Position probability density function for a system of mutually exclusive particles in one dimension

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1 Position probabiity density function for a system of mutuay excusive partices in one dimension Rasoo Kheiry Department of physics, Isfahan University of Technoogy, Isfahan , Iran Shahram Saehi (Dated: May, 4) Abstract Position probabiity distribution of a set of massy mutuay excusive partices in one dimension has been defined. Exampes with a given two mutuay excusive partices system are considered. It is emphasized that quantum partices at finite potentias can not be regarded as a mutuay excusive system or they are indistinguishabe. Afterward it is attempted to ascribe a mutuay excusive system to continuous mass densities of a rigid body to cacuate average vaues. We do this by appying correspondence principe with regard to probabiity densities. It is found interesting to divide quantum probabiity density by that of cassica, both for the same potentia. On cacuating, computationa anguage Mape is being used. Finay it is argued that such systems are incuding semicassica partices. This agrees with investigations that emphasizes; statistica nature of wave function in its cassica imit shoud sette in an ensembe instead of one partice. I. INTRODUCTION From beginning years of formuation of quantum mechanics, there were many admired strugges that tried to approximate the nove quantum notion to the od cassica sense. Among them are Bohr and great correspondence principe, Ehrenfest with bod cassica path, Wigner with a genius distiribution function for position and mpmentum, 3 Dirac with his anaogy 4 and go on. These papers sti are foowed, for exampe and a ot ese.inversey some instructive papers have attempted to buid a probabiistic structure for inherenty predetermined variabes of cassica mechanics to compare with essentiay statistica quantum observabes. One use of this comparison is pedagogica, as Curtis and Eis have mentioned in, 8 introductory physics course utiize historica Newtonian approach which is identified by instantaneous vaues for position, speed, and acceeration, whereas quantum course by conceptuay probabiistic observabes. Aso this approach brings us remarkabe comparisons in semicassica state by appying correspondence principe; squared wave function { ψ n (x) } and cassica position probabiity density p c (x) of one partice for arge n must become indistinguishabe. 9 This point of view considered mainy by Robinett with severa exampes. Further, with not ony one partice, it has been proposed that, since a quantum mechanica wave function inherits an intrinsic statistica

2 behavior, its cassica imit must correspond to a cassica ensembe not an individua partice. This approach has been mentioned by Baentine et a They demonstrated that the cassica imit of quantum mechanics arising from Ehrenfest s theorem is neither necessary nor sufficient to identify the cassica regime; The cassica imit of a quantum state is not a singe cassica orbit, but an ensembe of orbits. This statement has taken into consideration, Boivar has mentioned this and aso brings severa objections on WKB semicassica method and works on another cassica imit for bosons 5 and set up a nove cassica imiting process ħ. 6 In an iuminating paper, Huang 7 drew two concusions: (i) A wave function does not describe an individua partice but a cassica pure ensembe. (ii) Given a quasicassica wave function, we can te which cassica pure ensembe is described by it.jin this presentation after remembering position probabiity density for one partice in one dimension, the concept of mutuay excusive systems incuding isoated partices is defined. We continue with exempification about discrete partices in some probabiity densities and accentuate that quantum partices can not be regarded as a mutuay excusive system in finite (rea) potentias which refers to indistinguishabity. Afterward it has been attempted for repacing continues mass densities of rigid bodies by mutuay excusive systems by means of a test function with severa zeroes and further by correspondence principe. It foows necessariy that, we must make a remarkabe function as ψ n (x) p c (x) in a given potentia for our purpose. We've made some anaytic and numerica probems with this point of view for cacuating moments. At ast it concuded that regained probabiity density in this way is associated with semicassica partices. ӀӀ. REVIEW OF POSITION PROBABILITY DENSITY FUNCTION FOR ONE PARTICLE Consider a cassica partice of mass m moving in a one dimensiona finite region with returning points. If the veocity of the partice at point x be denoted by v(x), the time it spends near the x and hence the probabiity of our catching it there during a random spot check, varies inversey with v(x) 9 p(x) v(x) E U, () where p(x) stands for position probabiity density function and U is a of the non-kinetic energies with tota amount of E. By normaizing p(x) = τ v(x), () where τ is the whoe time partice once sweep the route. Another we defined approaches coud be p(x) = δ[x x(t)], or p(x) = dt τ as mentioned in. 8, Aso probabiity density means for normaized quantum wave function ψ(x) of a partice as p(x) = ψ(x). The average vaues, for position x, are evauated by x k = dx p(x) x k, (3) where k = gives normaization formua.

3 III. SYSTEMS INVOLVING MORE THAN ONE PARTICLE Here we wi deiberatey ascribe one variabe x to either one partice or one system for position probabiity density in one dimension, instead of p(x, x,, x N ) refers to joint probabiity density of N partices. This change can be usefu for cacuating average vaues for a system as one body. III- Dirac Deta Density Consider N point masses m i in stationary state ocated at x i, then Now if joint position probabiity density of this system is defined as p i (x) = δ(x x i ). (4) Then we have p(x) = N i m iδ(x x i ) N i m i (5) x k = n m k i ix i n. (6) m i i III- One Given Two Partices System In this part, probems invoved a given system incuding tow partices in one dimension is being regarded and different cassica and quantum position probabiity densities have been studied. (i) Cassica partices m and m moving in their apart domain with constant veocity in the boxes of ength (β i α i ), i=,. The probabiity density can be written as foow By normaizing, p i (x) = τ i v i = constant. (7) p i (x) = β i α i. (8) A specia case is iustrated in Fig.. 3

4 m m b a a b x FIG.. Tow given partices at an specia configuration with masses m and m. The position probabiity density of this system wi be m b a p(x) = m m + m b a { b < x < a, a < x < b. (9) The average vaue is as foow x = [(m m + m m ) a + m b m b ]. () If b = b, then x = [(m m ) (m + m )] (a + b) or if b = b and m = m, then x =. It is remarkabe that for unsymmetrica high veocities x wi be changed according to specia reativity. On the other hand, this resuts check against the eementary resuts, for exampe two point masses m and m ocated at x = (a + b) and x = (a + b ), the position of centra mass is given by x cm = m x + m x m + m = [(m m + m m ) a + m b m b ]. () (ii) The probabiity density of quantum partices m and m in the infinite wes of ength simiar to Fig. wi be m nπ b a sin (x + b) b < x < a, b a p(x) = m + m m nπ b a sin (x a) a < x < b. b a { () and the first moment is 4

5 x = which is equas to x in the cassica conditions. [(m m + m m ) a + m b m b ], (3) (iii) Another probabiity density coud be that of cassica harmonic osciator with ampitude x. For one partice centered at origin, p singe (x) = π x x, (4) see references. 9, In the configuration of Fig. we have, p(x) = m + m π { x m b x a, x [x + (a + b) ] m a x b. [x (a + b ) ] (5) where x = (b a) and x = (b a). The centra mass is again equa to Eq. (). (iv) Now it is time for exit from the box and have a gance to infinity but normaized densities the same as quantum harmonic osciator. Compare with Fig., two partices of masses m and m, for ease in the ground state, centered on x = (a + b) and x = (a + b ). p(x) = m + m π [m α e α (x+x ) + m α e α (x x ) ], α i = m iω i ħ, i =,. (6) Once again Eq. () coud be checked. Evidenty every arbitrary normaized function can be used to make position probabiity density for cacuating moments x k of such systems. Two pots with m m = and ω ħ = ω ħ = are given by Fig. a and Fig. b. Of course this density is not exacty in agreement with two mutuay excusive partices so the first important resut is that quantum partices in finite potentias can not be as a mutuay excusive system, simiary, they are indistinguishabe. This consequence is an outcome from the fact that in finite (rea) potentias, wave functions vanish ony at infinity. Aso, even if rea partices interact with each other, potentia is finite. Second consequence is that probabiity density of a quantum system such as Eq. (6) depends on mass of individua partices bring about coefficient α, whereas probabiity density of a mutuay excusive system depends just on the ratio of masses of partices invoved. We use these resuts in next section. 5

6 Probabiity Density (in SI inverse meter) Probabiity Density (a).6.5 (b) Position (in SI meter) Position FIG.. Two quantum harmonic osciators both in ground state with α = and α =, (a) centered on x= - and x=, (b) centered on x = - and x =. IV. MUTUALLY EXCLUSIVE PARTICLES IN PLACE OF CONTINUOUS DISTRIBUTIONS Here we attempt to find a certain function as joint probabiity density for N mutuay excusive partices which in average vaues x k, tends to a certain continuous mass density of a hypothetica soid body as N becomes greater. We start utiizing a test function with severa zeros to buid a mutuay excusive system. Then we use correspondence principe, wherein wave functions be regarded as test functions. Finay we discuss what kinds of partices are invoved in this probem. IV- Making use of a positive periodic test functions with constant ampitude which has severa zeros on a given interva mutipied by continuous mass density. Such a satisfactory (not piecewise) function is that of a sinusoida function on the interva [, ]. Such osciating function can break the continuous mass density into severa sices reated to number of zeroes. Hence joint position probabiity density incuding N partices can be written as p(x; n) = C λ(x) sin nπ x, (7) where λ(x) denoting inear(one dimensiona) mass density and the domain of p(x; n) is equa to the domain of λ(x). In which foows there are some exampes. (i) A bar of ength with uniform mass density λ(x) = A and p(x; n) = sin nπ x (8) x n =. (9) 6

7 Probabiity Density Probabiity Density Probabiity Density Here there are N = n partices, as shown in Fig. 3a. and domain interva of i-th partice is ((i ) n, i n). (ii) A bar of ength with inear mass density λ(x) = Ax and p(x; n) = 4 x sin nπ x () x n = 3 [ 3 n π ], im x n = n 3. () (a) (b) (c) Position Position Position FIG. 3. (a) Uniform (b) Linear (c) Paraboic, mass density of five discrete partices of a unit ength bar iustrated by the squared sinusoida test function or infinite we s squared wave function. Masses become heavy through ine, see Fig. 3b. We see that p(x; n) does not depend on the coefficient A, which conforms to mutuay excusive system s characteristic expained above about the dependence of probabiity density on the ratio of masses invoved. It is obvious that, even for a few partices, centra mass tends to continuum hypothesis. (iii) A bar of ength with paraboic mass density λ(x) = Ax and nπ p(x; n) = 3 ( 6 x sin x () 4n π ) x n = 3 4 [ 3 n4 π 4 3 n π ], im x n = 3 n 4. (3) See Fig. 3c. Foowing this term we can see that im n xr sin nπ x dx = xr dx = x r+ r +. (4) So if λ(x) = Ax r, then im x n = r + n r +. (5) 7

8 Generaizing this consequence to every anaytic function f(x) in the interva (, ) by using Tayor series gives rise to f(x) = a r x r, (6) r= im f(x) nπ n sin x dx = f(x) dx. (7) Considering f(x) = x k λ(x), (8) one can write im n xk n = im x k Cλ(x) sin nπ n x dx = xk λ(x) dx. (9) λ(x) dx IV-. Since the soutions of the Schrödinger equations fuctuate in their domain, it might be not incorrect, in restricting conditions, ascribing them as test functions with severa zeros to make mutuay excusive partices. According to correspondence principe, squared wave function { ψ n (x) } and cassica position probabiity density p c (x) of one partice for arge n must become indistinguishabe. 9 One forma statement might be x x im ψ n(x) dx = p c (x) dx. (3) n x x This principe shoud works for averages too, namey in position space x x im n xk ψ n (x) f ψ n (x) dx = x k f(x) p c (x) dx, (3) x x provided that normaization be appied. Taking then λ(x) f(x) p c (x), (3) 8

9 im n xk C ψ n (x) f ψ n (x) dx = xk λ(x)dx λ(x)dx. (33) If we were ooking for x k f(x), normaization of wave function was satisfactory, but here we are ooking for x k, hence the functions ψ n (x) f ψ n (x) shoud be normaized, this is the reason of appearing C in the above equation. In this way, with some excusive conditions expained beow, a satisfactory function with severa zeros reated to n can be obtained for ascribing N partices to a certain mass density. At first, obtaining correct moments x k in arge numbers, offers p(x; n) = C ψ n (x) f ψ n (x). (34) But not a we-behaved functions as Eq. (34) coud be probabiity density for our purpose. The first condition that restricts suitabe functions is normaization and the second one is that it must be absoutey positive. + p(x; n) dx =, p(x; n). (35) The third condition is about domain (D); probabiity density must not exist out of continuous mass density domain: D p(x;n) = D λ(x), or p(x; n) dx D λ(x) =. (36) Forth condition makes p(x; n) suitabe for ascribing compete partices to it, that is about cassica returning points ±, where >. We shoud have im x p(x; n) = im p(x; n) =. (37) + x These conditions have the foowing resuts: - In the most genera situation operators produce abnorma functions or negative vaues and functions with domains unreated to mass density. For instance eading to P = iħ d dx, ψ n(x) sin nπ x, (38) ψ n (x) P ψ n (x) = iħψ n (x) d ψ n(x) dx 9 sin nπ x. (39) This is ceary not a normaizabe function, so it is not a suitabe candidate for probabiity density. Therefore, for generaization, f ought to be a positive function of x(i.e. not every operator) in mass density domain at both sides of the Eq. (3). That is

10 p(x; n) = C f(x) ψ n (x), x D λ(x), f(x). (4) - It seems straightforward to make a probabiity functions from one specific case associated with λ(x) =constant, as then f(x) p c (x), (4) p(x; n) = C ψ n(x) p c (x). (4) One can see that: For arge n, the function ψ n (x) p c (x) made of a given potentia is a confined function in the domain of p c (x), which is approximatey periodic with neary constant ampitude. This assertion is cear from the figures. In the genera case of mass density we have p(x; n) = C λ(x) ψ n(x) p c (x). (43) This suggestion impicity et us normaize p(x; n) in the domain of mass density by using the divergence property of p c (x) constant at returning points, so that it desired for thinking of partices and this naturay gives Eq. (37). Aso when p c (x) =constant, Eq. (37) simpy arises from continuity, because ψ n (±) =. 3- This probabiity density is suitabe with the exceptions in which the initia domain of p(x; n) produced by the presence of expicit form of ψ n (x) p c (x) is changed, such as λ(x) p c (x). (44) This resut does not introduce in the case of uniform cassica probabiity density, since it does not impose additiona properties over domain of p(x; n). 4- In the case of non symmetric mass densities with respect to origin, that is Condition 4 turns into λ(x) [x, x ], x x rather than [, ], (45) im p(x; n) = im p(x; n) =. (46) x x+ x x This point especiay woud be important if λ(x ) or λ(x ). In this case, the chosen wave functions shoud be zero at x and x. We iustrate this in the foowing exampes. IV-III Exampes

11 Probabiity Density Probabiity Density (i) Drawing out discrete partices from an infinite we as < x <, U(x) = { otherwise., p c (x) = constant, { ψ n + (x) cos ( nπx ) for odd n, ψ n (x) sin ( nπx ) for even n. (47) With a mass density in the interva [, ], we have p + (x; n) = Cλ(x) cos ( nπx ) for odd n, { p (x; n) = Cλ(x) sin ( nπx (48) ) for even n. Figure 4a with n = 9 and Fig. 4b. with n = show partices for uniform mass density. (a) (b) Position Position FIG. 4. Partices iustrated by the symmetric infinite we, λ [,]. (a) n=9, (b) n=. According to resut 4, if one wants to buid the probabiity density p(x; n) in the interva [, ] and we have the condition λ(), then ψ() must be zero. Then p(x; n) = Cλ(x) sin nπ x, n =,,3, (49) im n xk n = im x k Cλ(x) sin nπ n x dx = xk λ(x) dx λ(x) dx. (5) It is identica to Eq. (9). This means that, a things there have ony been one specific resut of corresponding principe. (ii) Consider the harmonic osciator potentia U(x) = mω x. (5)

12 Soving Schrodinger equation and energy equation yieds ψ n (x) e mω ħ x H n [( mω ħ ) x], pc (x) where H n (x) are Hermit Poynomias of degree n. For constant λ we have x x, (5) The energy reation is p(x; n) = C(x x ) e mω ħ x {H n [( mω ħ ) x]}. (53) Thus E n = (n + ) ħω = mω x. (54) p(x; n) = Cλ(x)(x x ) x (n+) e x {H n [(n + ) x ]}. (55) x The corresponding principe guarantees that im n xk n = xk λ(x) dx λ(x) dx. (56) Figure 5a and Fig. 5b. iustrate p(x; 9), p(x; ) respectivey for a constant symmetric continuous mass density in the interva (,). The number of partices becomes N = n +. Athough probabiity density of ending points do not exist, but since probabiity in one singe point becomes zero it may sound not to be important to ascribe p(x; n) to the λ [, ]. In the interva (,), even wave functions may not be suitabe, because p(x; n) isn t zero at the origin. Figure 5c exhibits one correct probabiity density refer to this interva for uniform mass density and the number of partices for odd n is N = (n + ).

13 FIG. 5. Partices iustrated by the harmonic osciator. (a) λ(x) = A and n=9 ( discrete partices). λ(x) = A and n= ( discrete partices). (c) λ(x) = A and n=9 (5 discrete partices). (d) λ(x) = Ax and n=9 (5 discrete partices). (e) λ(x) = Ax and n= (6 discrete partices). (f) λ(x) = Ax and n=9 (5 discrete partices). Other comparative exampes for λ [, ] are λ(x) = Ax r p(x; n) = C x r (x x ) x (n+) e x {H n [(n + ) x ]}. (57) x Figure 5d and Fig. 5e. iustrate p(x; 9) and p(x; ) respectivey for inear mass density (r = ). Figure 5f iustrates p(x; 9) for paraboic mass density (r = ). (Figures 5d, 5e and 5f are for a bar with unit ength). According to resut 3 densities such as λ(x) = A x, x or A ± x can not be presented by harmonic osciator as discrete partices. However, correspondence principe sti is true for averages by ascribing an imaginary probabiity density p im (x; n) to them. For instance, et (x) = A x, where again stands for ength of a bar ocated at positive axes of x from zero to. Then and f(x) = constant, (58) p im (x; n) = ψ n (x) = (n + ) π n n! 3 e (n+)x {H n [(n + ) x ]}. (59) We notice that wave functions have normaized in the interva [,). The moments are as foow

14 x k = k (n + ) k π n n! yk e y [H n (y)] dy, y = (n + ) x (6) where x k n = C n,k ( n ) k, (6) C n,k = There is a usefu tabed integra 8 k (n + ) π n n! yk e y x m e αx dx = [H n (y)] dy. (6) Γ[(m + ) ] α m+, (63) where Γ(x) is denoting the Gamma function. The coefficients C n,k can be written by a summation of these integras. Athough in this way mean vaues are attainabe, they woudn t be for genera term in Eq. (63). One can, instead, see that for even numbers k, integrand is an even function. Hence (n + ) m C n,m = π n n! y m e y [H n (y)], k = m, m integer. (64) They can be evauated using orthogonaity and recurrence reations e x H n (x)h m (x)dx = π n n! δ n,m, H n+ (x) = x H n (x) n H n (x). (65) For exampe see, 9 then we can write x e x H n (x)h m (x)dx = n π (n + )n! δ n,m + n π (n + )! δ n+,m + n π n! δ n,m. (66) Thus x n = ( n). (67) Substituting recurrence reation in the reated integra for x 4 n and using Eq. (66) resut in 4

15 x 4 n = 3 n + n + (n + ) ( n) 4, im x 4 = 3 n 8 ( n) 4. (68) In this form a even averages can be evauated. Aso a way is offered for the specia case of centra mass by using derivative formuas 8 d dx [e x H n (x)] = e x H n+ (x), d dx H n(x) = nh n (x) (69) and recurrence reation x e x [H n (x)] dx = e x H n+ (x)h n (x)dx + n e x H n (x)h n (x)dx. (7) Appying derivative formuas and integrating by parts, we have x e x [H n (x)] dx By knowing that n = [H n()] + n! k (n k)! [H (n k)()] k= n = [H n()] + n! k (n k)! [H (n k)()]. (7) k= And (n)! H n () = ( ) n, H n! n+ () = (7) n k= [(n k)]! [(n k)!] k = (n + )! (n!), (73) eading to x e x [H n (x)] dx = [(n)! ] n! n + (n)! k= [(n k)]! [(n k)!] k and = [(n)! ] n! + (n + ) [ (n)! ] = (n + n! ) [H n()] (74) 5

16 x e x [H n+ (x)] dx = (n + )! Now the coefficients can be written as n k= n = (n + )! k= (n + )! = [ ] n! [(n k)]! [(n k)!] k+ [(n k)]! [(n k)!] k = (n + ) [H n ()]. (75) C n, = π (n + ) (4n + ) (n)! n (n!), (76) C n+, = π (n + ) (4n + 3) (n)! n (n!) (77) With the hep of Mape, we have thus im n n (n)! n (n!) = π, (78) im C n, = im C n+, = n n π n (n)! im n (n!) = π = x c. (79) n (iii) Another bound potentia referred to a finite mass density is that of inear potentia U(x) = F x E c = F. (8) Reference [] has a brief but satisfactory discussion of this potentia (for eaborated soution one can see for exampe and ) x σ ψ(x) Ai ( ρ ) for x >, p c(x) x, (8) where Ai referred to one of the two soutions of Airy differentia equation (another soution that diverges at positive axis is Bi) and 6

17 σ = E F, 3 ħ ρ = ( mf ). (8) Because of the symmetry, we have E n (±) = yn (±) ρf, (83) where pus and minus signs are referred to even and odd wave functions respectivey and zeros of the A i(y) and Ai(y) are y n ( ) and yn (+) respectivey. We note that zeroes of A i(y) are identica to zeroes of Bi(y). Hence and ψ (±) (±) x F n (x) Ai [y n ( )], x > (84) C p(x; n) = λ(x)( (±) x F x ) {Ai [y n ( )]} C { λ(x)( x ) {Ai [y (±) x F n ( )]} x <, x. (85) Pots of uniform mass density in the interva [,] with F= are Fig. 6a and Fig. 6b. Once again in the interva [,] even wave functions, i.e. ψ n (+) (x), wi not be suitabe for every non zero mass density at the origin. Figure 6c, Fig. 6d. and Fig. 6e. are partices for constant, inear and paraboic mass density in the interva [,] respectivey. and p(x; n) = C λ(x)( x ) ( ) x F {Ai [y n ( )]}, λ [,], (86) p(x; n) = C λ(x)( x ) {Ai [y (+) x F n ( )]}, λ [,], λ() =. (87) 7

18 FIG. 6. Partices iustrated by the symmetric inear potentia. (a) λ(x) = A and (y 5 ( ) ). (b) λ(x) = A and (y5 (+) ) (c) λ(x) = A, in the interva [,] and (y 5 ( ) ). (d) λ(x) = Ax and in the interva [,] and (y5 ( ) ). (e) λ(x) = Ax in the interva [,] and (y 5 ( ) ). IV-IV. NUMERICAL SOLUTIONS In order to show vaidity of Eq. (56), we've soved some integra numericay with Mape reated to centra mass of our system with λ(x) = Ax utiizing Harmonic osciator and inear potentia. 8

19 TABLE I: Mape cacuating of centra mass for a bar of entgh =, λ(x) = Ax with utiizing harmonic osiator potantia. Continuous distribution gives x = 3. state x n = x p(x; n) dx amount n= evaf ( x. x. e.x. (H(,. x)) dx. x. x. e.x. (H(,. x)) dx ).539 n= evaf ( x. x. e 3.x. (H(, 3. x)) dx. x. x. e 3.x. (H(, 3. x)) dx ).693 n= evaf ( x. x. e.x. (H(,. x)). x. x. e.x. (H(,. x)) dx) dx n= evaf ( x. x. e 4.x. (H(, 4. x)). x. x. e 4.x. (H(, 4. x)) dx) dx.6798 TABLE II: Cacuating of centra mass for a bar of entgh =, λ(x) = Ax utiizing inear potantia. Continuous distribution gives x = 3. state y ( ) y ( ) x n = x p(x; n) dx evaf (. x. x. (AiryAi( AiryAiZeros(). (x ))) dxdx) x. x. (AiryAi( AiryAiZeros(). (x ))) dx evaf (. x. x. (AiryAi( AiryAiZeros(). (x ))) dxdx) x. x. (AiryAi( AiryAiZeros(). (x ))) dx amount y 3 ( ) evaf (. x. x. (AiryAi( AiryAiZeros(3). (x ))) dxdx) x. x. (AiryAi( AiryAiZeros(3). (x ))) dx y 5 ( ) evaf (. x. x. (AiryAi( AiryAiZeros(5). (x ))) dxdx) x. x. (AiryAi( AiryAiZeros(5). (x ))) dx.673 ( ) y 5 evaf (. x. x. (AiryAi( AiryAiZeros(5). (x ))) dxdx) x. x. (AiryAi( AiryAiZeros(5). (x ))) dx.67 9

20 V. DISCUSSION AND CONCLUSION One may ask: Which entity beongs to probabiity density of p(x; n) or what kind of partices it refers to, cassica or quantum? This matter is discussed here according to consequences at the end of section III. We see that p(x; n) is the same for a of the coefficients of a certain mass density. It means that joint probabiity density depends on the ratio of masses invoved, rater than, the mass of each partice individuay, as occurred in quantum cases. Therefore p(x; n) do not show quantum mechanica partices. Aso we saw that none of discrete systems in finite potentias are reated to quantum mechanica partices. Hence, if information about the ocations of partices invoved is compete, every discrete system wi be cassica and it is semicassica if the oca information be a finite interva. We thus find out that p(x; n) can be regarded as position probabiity density of semicassica partices from the fact that we have a confined probabiity density for one partice in contrast with quantum partices in finite potentias, on the other hand we have one peak in the center of probabiity density s domain in contrast with cassica position probabiity distributions in bound potentias. Therefore, if we want to ascribe our figures to a system of partices, the ony candidate wi be semicassica as it can intuitivey be understood from using correspondence principe and the form of ψ n (x) p c (x). As it expected, when n (or N) becomes arge the interva for one partice becomes smaer and its oca information increase. Hence p(x; n) shows an ensembe of semicassica partices The imit of p(x; n) as n shows continues mass density incuding infinite cassica point partices It is in agreement with researches which pointed out that; since a quantum mechanica wave function inherits an intrinsic statistica behavior, its cassica imit must correspond to a cassica ensembe. Through a efforts for repicate quantum oose tissue on cassica soid edges or vice versa, we suggest a dua nature function as ψ n (x) p c (x) which can be checked in the high quantumnumber imit (n ), besides other cassica imits and hope for other appications in the future. ACKNOWLEDGMENTS The authors woud ike to thank Dr. Keivan Aghababaei Samani and for insightfu comments, enightening discussions and suggestions in the ight of which this paper has been revised. The authors aso thank V. Monfaredi for variety of heps.

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