hB, 0 : ~C,

4 . - C o n c l u s i o n s .

In an earlier work (ix) related to deep inelastic and fission processes in heavy-ion collisions we showed that one may use a path integral formalism in order to describe open dissipative systems with an arbitrary coupling strength to the outside. Starting from the explicit form of the propagation kernel/~ eq. (1) which contains up to quadratic terms in the collective path variables, we have shown that one is able to derive a closed algebraic expression for the probability of presence at a given time and a given point of the open quantum system. This was possible because we chose the conservative potential which drives the system to be of the harmonic-oscillator type.

Except for the harmonic approximation of the potential and the assump- tion of constant-mass correction and friction coefficients, the situation which was investigated typically corresponds to the conditions which prevail in heavy-ion collision processes, characterized by a strong coupling to an external system (the intrinsic subsystem of the nucleonic degrees of freedom) and finite-

170 T. SAMI a n d j . RICH~RT

time effects induced by the coupling force. I t comes out very naturally that the quantum propagation kernel has two different expressions depending on the regime of the corresponding classical system which might be overdamped or oscillatory.

We aim to use expressions (15)-(17) of the quantum probabihty in or- der to pursue different investigations. First we want to compare numerical apphcations of these expressions with the ones obtained by using the discre- tized expression of the probability in order to test numeric~ accuracy and the stabihty of the result as a function of the number of mesh points taken in the time interval over which the system moves. This might be an interesting point since analytical expressions of the probability are generahy not available in more realistic situations in which the potential is no longer harmonic. Second the present closed-form result gives us the opportunity to compare the clas- sical and quantum-mechanical limits of a dissipative system. We aim to com- pare numerically the probability distributions in these cases and analyse the effects of the friction force and the time correlations induced by the finite response times generated through the correlation function B(t, t'). Finally it seems to us very easy to study the tunnelling probability through the potential barrier of an inverted parabolic shape and analyse it in terms of the physical parameters which govern the evolution of the dissipative system.

We should like to thank Prof. H. A. WEIDENM'(J-LLER for a discussion which clarified some fundamental point and Dr. D.M. B~INK for interesting sugges- tions concerning the asymptotic behaviour of the distribution functions.

APPENDIX A

We work out the normalization factor which stands in front of the expres- sion (9) of U. We start from

(i.1) I~v(q,, t,)l ~ -- fdq0 dq~J~(qo, q'0, tolqx,

Introducing

and

q~, t~)~o*(qo, to)v,(qo, to).

tl ---- to ~- ~ ,

QUANTI:M PI~OBABILIT]}]S IN Oi']~N DISSIPATIV}] SYST~]bI$ :ETC.

we have for one dimension and ~ inf ini tesimal

(A.2)

and

(A.3)

171

g(qo, q'o, tolq,, q~, t~) = g ( q ~ - ~, q l - ~, tolq,, q~, ~o + ~) =

1[

? lY~(q,, t~)p = ]~v(q,, to -.- e.)['~ = J d v d ~ R ( q , - - V , q l - - ~ , to[ql, to ~)"

�9 v ,*(q~- ~, to) v,(q,-- ~, to).

E x p a n d i n g the l e f t -hand side and the r igh t -hand side of eq. (A.3), we obta in

(A.4)

�9 I~o(q~, to)l ~-- ~ o ~ - - - ~ - ~ + ~ a---~ c~-~ . . . .

Then for h r t ime in te rva ls

(A.7) ~ - - IA,I = ~ - 2:th~ J

The leading t e rms on bo th sides have to agree in the l imi t e --> 0, hence it is ~mcessary to have

(A.5) [w(q,,to)l~-_lA, i.. d ~ d ~ e x p ~ ,~-~-}- (~1~--~2)--~,q1(~--~) �9

�9 Iw(ql, to)l 2 ,

which implies

2 ~ = o

(1 + (r/2~,)~)

172 r. SAMI and J. RICI.IE1KT

_A_PPEINDIX B

We give here the explicit expression of the terms which enter the action 6f' defined in (8b).

Conservative par t :

(B.~) ~f ~f

1 fat IMam-- ao;o u ~3 f dt ~f(u, ie) : 72 ti ti

and a similar expression in terms of the variable v. Inert ial contr ibut ion:

(B.2) tl

8m f d t ( ~ + ~ ) ( u - - v ) : 3m 2 2

ti

since u and v obey (7a). Dissipative contribution :

t[

{,++ +,(+ +) +2)} =

t i

8mfd = T t(~-'-,)9,

(B.3) t f t f

2 7 f dt(ie + b ) ( u - v ) = - -~ Y f d t ( O u - - i w ) . $1 tl

Put t ing (A.1) and (k.2) together the introduct ion of the discretization procedure leads to the following expressions:

(B.~) ',+~-ZII + + - - ++~ O+o,

) ' 1 _ Oo (~M + + v~)

for (B.1) and (B.2),

(B.5) 2?--I

j=0

for (B.3) and finally

i E 2 2~--1 (B.6) § ~-~ ~ (u+--v~)Bk~(u~- v~) ,

where

(B.7) Bk~ ~ o exp [ - -a(k - - l ) 2] cos [b(k--1)] (k~l z O~ . . . , N - - 1 ) ~

where c, a and b are constants, the explicit expressions of which are given in (u). Using these diseretized expressions for the action Sf' one gets expression

Q U A N T l J M P R O B A B I L I T I I E S I N O l d E N D I S S I P A T I V E SYST~EI~IS : E T C .

(9) for C in which the finite matrix P takes the following form:

p = i# Q +i#~ iy e~

where Q, V, F ~re bloekwise tridiagonal matrices:

zV--I

m

- - 2 1 0 . . . . . . . 0

1 - - 2 1 O- �9 . 0

o ~ ~

' ~ ~ 0

" " " 1

0 . . . . . . . 0 1 - - 2

V ~

Q _= 2 - - 1 . . . . . . . . . �9 . . . . . 0

- -1

0

2 1 0 . . . . . . . . . . . 0

, , ,

~ 1 7 6 0

. . . . ] .

0 . . . . . . . . . . . . . . . 0 - - 1 2

2

0

�9 b ~

.-'Y - - 1

�89 0 . . . . . . . . . . . 0

2 ~ o . . . . . . . o

, , ~ ,

0 0

~ 1 2

o . . . . . . . . . . . o �89 2

- 2 - � 8 9 o . . . . . . . o

- -2 --�89 0 . . . . 0 - - 2

, , ,

0 " ~ 1 7 6 0

o ~ __21~

o . . . . . . . o _ 1 - 2

1 7 3

.N'q-i

174

~md

. / - t - - ~

T . S A M I and Z . R I C H E R T

m m

o ~ o . . . . . . . . . . . o

-�89 o �89 o . . . . . . . o

O . . . 0

' ' * O

" * " �89

o . . . . . . . . . o - � 8 9 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 0 0 . . . . . . . 0 0 - - 2

�89 o - � 8 9 o . . . . . . . o

. . . :

0 ~ 0

�9 " " ~ � 8 9

o . . . . . . . . . . . . . o ~ o

B----- Bkl --B~-z

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. --Bk~ Bkl

where Bk~ has been def ined above .

�9 RIASSUNTO (*)

Si definisce e si deriva la probabilit~ della presenza in un determinato punto dello spazio e del tempo di un sistema quantico tmidimensionale ehe interagisce con un si- sterna esterno. Si esprime la probabilit~ in termini di un doppio integrale di percorso che contiene effetti di memoria. Si mostra che b possibile ottenere una semplice espres- sione algebrica compatta della probabiliti~ come l imits continuo di una forma disere- t izzata degli integrali di percorso.

(*) T v a d u z i o n e a cura del la Redaz i~nv .

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