corresponding velocity eld to consider in the Fokker-Planck equation shows singularities in the locations of the nodes of the wave function. These sin

Transcription

1 Exact solutions of Fokker-Planck equations associated to quantum wave functions Nicola Cufaro Petroni, Salvatore De Martino and Silvio De Siena y ABSTRACT: We analyze the non stationary solutions of the Fokker-Planck equations associated to quantum states by the stochastic mechanics. In particular we study the exact solutions for the stationary states of the harmonic oscillator and the potentials which realize new possible evolutions ruled by the same equations. To appear on Physics Letters A 1. Introduction In a few recent papers [1] the analogy between diusive classical systems and quantum systems has been reconsidered from the standpoint of the stochastic mechanics (SM) [],[3]. Particular attention was devoted there to the evolution of the classical systems associated to a quantum wave function when the conditions imposed by the stochastic variational principle are not satised (non extremal processes) in order to check if and how the evolving distribution converges in time toward the quantum distribution. This hypothesis constitutes an important point in the discussion engaged by Bohm and Vigier several years ago about some criticisms to the assumptions of the Causal Interpretation of the Quantum Mechanics (CIQM) [4]. In the quoted papers [1] it was pointed out that the right convergence was achieved for a few quantum examples, but no general results were available to this eect. Moreover there were also a few counterexamples: in fact not only for non stationary wave functions (as for a minimal uncertainty packet) there may be no convergence at all, but also in the case of stationary states with nodes (namely with zeros) we do not get the right asymptotic behaviour. The problem is that for stationary states with nodes the INFN Sezione di Bari and Dipartimento di Fisica dell'universita di Bari, via Amendola 173, 716 Bari (Italy); CUFARO@BARI.INFN.IT y Dipartimento di Fisica dell'universita di Salerno, Via S.Allende, 8481 Baronissi, Salerno (Italy); DEMARTINO@PHYSICS.UNISA.IT, DESIENA@PHYSICS.UNISA.IT 1

2 corresponding velocity eld to consider in the Fokker-Planck equation shows singularities in the locations of the nodes of the wave function. These singularities eectively separate the available interval of the space variables into (probabilistically) non communicating sections which trap any amount of probability initially attributed and make the system non ergodic. The rst new result of this paper is well understood in the light of these opening remarks: we show, by means of a general method (the eigenfunction expansion for the diusion equations) that for transitive systems with stationary velocity elds (as, for example, a stationary ground state) we always have the right exponential convergence to the right quantum probability distribution associated to the extremal process, even if we initially start from an arbitrary non extremal process. These results can also be extended to an arbitrary stationary state if we separatedly consider the process as conned in every conguration space region between two subsequent nodes. However this is not the unique path we trod along in the following pages: the second new feature of this article starts from the remark that non extremal processes can be considered virtual, as trajectories in the classical Lagrangian mechanics, but that in the same way they can also be turned real if we modify the potential in a suitable way. The interest for that lies not only in the fact that non extremal processes are exactly what is lacking in quantum mechanics in order to interpret it as a classical stochastic process theory (for example in order to have a classical picture of a double slit experiment [5]), but also to engeneer some new controlled real evolutions of quantum states. In particular this could be useful to study (a) transitions between stationary states (b) possible models for measure theory [3] and (c) control of the particle beam dynamics in accelerators [6]. In fact it should be pointed out that the stochastic mechanics is also a theory, independent from quantum mechanics, which has applications in several physical elds, in particular for systems not perfectly described by the quantum formalism, but whose evolution is corectly controlled by quantum uctuation: the so called mesoscopic or quantum-like systems. This behaviour characterizes, for example, the beam dynamics in particle accelerators and there is evidence that it could be well described by the stochastic formalism of Nelson diusions [6]. Of course in these systems trajectories and transition probabilities always are perfectly meaningful. Since to study in detail the evolution of the probability distributions in these cases, and in particular to try to understand if and how it is possible to make a controlled transition between two quantum states, it is necessary to determine the fundamental solutions (transition probability densities) associated by SM to every quantum state in consideration, we eventually devoted the second part of this letter to a sketch of the way we could undertake this task. SM is a generalization of classical mechanics based on the theory of classical stochastic processes []. The variational principles of Lagrangian type provide a solid foundation for it, as for the classical mechanics or the eld theory [3]. In this scheme the deterministic trajectories of classical mechanics are replaced by the random trajectories of diusion processes in the conguration space. The surprisig feature is that programming equations derived from the stochastic version of the lagrangian principle are formally identical to the equations of a Madelung uid [7], the hydrodynamical equivalent of the Schrodinger equation in the Stochastic Interpretation of the

3 Quantum Mechanics (SIQM) [8]. On this basis, it is possible to develop an interpretative scheme where the phenomenological predictions of SM coincide with that of quantum mechanics for all the experimentally measurable quantities. Within this interpretative code the SM is nothing but a quantization procedure, dierent from the ordinary ones only formally, but completely equivalent from the point of view of the physical consequences. Hence we interpret here the SM as a probabilistic simulation of quantum mechanics, providing a bridge between this fundamental section of physics and the stochastic dierential calculus. However it is well known that the most peculiar features of the involved stochastic processes, namely the transition probability densities, seem not always enter into this code scheme: in fact, if we want to check experimentally if the transition probabilities are the right ones for a given quantum state, we are obliged to perform repetead position measurements on the quantum system, so that, according to quantum theory, the quantum state changes in every measurement (wave packet reduction). On the other hand our transition probabilities are associated to a well dened wave function: hence it will be practically impossible in general to experimentally observe these transition probabilities. Several ways out of these diculties can be explored: for example stochastic mechanic scheme could be modied by means of non constant diusion coecients [1], or alternatively it would be possible to modify the stochastic evolution during the measurement [9]. Here we rather assume the standpoint of the convergence (in time) of the processes which do not satisfy the stochastic variational principle toward the processes associated to quantum states. In this way any departure from the distributions of quantum mechanics is quickly reabsorbed in the time evolution [1], at least in many meaningful cases. However the possibility is also considered here that the non standard evolving distributions can be realized by suitable quantum systems for modied, time dependent potentials which, on the other hand, asymptotically in time rejoin the usual potentials. Starting from [1] we will study the corrispondence between stochastic and quantum mechanics in several cases of stationary states, and in section we will in particular recall the well known technique of the eigenfunction expansion for Fokker-Planck equations in order to approach these problems in a general way in section 3. In section 4 we will discuss a few explicit examples of Fokker-Planck equations with singular drifts associated to particular wave functions of the quantum harmonic oscillator. Finally in section 5 we will briey discuss how it is possible to consider any classical evolution as produced by a suitable quantum system: a step instrumental to point out that also non extremal evolutions could be in principle produced by means of a suitable (time-dependent) deformation of the potential.. Eigenvalue problem for the Fokker-Planck equation Let us recall here (see for example [1] pag. 11) a few general facts about the pdf's (probability density functions) f(x; t) solutions of a one-dimensional Fokker-Planck equation of the t f x (vf) (Df)? vf (:1) 3

4 dened for x [a; b] and t t. Here D(x) and v(x) are two time independent functions such that D(x) > and v(x) has no singularities in (a; b); moreover they are both continuous and dierentiable functions. The conditions imposed on the probabilistic solutions are of course Z b a f(x; t) ; a < x < b ; t t ; f(x; t) dx = 1 ; t t ; and from the form of (.1) the second condition also takes the (Df)? vf a;b = ; t t : (:3) (:) Suitable initial conditions will be added to produce the required evolution: for example the transition pdf p(x; tjx ; t ) will be selected by the initial condition It is also possible to show by direct calculation that lim f(x; t) = f(x; t + ) = (x? x t!t + ) : (:4) h(x) = N?1 e?r [D (x)?v(x)]=d(x) dx ; N = Z b a e?r [D (x)?v(x)]=d(x) dx dx (:5) is an invariant (time independent) solution of (.1) satisfying the conditions (.). Remark that (.1) is not in the standard self-adjoint form (see [11] pag.114). However if we dene the function g(x; t) by means of f(x; t) = p h(x) g(x; t) (:6) it would be easy to show that g(x; t) obeys now an equation of the form where L dened by with L' = d dx p(x) = D(x) > ; q(x) t g = Lg (:7) p(x) d'(x) dx D (x)? v(x) 4D(x)?? q(x)'(x) ; (:8) D (x)? v(x) is now self-adjoint. Then, by separating the variables by means of g(x; t) = (t)g(x) we have (t) = e?t while G must be solution of a typical Sturm-Liouville problem associated to the equation ; (:9) LG(x) + G(x) = (:1) with the boundary conditions D (a)? v(a) G(a) + D(a)G (a) = ; D (b)? v(b) G(b) + D(b)G (b) = : (:11) 4

5 It easy to see that = is always an eigenvalue for the problem (.1) with (.11), and that the corresponding eigenfunction is p h(x) dened in (.5). For the dierential problem (.1) with (.11) we have that (see [11] pag. 138) the simple eigenvalues n will constitute an innite, increasing sequence and the corresponding eigenfunction G n (x) will have n simple zeros in (a; b). For us this means that =, corresponding to the eigenfunction G (x) = p h(x) which never vanishes in (a; b), is the lowest eigenvalue so that all other eigenvalues are strictly positive. Moreover the eigenfunctions will constitute a complete orthonormal set of functions in L? [a; b] (see [1] pag. 134). As a consequence the general solution of (.1) with (.) will have the form f(x; t) = 1X n= c n e?ntp h(x)g n (x) (:1) with c = 1 for normalization (remember that = ). The coecients c n for a particular solution selected by an initial condition are then calculated from the orthonormality relations as c n = and in particular for the transition pdf we have from (.4) that f(x; t + ) = f (x) (:13) Z b a f (x) p G n(x) dx ; (:14) h(x) c n = G n(x ) p h(x ) : (:15) Since = and n > for n 1, the general solution (.1) of (.1) has a precise time evolution. In fact all the exponential factors in (.1) vanish with t! +1 with the only exception of the term n = which is constant, so that exponentially fast we will always have p lim f(x; t) = c h(x)g (x) = h(x) ; (:16) t!+1 namely: the general solution will always relax in time toward the invariant solution h(x). 3. Fokker-Planck equations for Stochastic Mechanics We will now examinate the consequences of the previous remarks on the evolution equations of the SM, and for the sake of brevity in the presentation of the notation we will take a shortcut borrowed from the SIQM which, as it is well known, is formally ruled by the same dierential equations as the SM. Let us consider the (one dimensional) Schrodinger equation h ih@ t = ^H =? x + V (3:1) 5

6 for a time-independent potential V (x) which gives rise to a purely discrete spectrum and bound, normalizable states, and let us use the following notations for stationary states, eigenvalues and eigenfunctions: n(x; t) = n (x) e?ient=h ^H n =? h m n + V n = E n n (3:) For later convenience we will also introduce the constant D = h m (3:3) which will play the role of a diusion coecient when we will associate a stochastic process to every quantum state. As a consequence the previous eigenvalue equation can be recast in the following form D n = V? E n h n : (3:4) It is also well-known that for these stationary states the pdf is the time independent, real function With the Ansatz usual in SIQM (see for example [7]) f n (x) = j n (x; t)j = n(x) : (3:5) (x; t) = R(x; t) e is(x;t)=h (3:6) where R and S are real functions, the function R = j j comes out to be a particular solution of a Fokker-Planck equation (with constant diusion coecient and time-independent forward velocity eld) of the form t f = D@ x x(vf) x (D@ x f? vf) ; (3:7) v(x; t) = 1 xs + h x(ln R ) : (3:8) Remark that the explicit dependence of v on the form of R clearly indicates that to have a solution of (3.7) which makes quantum sense we must pick-up only one, suitable, particular solution. In fact in the stochastic mechanical framework the system is ruled not only by the Fokker-Planck equation (3.7), but also by the second, dynamical t S + (@ xs) m + V? x R m R = (3:9) namely the so-called Hamilton-Jacobi-Madelung equation deduced by separating the real and imaginary parts of (3.1) (see [7]). The analogy between (3.7) and the Fokker-Planck equation (.1) is more than formal in the sense that the SM (see []) shows how to recover both the equations (3.7) and (3.9) (namely the Schrodinger equation (3.1)) in a classical, dynamical stochastic context. This dynamics can also be connected to a stochastic variational principle (see [3]) which, as usual in the dynamical theories, select the actual trajectories of the system among all the virtually possible 6

7 evolutions. In particular from this standpoint the velocity eld v is not a given eld, but becomes a dynamical variable to be determined by the equations of the system. We will now x our attention on a given stationary solution n (x; t) and we will remark that in this case we have so that for our state the velocity eld is S(x; t) =?E n t ; R(x; t) = n (x) ; (3:1) v n (x) = D n(x) n (x) : (3:11) This means that now v n is time-independent and it possibly presents singularities in the zeros (nodes) of the eigenfunction. Since the n-th eigenfunction of a quantum system with bound states has exactly n simple nodes (see [11] pag. 138) that we will indicate with x 1 ; : : : ; x n, the coecients of the Fokker-Planck equation (3.7) are not dened in these n points and we will be obliged to solve it in separate intervals by imposing the right boundary conditions connecting the dierent sections. In fact these singularities eectively separate the real axis in n + 1 sub-intervals with impenetrable (to the probability current) walls. Hence the process will not have an unique invariant measure and will never cross the boundaries xed by the singularities of v(x): if we start in one of the intervals in which the axis is so divided we will always remain there (see [13]). As a consequence we must think the normalization integral (.) (with a =?1 and b = +1) as the sum of n + 1 integrals over the sub-intervals [x k ; x k+1 ] with k = ; 1; : : : ; n (where we understand, to unicate the notation, that x =?1 and x n+1 = +1). Hence for n 1 we will be obliged to solve the equation (3.7) in every interval [x k ; x k+1 ] by requiring that the integrals Z xk+1 x k f(x; t) dx (3:13) be kept at a constant value for t t : this value is not, in general, equal to one (only the sum of these n + 1 integrals amounts to one) and, since the separate intervals can not communicate, it will be xed by the choice of the initial conditions. The boundary conditions associated to (3.7) are hence imposed by the conservation of the probability in [x k ; x k+1 ] and that means the vanishing of the probability current at the end points of the interval: D@x f? vf = x k ;x ; k+1 t t : (3:14) To have a particular solution we must moreover specify the initial conditions: in particular we will be interested in the transition pdf p(x; tjx ; t ), which is singled out by the initial condition (.4), since (see [1]) the asymptotic approximation in L 1 among solutions of (3.7) is ruled by the asymptotic behavior of p(x; tjx ; t ) through the Chapman-Kolmogorov equation f(x; t) = Z +1?1 p(x; tjy; t )f(y; t + ) dy : (3:15) It is clear at this point that in every interval [x k ; x k+1 ] (both nite or innite) we can solve the equation (3.7) along the guidelines sketched in the Section 1 by keeping in mind that in [x k ; x k+1 ] 7

8 we already know the invariant, time-independent solution n(x) (or, more precisely, its restriction to the said interval) which is never zero in this interval with the exception of the extremes x k and x k+1. Hence, as we have seen in the general case, with the position f(x; t) = n (x)g(x; t) (3:16) we can reduce (3.7) to the t g = L n g (3:17) where L n is now the self-adjoint operator dened on [x k ; x k+1 ] by where we have now L n '(x) = d dx p(x) = D > ; p(x) d'(x) dx q n (x) = v n(x) 4D? q n (x)'(x) (3:18) + v n(x) : (3:19) To solve (3.17) it is in general advisable to separate the variables, we then immediately have (t) = e?t while G must be solution of the Sturm-Liouville problem associated to the equation with the boundary conditions L n G(x) + G(x) = (3:) DG (x)? v n (x)g(x) x k ;x k+1 = : (3:1) The general behaviour of the solutions obtained as expansions in the system of the eigenfunctions of (3.) has already been discussed in Section Harmonic oscillator states To see in an explicit way how the pdf's of SM evolve, let us consider now in detail the particular example of a quantum harmonic oscillator (HO) characterized by the potential It is well-known that its eigenvalues are E n = h! n + 1 V (x) = m! x : (4:1) ; n = ; 1; : : : (4:) while, with the notation the eigenfuncions are n (x) = = h m! ; (4:3) 1 p p n n! e?x =4 Hn x p (4:4) 8

9 where H n are the Hermite polynomials. The corresponding velocity elds are easily calculated and are for example v (x) =?!x ; v 1 (x) =! x?!x ; x v (x) = 4!?!x ; x? with singularities in the zeros x k of the Hermite polynomials. If we now keep the form of the velocity elds xed we can consider (3.7) as an ordinary Fokker-Planck equation for a diusion process and solve it to see the approach to the equilibrium of the general solutions. When n = the equation (3.7) takes the form t f x f +!x@ xf +!f (4:7) and the fundamental solution comes out to be the Ornstein-Uhlenbeck transition pdf where we used the notation p(x; tjx ; t ) = 1 (t) p e?[x?(t)] = (t) ; (t t ) (4:8) (t) = x e?!(t?t) ; (t) = 1? e?!(t?t ) ; (t t ) : (4:9) The stationary Markov process associated to the transition pdf (3.1) is selected by the invariant pdf f(x) = 1 p e?x = (4:11) which is also the asymptotic pdf for every initial condition when the evolution is ruled by (4.7) (see for example [1]) so that the invariant distribution plays also the role of the limit distribution. It is remarkable that this invariant pdf also coincides with the quantum stationary pdf = j j ; in other words, the process associated by the SM to the ground state of a quantum HO is nothing but the stationary Ornstein-Uhlenbeck process. For n 1 the solutions of (3.7) are no more so easy to nd and, as discussed in the previus section, we will have to solve the eigenvalue problem (3.) which, with = h, an be written as? h m m G (x) +! x? h! n + 1 G(x) = G(x) (4:1) in every interval [x k ; x k+1 ] with, k = ; 1; : : : ; n, between two subsequent singularities of the v n eld. The boundary conditions at the endpoints of these intervals are deduced from (3.14) through (3.16) and are [ n G? n G] x k ;x k+1 = (4:13) and since n (but not n) vanishes in x k ; x k+1, the boubdary conditions to impose are G(x k ) = G(x k+1 ) = (4:14) 9

10 where it is understood that for the conditions in x and x n+1 we respectively mean lim x!?1 G(x) = ; lim G(x) = : (4:15) x!+1 It is also useful at this point to give the eigenvalue problem in an adimensional form by using the new adimensional variable x= (which will still be called x) and the eigenvalue = =! = =h!. In this way the equation (4.1) with the conditions (4.14) becomes y x (x)? 4? n + 1? y(x) = y(x k ) = y(x k+1 ) = (4:16) where x; x k ; x k+1 are now adimensional variables. If m and y m (x) are the eigenvalues and eigenfunctions of (4.16), the general solution of the corresponding Fokker-Planck equation (3.7) will be f(x; t) = 1X x c m e?m!t n (x)y m m= : (4:18) Of course the values of the coecients c m will be xed by the initial conditions and by the obvious requirements that f(x; t) must be non negative and normalized (on the whole x axis) along all its evolution. Two linearly independent solutions of (4.16) are y (1) = e?x =4 M? + n ; 1 ; x ; y () = xe?x =4 M where M(a; b; z) are the conuent hypergeometric functions.? + n? 1 ; 3 ; x ; (4:) We consider now the case n = 1 (x =?1, x 1 = and x = +1) so that (4.16) will have to be solved separately for x and for x with the boundary conditions y() = and lim y(x) = lim y(x) = : (4:1) x!?1 x!+1 The eigenvalues are m = m with m = ; 1; : : : and the complete set of eigenfunctions is given by 3 y m (x) = xe?x =4 x M?m; ; = (?1)m m! xp p e?x =4 H m+1 : (4:3) (m + 1)! In fact this is the form of the eigenfunctions for both x and x. In particular it is easy to see that y (x) = xe?x =4 and its relation with the quantum eigenfunction 1 is (4:4) 1 (x) = y (x= ) p p ; (4:5) and the general solution of (3.7) with v = v 1 is f(x; t) = 1X m= c m e?m!(t?t) 1 (x)y m (x= ) : (4:8) 1

11 To determine the c m 's we must now impose an initial condition: in particular, the initial condition for the transition pdf's requires As a consequence we will have p(x; tjx ; t ) = 1(x) xx c m = 1X m= A long calculation shows that this series can be summed up to (m + 1)!! y p m (x = ) p (m)!! y (x = ) : (4:3) e?m!(t?t) x m (m + 1)! H x m+1 p H m+1 p : (4:33) p(x; tjx ; t ) = x e?[x?(t)] = (t)? e?[x+(t)] = (t) (t) (t) p (4:35) where (t) and (t) are dened in (4.9). It must be remarked still once that the form (4.35) of the transition pdf must be considered as restricted to x when x > and to x when x <, and that only on these intervals it is suitably normalized. In order to take into account at once both these possibilities we can also introduce the Heavyside function (x) so that for every x 6= we will have p(x; tjx ; t ) = (xx ) x e?[x?(t)] = (t)? e?[x+(t)] = (t) (t) (t) p : (4:36) This completely solves the problem for n = 1 since from (3.15) we can now deduce also the evolution of every other initial pdf. In particular it can be shown that = lim p(x; tjx ; t ) = (xx ) x e?x p = (xx ) t!+1 1(x) ; (4:37) 3 and hence, if f(x; t + ) = f (x) is the initial pdf, we have for t > t lim f(x; t) = lim t!+1 t!+1 = 1(x) Z +1?1 Z +1?1 p(x; tjy; t )f (y) dy (xy)f (y) dy =?(q; x) 1(x) ; (4:38) where we have dened the function?(q; x) = q(x) + (? q)(?x) ; q = Z +1 f (y) dy : (4:39) Remark that when q = 1 (namely when the initial probability is equally shared on the two real half-axis) we have?(1; x) = 1 and the asymptotical pdf coincides with the quantum stationary pdf 1(x); if on the other hand q 6= 1 the asymptotical pdf has the same shape of 1(x) but with dierent weights on the two half-axis. 11

12 If nally n = we have x =?1, x 1 =?1, x = 1 and x 3 = +1, and the equation (4.16) must be solved in the three intervals (?1;?1], [?1; 1] and [1; +1). The two linearly independent solutions are now y (1) = e?x =4 M? + ; 1 ; x ; y () = xe?x =4 M? + 1 ; 3 ; x ; (4:4) and it is easy to verify that = is an eigenvalue for all the three intervals with eigenfunction y (x) = e?x =4 M?1; 1 ; x = e?x =4 H xp so that the relation with the quantum eigenfunction is = e?x =4 (x? 1) (4:41) (x) = y (x= ) p p : (4:4) 8 As for the other eigenvalues and eigenfunction they are not too easy to nd so that a complete analysis of this case has still to be elaborated. A few indications can be obtained numerically: for example it can be shown that, beyond =, the rst eigenvalues in the interval [?1; 1] can be calculated as the rst values such that M? + 1 ; 3 ; 1 = (4:5) and are 1 7:44, 37:6, 3 86:41. Also for the unbounded interval [1; +1) (the analysis is similar for (?1;?1]) the eigenvalues are derivable only numerically. 5. Conclusions We want to conclude this paper by remarking that the processes discussed in the previous sections, namely the processes solutions of (3.7) but not associated to quantum mechanical states solutions of (3.1) (in other words, the processes that do not satisfy either the stochastic variational principle [3] or the Nelson dynamical equation []), are not without a deep relation with the quantum mechanics. In fact we will show here that to every solution f(x; t) of a Fokker-Planck equation (.1), with given v(x) and constant diusion coecient (3.3), it is possible to associate the wave function of a quantum system with a suitable time-dependent potential. This means in practice that even the virtual (non optimal) processes discussed in this paper can be associated to proper quantum states, namely can be made optimal albeit with a potential which is dierent from the V (x) of (3.1). Let us take a solution f(x; t) of the Fokker-Planck equation (.1), with given v(x) and constant diusion coecient (3.3): if we dene the functions R(x; t) and W (x) by p R(x; t) = f(x; t) ; v(x) = W (x) ; (5:1) 1

13 and we remember that [] the following relation holds mv x S + h ln R x S + h ln f (5:) if S(x; t) is supposed to be the phase of a wave function as in (3.6), we immediately get S(x; t) = mw (x)? h ln f(x; t)? (t) (5:3) which allows us to determine S from f and v (namely W ) up to an arbitrary function of the time (t). However, in order that the wave function (3.6) with our R and S be a solution of a Schrodinger equation, we must be also sure that the Hamilton-Jacobi-Madelung equation (3.9) be satised. Since S and R are now xed, this equation (3.9) must be considered as a relation dening the new potential which, after a short calculation, becomes V (x; t) = h x ln f + h (@ t ln f + v@ x ln f)? mv + _ : (5:4) Of course if we start with a quantum wave function for a given potential and if we pick up as a solution of (3.7) exactly f = R the formula (5.4) will correctly give back the initial potential, as can be seen for the ground state and the rst excited state of the harmonic oscillator which (by choosing respectively (t) = h!t= and (t) = 3h!t=) give as result the usual harmonic potential (3.1). On the other hand let us consider now the (non stationary) fundamental solution (4.8) associated to the velocity eld v (x) of (4.6) for the case n = of the harmonic oscillator (we put t = to simplify the notation): a short calculation shows that, by choosing _(t) = h! (t)? 1 = h! 1 tanh!t! h! ; (t! +1) ; (5:5) we get the time-dependent potential V (x; t) = h! x? (t) (t) (t)? m! x! m! x ; (t! +1) : (5:6) Of course the fact that for t! +1 we recover the harmonic potential is associated to the fact, aleady remarked, that the usual quantum pdf (x) is also the limit distribution for every initial condition. As for the case n = 1, with v 1 (x) from (4.6) and the transition probability (4.36) as given non-stationary solution, the calculations are lenghtier. However if we dene F (x; t) = e?[x?(t)] = (t) (t) p T x(t) (t) = x(t) (t) ; G(x; t) = e?[x+(t)] = (t) (t) p ; (5:7) F (x; t) + G(x; t) F (x; t)? G(x; t) ; T (x) = x tanh x ; (5:8) and if we choose _(t) = h! 4 (t)? (t)? 1! 3 h! ; (t! +1) (5:9) 4 (t) 13

14 we have as time dependent potential for every x 6= V (x; t) = m! x 4? 1 + h! 4! m! x ; (t! +1) : 1? T x? h 4mx h x i 1? T (5:1) In this case the asymptotic potential is the usual harmonic potential, but we must consider it separately on the positive and negative x semi axis since in the point x = a singular behaviour would show up. This means that, also if asymptotically we recover the right potential, this will be associated with new boundary conditions in x = since we will be obliged to keep the system bounded on the positive (for example) semiaxis. These simple examples show that we can always design a sutable (time dependent) potential which realizes arbitrary diusive evolutions of classical systems. This has more than a purely formal interest since it can allow one, for instance, to produce a quantum evolution between two stationary states of a system by means of a deformation of the original potential. Since also the measurement processes (which can not be described along a time evolution in the usual formalism of quantum mechanics) can be considered as a transition between two states, these techniques seem to indicate a way to simulate a continuous reduction of a wave packet controlled by a time dependent potential. A subsequent paper will be devoted to a discussion of this point which, however, will require a generalization of the results on the solutions of the Fokker-Planck equation to the case of time dependent velocity elds since in this case we will have to do with non stationary quantum wave functions. REFERENCES 1. N.Cufaro Petroni and F.Guerra: Found.Phys. 5 (1995) 97; N.Cufaro Petroni: Asymptotic behaviour of densities for Nelson processes, in Quantum communications and measurement, V.P.Belavkin et al. Eds., Plenum Press, New York, 1995, p. 43; N.Cufaro Petroni, S.De Martino and S.De Siena: Non equilibrium densities of Nelson processes, in New perspectives in the physics of mesoscopic systems, S.De Martino et al. Eds., New Scientic, Singapore, 1997, p E.Nelson: Phys.Rev. 15 (1966) 179; E.Nelson: Dynamical Theories of Brownian Motion (Princeton U.P.; Princeton, 1967); E.Nelson: Quantum Fluctuations (Princeton U.P.; Princeton, 1985); F.Guerra: Phys.Rep. 77 (1981) F.Guerra and L.Morato: Phys.Rev. D7 (1983) 1774; F.Guerra and R.Marra: Phys.Rev. D8 (1983) 1916; F.Guerra and R.Marra: Phys.Rev. D9 (1984) D.Bohm and J.P.Vigier: Phys.Rev. 96 (1954) N.Cufaro Petroni: Phys.Lett. A 141 (1989) 37; N.Cufaro Petroni: Phys.Lett. A 16 (1991) 17; N.Cufaro Petroni and J.P.Vigier: Found.Phys. (199) 1. 14

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