ISSN 1715-7862 [PRINT] Advances in Natural Science ISSN 1715-7870 [ONLINE] Vol. 5, No. 4, 2012, pp. 36-51 www.cscanada.net DOI:10.3968/j.ans.1715787020120504.1977 www.cscanada.org
The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
PANG Xiaofeng[a],*
[a] Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu, China. INTRODUCTION * Corresponding author. The Features of Microscopic Particle Described by the Linear Schrödinger Equation. Received 25 August 2012; accepted 14 December 2012 As it is known, the states of microscopic particles in quantum systems were as yet described by quantum Abstract mechanics, which was established by several great The dynamic and collision features of microscopic scientists, such as Bohr, Born, Schrödinger and particles described by nonlinear Schrödinger equation are Heisenberg, etc., in the early 1900s[1-10]. In quantum investigated deeply using the analytic and the Runge-Kutta mechanics the dynamic equation of microscopic particles method of numerical simulation. The results show that is the following Schrödinger equation: the microscopic particles have a wave-corpuscle duality 2 ∂ψ 2 and are stable in propagation. When the two microscopic i =−∇+ψψV( rt, ) (1) ∂tm2 particles are collided, they can go through each other → 22 and retain their form after their collision of head-on from Where ∇ /2m is the kinetic energy operator, V( r , t) is the externally applied potential operator, m is the mass opposite directions, This feature is the same with that of → of the particles, ψ (rt ,) is a wave function describing the the classical particles. However, a wave peak of large → amplitude, which is a result of complicated superposition states of the particles, r is the coordinate or position of of two solitary waves, occurs in the colliding process. the particle. Equation (1) is a wave equation, if only the externally applied potential is known, we can find the This displays the wave feature of microscopic particles. [7-9] Therefore, the collision process shows clearly that the solutions of the equation . However, for all externally applied potentials, its solutions are always a linear or solutions of the nonlinear Schrödinger equation have a both → corpuscle and wave feature, then the microscopic particles dispersive wave, for example, at V( r , t) = 0, the solution represented by the solutions have a wave-corpuscle is a plane wave as follows: →→ duality. Obviously, this is due to the nonlinear interaction ψω(rt,) = A 'exp[ ikr ( ⋅− t )] (2) b|φ |2φ . Thus we can determine the nonlinear Schrödinger Where k is the wavevector of the wave, ω is its equation can describe correctly the natures and properties frequency, and A' is its amplitude. This solution denotes of microscopic particles in quantum systems. the state of a freely moving microscopic particle with an Key words: Microscopic particle Schrödinger eigenenergy of equation; Wave-corpuscle duality; Nonlinear interaction; 2 Collision; Propagation; Quantum mechanics p 1 222 E= =( pppx + y + z),( −∞ < pppxyy , , < ∞ ) (3) PACS numbers: 02 .60.Cb, 03.65.-w 22mm This energy is continuous, this means that the PANG Xiaofeng (2012). The Dynamic and Collision Features of probability of the particle to appear at any point in the Microscopic Particles Described by the Nonlinear Schrödinger Equation space is a constant, thus the microscopic particle cannot be in the Nonlinear Quantum Systems. Advances in Natural Science, 5(4), 36-51. Available from: http://www.cscanada.net/index.php/ans/article/ localized, can only propagate freely in a wave in total space. view/j.ans.1715787020120504.1977 DOI: http://dx.doi.org/10.3968/ Then the particle has nothing about corpuscle feature. j.ans.1715787020120504.1977
Copyright © Canadian Research & Development Center of Sciences and Cultures 36 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
If the potential field is further varied, i.e., V(r,t) particles in the system of many bodies and many particles, ≠ 0, the solutions of Equation (1) is still some waves involving condensed matter, atoms and molecules[14-16]. with different features[5-12]. This shows clearly that the These problems not only awake and evoke us that microscopic particles have only a wave feature and not quantum mechanics must develop forward but also corpuscle feature, which is inherent nature of particles in indicate the its direction of development. the quantum mechanics. These features of microscopic In view of the above problems of quantum mechanics, particles are not only incompatible with de Broglie we should take into account of the nonlinear interactions relation of wave-corpuscle duality of E = hv =ω and of the particles, which was ignored in quantum mechanics in Equations (1) and (4). As a matter of fact, the nonlinear pk= and Davisson and Germer’s experimental result of [10-13] interactions exist always in any realistic physics systems electron diffraction on double seam in 1927 but also including the hydrogen atom, which is generated by the contradictory with regard to the traditional concept of [14-16] interaction between the particles and another particles or particles . These are just the limitations and difficulties background field[17-28]. Therefore, once the real motions of the quantum mechanics, which result in a duration [8-12] of the microscopic particles and background fields and controversy about a century in physics and have not their true interactions are considered, then the properties been solved up to now. This is very clear that the reasons and states of microscopic particles cannot be described having rise to these difficulties are the simplicity and by Schrödinger Equation (1), but should be depicted by approximation of quantum mechanics. As a matter of fact, the following nonlinear Schrödinger equation in nonlinear the Hamiltonian operator of the system corresponding quantum systems Equation (1) in quantum mechanics is represented by 2 ∂φ 2 2 ^ → i ==−∇+φVrt(,) φ − b φφ, (5) Ht()= 22∇ /2m + V( r , t) , (4) ∂tm2 → Which is only composed of kinetic and potential Where φ(rt ,) is now a wave function representing operator of particles. The latter is not related to the wave the states of microscopic particles in the nonlinear function of state of the particle, thus it can only change systems, b is nonlinear interaction coefficient, the the states of particles, such as amplitude, velocity and nonlinear interaction is now denoted by b|φ |2 φ , which frequency, cannot influence their natures. The natures of is generated by interaction between the moved particle particles can be only determined by the kinetic energy and background field. Therefore, Equation (5) represents term in Equation (4), which but has a dispersive feature. correctly motion features of microscopic particle and its Thus the microscopic particles have only a wave feature, interactions with other particles or background field. In not corpuscle feature. This is just the root and reason of this case the nonlinear interaction and dispersive effect the above limitations and difficulties quantum mechanics occur simultaneously in the dynamical equation, the possesses. deformational effect of nonlinear interaction on the wave On the other hand, the Hamiltonian of the systems and can suppress its dispersive effect, thus dynamic Equation dynamic equation in Equations (1) and (4) contain only (5) have a soliton solution[29-30], which has both wave the kinetic energy and externally applied potential term. and corpuscle feature. Thus the microscopic particles This means that we must incorporate all interactions, have a wave-corpuscle duality in such a case. Therefore, including nonlinear and complicated interactions, Equation (5) not only describes really the motions of the among particles or between particle and background particles and their interactions with other particles or field, such as the lattices in solids and nuclei in atoms background field but also represents correctly their wave- and molecules, into the external potential by means of corpuscle duality. Then we have the sufficient reasons to various approximate methods, such as the free electron believe the correctness of nonlinear Schrödinger equation and average field approximations, Born-Oppenheimer for describing the properties of microscopic particles in approximation, Hartree-Fock approximation, Thomas- quantum systems. Fermi approximation, and so on. This not only denotes the approximation of quantum mechanics but also is obviously incorrect[10-13]. The essence and substance of this 1. THE CHANGES OF PROPERTY OF method, in which these real interactions between them are replace by an average field, are that quantum mechanics MICROSCOPIC PARTICLES DESCRIBED freezes or blots out real motions of the microscopic BY THE NONLINEAR SCHRÖDINGER particles and background fields and ignore completely real EQUATION interactions between them, including nonlinear and other complicated interactions, which can influence the natures 1.1 The Wave-Corpuscle Duality of Solutions of of the particles. This is just the essence and approximation Nonlinear Schrödinger Equation of quantum mechanics. Therefore, quantum mechanics As it is known, the microscopic particles have only the cannot be used to study the real properties of microscopic wave feature, but not corpuscle property in the quantum
37 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems mechanics. Thus, it is very interesting what are the Schrödinger Equation (6) are solitons. The envelop φ(x, properties of the microscopic particles in the nonlinear t) is a slow varying function and is a mass centre of the quantum mechanics? We now study firstly the properties particles; the position of the mass centre is just at x0, A0 is of the microscopic particles described by nonlinear its amplitude, and its width is given by W'2= π / Am0 2. Schrödinger equation in Equation (5). In the one- Thus, the size of the particle is AW'2= π /2 m and dimensional case, the Equation (5) at V(x, t) = 0 becomes 0 a constant. This shows that the particle has exactly a as determinant size and is localized at x0. Its form resemble 2 iφt' + φx'x' + bφ φ = 0 (6) a wave packet, but differ in essence from both the wave 2 solution in Equation (1) and the wave packet mentioned Where x'= x / / 2m , t'= t / . We now assume the solution of Equation (11) to be of the form above in linear quantum mechanics due to invariance of iθ (x',t') form and size in its propagation process. According to the φ(x',t') = ϕ(x',t')e (7) soliton theory[29-30], the bell-type soliton in Equation (14) substituting Equation (7) into Equation (6) we get can move freely over macroscopic distances in a uniform 2 2 x'x' t' x' b 0,(b 0) (8) velocity v in space-time retaining its form, energy, momentum and other quasi-particle properties. However, 2 0 (9) x'x' x' x' t' the wave packet in linear quantum mechanics is not so If let and will be decaying and dispersing with increasing time. (x'v t'), (x'v t'), x ' vt ', → c e c Just so, the vector r or x in the representation in Equation
'x ' vte ', (14) has definitively a physical significance, and denotes exactly the positions of the particles at time t. Thus, the then Equations (8)-(9) become → 23 wave- function φ(rt ,) or φ(x, t) can represent exactly the ϕxx''+vb c ϕθ x ' −−= ϕθ x ' ϕ 0 (10) → states of the particle at the position r or x at time t. These ϕθxx''+20 ϕ x 'θ x ' −=v e ϕ x ' (11) features are consistent with the concept of particles. Thus the microscopic particles depicted by Equation (6) display If fixing the time t' and further integrating Equation (11) [31-38] with respect to x' we can get outright a corpuscle feature . 2 Using the inverse scattering method Zakharov and ϕθ(2−=v ) At ( ') (12) xe' Shabat[39-40] obtained also the solution of Equation (6), Now let integral constant A(t') = 0, then we can get which was represented as θx' = ve / 2. Again substituting it into Equation (10), and 1 2 further integrating this equation we obtain ′′ 2 ′′′ 22′ ϕ ( xt,) = 2η sech 2 η( x−+ x0 ) 8 ηξ t × exp − 4 i(ξ − η) t − i2 ξ x ' − i θ ϕ dϕ 1 b =x'' − vt 2 ∫ϕ e 2 (13) 0 Q()ϕ ′′ ′′′ 22′ ϕ ( xt,) = 2η sech 2 η( x−+ x0 ) 8 ηξ t × exp − 4 i(ξ − η) t − i2 ξ x ' − i θ (15) b ϕϕ=−+−42 ϕ 2 + Where Q( ) b /2 ( ve 2 vv ec ) c '. When c' = in the coordinate of (x', t'), where η is related to the 2 2 amplitude of the microscopic particle, ξ relates to 0, ve − 2vc ve > 0, then ϕ=±=− ϕϕ00, [(ve 2 vv ec ) / 2 b ] is the the velocity of the particle, θ = arg γ, λξ= + i η, roots of Q(φ)=0 except for φ=0. From Equation (13) Pang γ x′ = (2η )−1 log( ), obtained the solution of Equations (8)-(9) to be 0 2η γ is a constant. We now re-write it as [23-27] b following form : (x',t') 0 sec h[ 0 (x'vet')] . 2 iv[ x '−− x' v t '] 2 ' ec( 0 ) 2 [21-30] φ ( x', t ') = 2 k sec h 2 k( x '−− x0 ) vte ' e Pang represented eventually the solution of b { } nonlinear Schrödinger equation in Equation (6) in the (16) coordinate of (x, t) by Where ve is the group velocity of the particle, vc is the A0 bm i[ mv( x−− x0 ) Et]/ phase speed of the carrier wave in the coordinate of (x', φ ( x, t) = A00 sec h( x−− x) vt e (14) t'). For a certain system, ve and vc are determinant and 3/2 1/2 2 do not change with time. We can obtain 2 k/b = A0, Where A0 =( mv /2 − E )/2 b , v is the velocity v 2 − 2v v of motion of the particle, E = ω . This solution is A = e c e . 0 2b completely different from Equation (3), and consists [29-30] of an envelop and carrier waves, the former is According to the soliton theory , the soliton shown in Equation (16) has determinant mass, momentum and ϕ(xt , )= A sec hAbmxx−− vt / [21-28] 00{ ( 0) } and a bell- energy, which can be represented by ∞ 2 type non-topological soliton with an amplitude A0, the Ns =φ dx' = 22 A0 , latter is exp{i [ mv ( x−− x0 ) Et ] / }. This solution is shown in ∫−∞ Figure 1a. Therefore, the particles described by nonlinear
Copyright © Canadian Research & Development Center of Sciences and Cultures 38 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
* * ∂ϕ ∂∂ ϕθ ∂2 θ pi x' x' dx'2 2A0ve Nsve const , −+u2 +ϕ = 0 (20) ∂ξ ∂∂ ξξ ∂ ξ2 2 1 4 1 E dx' E M v 2 du x' 0 sol e (17) 2 2 Where u= ' . If 2 ∂θξ ∂−u0 ≠, Equation (20) may dt
Where M sol = N s = 2 2A0 is just effective mass of be denoted as ' ' the particles, which is a constant. Thus we can confirm 2 g(t ) ∂θ g(t ) u or = + (21) '2 that the energy, mass and momentum of the particle cannot ( u/2) ∂x2ϕ be dispersed in its motion, which embodies concretely Integration of Equation (26) yields the corpuscle features of the microscopic particles. This x′ dx'' u θ ′′= ′ ++ ′ ′ is completely consistent with the concept of classical (x ,) t gt () 2 x ht() (22) ∫0 ϕ particles. This means that the nonlinear interaction, 2 2 ' b φφ, related to the wave function of the particles, Where h(t ) is an undetermined constant of [41-43] balances and suppresses really the dispersion effect of the integration. From Equation (22) Pang can get kinetic term in Equation (6) to make the particles become ∂θ x′ dx'' gu gu u =′ −+ +′′ + → gt() 2 22x′=0 x ht() (23) ′ ∫0 eventually localized. Thus the position of the particles, r ∂t ϕϕϕ 2 or x, has a determinately physical significance. However, Substituting Equations (22)-(23) into Equation (17), the envelope of the solution in Equations (14)-(16) is a we have: solitary wave. It has a certain wavevector and frequency ∂22ϕ u u x′ dx'' gu g 2 = ++′′ + + +ϕϕ −3 + as shown in Figure 1(b), and can propagate in space- 2(ax ' c ) x h ( t )+ g 22x′=0 b 3 ∂()x′ 24∫0 ϕϕ ϕ time, which is accompanied with the carrier wave. Its feature of propagation depends on the concrete nature ∂22ϕ u u x′ dx'' gu g 2 of the particles. Figure= 1(b) ++ shows the′′ + width of + the +ϕϕ −3 + 2(ax ' c ) x h ( t )+ g 22x′=0 b 3 (24) frequency spectrum∂()x ′of the envelope24 φ(x, t) which has ∫a0 ϕϕ ϕ localized distribution around the carrier frequency ω0. 2 2 [21-28] ∂ ϕ ϕ This shows that the particle has also a wave feature . d Since 2 = 2 , which is a function of ξ Thus we believe that the microscopic particles described ∂()x′ dξ by nonlinear quantum mechanics have simultaneously a only. In order for the right-hand side of Equation wave-corpuscle duality. Equations (14)-(16) and Figure (29) is also a function of ξ only, it is necessary 1(a) are just the most beautiful and perfect representation ′ = = of this property, which consists also of de Broglie that gt( ) g0 constant , and 2 relation, Eh=υω = and pk= , wave-corpuscle u'' u gu (ax'+c)+ x+ h(t )+ +=' V ()ξ (25) duality and Davisson and Germer’s experimental result 24ϕ 2 x0= of electron diffraction on double seam in 1927 as well as Next, we assume that VV()ξ= () ξβ − , where β is real the traditional concept of particles in physics[10-13]. Thus 0 and arbitrary, then we have reasons to believe the correctness of nonlinear 2 [21-28] uu gu quantum mechanics proposed by Pang. ax'+= c V00 ()ξβ − x′′ + −x′= −h() t − (26) 24ϕ 2 1.2 The Wave-Corpuscle Duality of Solution Clearly, in the discussed case V0(ξ) = 0, and the of the Nonlinear Schrödinger Equation with function in the brackets in Equation (26) is a function of Different Potentials t¢. Substituting Equation (26) into Equation (4), we can We can verify that the nature of wave-corpuscle duality get: of microscopic particles is not changed when varying the 2 d ϕ 3 g externally applied potentials. As a matter of fact, if V(x') = =−+βϕ ϕ 0 23b (27) αx' + c in Equation (5), where α and c are some constants, dξ ϕ [31-38, 41-43] in this case Pang replaced Equation (8) by Where β is a real parameter and defined by 2 2 x'x' t' x' b x'c. (18) VV0 ()ξ= () ξβ − (28) Now let 2' with ϕ(',')x t = ϕξ (), ξ =−x ' ut ('),(') ut =−++α (t') v t d (19) 2 u u gu (αξxx '+c)+ '+ h(t')+ +== V() (29) Where u(t') describes the accelerated motion of φ(x', 24ϕ 2 x'0 t'). The boundary condition at ξ →∞ requires φ(ξ)to Clearly, in the discussed case,V0(ξ)=0. Obviously, approach zero rapidly, then Equation (9) can be written as ϕ= ϕξ()is the solution of Equation (27) when β and g are
39 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
1+∆ constant. For φlarge ξ ,= we η may assume η−+ that' ϕ≤α βξ/2 − ξ, × −ξ − α −+'ξ 2 − η 2 + α 23 − ξα2 + θ ( xt', ') 2 sec h{ 2( xx '0) (2 t ' 4 ' t ')} exp{ i [2( ' t ')( xx ' 00) 4( ' ) t ' 4 t ' / 3 4 ' t ' ] } φ= η η−+' αwhen2 − ξ ∆ is × a small − constant.ξ − α To −+ ensure' ξthat 2 − ηdd 222ϕξ/ + α 23, − ξα2 + θ ( xt', ') 2 sec h{ 2( xx '0) (2 t ' 4 ' t ')} exp{ i [2( ' t ')( xx ' 00) 4( ' ) t ' 4 t ' / 3 4 ' t ' ] } (39) [25- and ϕ approach zero when ξ → ∞ , only the solution At the same time, utilizing the above method, Pang 27,41] found also the soliton solution of Equation (5) at V(x) corresponding to g = 0 in Equation (27) is kept stable. 0 = kx2 + A(t)x + B(t), which could be represented as Therefore we choose g = 0 and obtain the following from 0 z iθ(x,t) Equation (26) =φ(x-u(t))e (40) ∂θ u Where = (30) ' β ∂x 2 2 ' ϕβ( x', t ') = sec h( x '−− x0 ) ut ( ') , thus, we obtain from Equation (34) b { } 2 uu u(') t = 2cos(2kt ' ++γ ) u (') t (41) αβx'+c=−+−x ' h(t') − , 0 t ' 24 '2 2 θ( x', t ') =− [ 2 αγ sin(2kt ' ++ ) u00 ( t ') / 2)]( x ' − x ) + {[ −α (2 cos(2kt ''++ γ ) u0 ( t '') / 2)] − ht( ')=− (β v2 /4 −− ct ) ' αα23 t ' /3 + v t '2 /2]} (31) ∫0 t ' t ' '2+−αγ + + 2'2+ + θ 2 Substitutingθ Equation( x', t ') =− [(31) 2 αγ into sin(2 Equationskt ' ++ ) (23) u ( tand ') / 2)] (29), xθ ' −( xx', t+ ')B=−( {[t [ '') − 2α αγ [ sin(2 (2 2 cos(2 sin(2kt ' ++ktk ''t )++ '' γ u )00 ) ( t u ') u (/00 t 2)]( '')t '') /( 2)]/ x 2]} ' − x− dtB) ''( +t'')+ Lt {[ ' −α (2 cos(2kt ''++ γ ) u0 ( t '') / 2)] − 00( ) ∫0 0 ∫0 we obtain 2 2 3 2 B( t '')+− [ 2αγ sin(2k t '' + ) + u ( t '') / 2]} dt ''+ Lt ' + θ θ=[(-αt'+v/2)Bx('+( t '')β+−-v [/4- 2αγc sin(2)t'-α t' k/3+ t '' +vαt )' +/2 u 00 ( t '') / 2]} dt(32) ''+ Lt ' + θ 00 — Finally, substituting the above into Equation (33), we Where L is a constant related to A(t’).When A(t) = B(t) can get = 0, the solution is still Equation (40), but 2 d { 3 - b{ +=b{ 0 (33) u( t ')= 2 cos(2kt ')+ u0 ( t ') , dp2 t ' '2 When β > 0, Pang gives the solution of Equation (38), θ ( x',' t) =− [2 k sin(2 kt ') + u (')/2)] t x ' −+ x {[ − k (2cos(2 kt '') + u ('')/2)] t + 00( ) ∫0 0 which is of the form t ' t ' '2'2 θ ( x', t ') =− [ 2 k sin(2 kt ') + u ( t ') / 2)] x ' −+ x[θ−(2 {[x −',k tk sin(2' (2) =− cos(2 [ 2 kt k '') kt sin(2 ++ '') u00 + ( tu '') kt ( t/ ') '')2]} + /u 2)]dt ( '' t ') +θ / 2)] x ' −+ x {[ − k (2 cos(2 kt '') + u ( t '') / 2)] + 2b 00( ) ∫0 000( ) ∫ 0 { = sec h()bg (34) 0 b [− 2k sin(2 kt '') ++ u00 ('')/2]} t dt '' θ [− 2k sin(2 kt '') ++ u ( t '') / 2]} dt '' θ [41-43] 00 Pang finally obtained the complete solution in this 2 2 For the case of V0(x') = α x' and b = 2, where α is condition, which is represented as [44-45] constant, Chen and Liu assume u(t') = (2ξ/α)sin(2αt'), 2b ' 2 z^x',t'h = sec hx" ba^ '(- xt0h + ''--vt d) , # thus they represent the soliton solution in this condition b 6 @ ' 223 by exp"it-ab'/+ vx2'^ - xv0h + (/--4)ct''- a t /3 + ' ' ' 2' 2 6 φ= η η− − ξα α× ξ − α −−+−− η vta '/2 , ( xt', ') 2 sec h{ 2( xx '0) (2 '/ )sin(2t ')} exp{ i [2 '( xx '00) cos 2 ( tt ' ) 4 ( tt ' 0 ) @ 2' ' (35) (ξα ' / )sin[4 α (tt '−+' ) θ ]} '' 2' ' ' 2' φ( xt', ') = 2 η sec h 2 η xx '− − (2 ξα '/ )sin(2 αt ')φ×( exp{xt', i') [2= ξ 2 ' η xx sec ' − h 2 cos00 η 2 xxα ' (− tt ' −−+−−0 − ) (2 ξα 4 '/ η ( )sin(2 tt ' ) αt ')× exp{ i [2 ξ ' xx ' −00 cos 2 α ( tt ' −−+−− ) 4 η ( tt ' 0 ) This is a soliton solution. If{ V(x’)( = c,0 )the solution can } ( 00{) ( ) 0} ( ) be represented2' as ξα2' α−+ θ (42) (ξα ' / )sin[4 α (tt '−+00 ) θ ]} ( ' / )sin[4 (tt '00 ) ]} 2β φβ= −'' − − ×Where −' 2hb −−=βis 2 the − amplitude of microscopic ( x',' t) k sec h{ ( x ' x00) vt (' t )} exp{[ iv( x ' x0) /2( v /4)']} ct b particles, 4ξ' is related to its group velocity in Equations β 2 '' ' 2 φβ( x',' t) = k sec h( x '− x00) − vt (' − t ) × exp{[ iv( x ' − x0) /2( −−β v /4)']} − ct (36) (39) and (42), and ξ'is the same as ξ in Equation (15). b { } At V(x') = 2αx' and b = 2, we can also get a From the above results we see clearly that these solutions corresponding soliton solution from the above process. of nonlinear Schrödinger Equation (5) under influences [44-45] of different potentials, V(x) = c, V(x') = αx', V(x') = However, Chen and Liu adopted the following 2 2 2 transformation αx'+c, V(x) = kx +A(t)x+B(t) and V(x') = α x' , still consist of envelop and carrier waves, which are analogous to 23 2 φ( xt',') = φ '(',')exp[2 xt −+ ixtit αα ''8 '/3],' xx =−= '2 α tttEquation ',' ' (14), some bell-type solitons with a certain amplitude A , group velocity v and phase speed v , and φxt','= φ '(',')exp[2 xt −+ ixtit αα ''8 23 '/3],' xx =−= '2 α ttt ','2 ' (37) 0 e c ( ) have a mass center and determinant amplitude, width, to make Equation (5) become size, mass, momentum and energy. If inserting these ′ ′′2 i'φt′ ++ φxx′′ 20 φφ = (38) solutions, Equations (35), (38), (39), (41) and (42) into [44-45] Equation (16) we can find out the effective masses, Thus Chen and Liu represented the solution of moments and energies of these microscopic particles, Equation (5) at V(x') = ax', b = 2 by respectively, which all have determinant values. Therefore φ= η η−+' α2 − ξ × −ξ − α −+'ξ 2 − η 2 + α 23 − ξα2 + θ ( xt', ') 2 sec h{ 2( xx '0) (2 t ' 4 ' t ')} exp{ i [2(we can ' determine t ')( xx ' 00) that 4( the ' microscopic ) t ' 4 t 'particles / 3 4 ' described t ' ] } by these dynamic equations still possess a wave-corpuscle
Copyright © Canadian Research & Development Center of Sciences and Cultures 40 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
duality as shown in Figure 1, although they are acted corpuscle feature, their combination results in its wave- by different external potentials. These potentials change corpuscle duality, but the external potentials influence only amplitude, size, frequency, phase and group and only wave form, phase and velocity of particles, but phase velocities of the particles, in which velocity and cannot affect the wave-corpuscle duality. These results frequencies of some particles are further related to time verify directly and clearly the necessity and correctness and oscillatory[47-50]. These results indicate that in Equation for describing the properties of microscopic particles (5) the kinetic energy term decides the wave feature of using the nonlinear Schrödinger equation, or the nonlinear the particles, the nonlinear interaction determines its quantum mechanics proposed by Pang[17-24].
Figure 1 The Solution of Equation (6) and Its Features
2 2. THE CLASSICAL FEATURES OF ∂φ * 2 2 −i =−∇±φ*b φφ * + V( rt ,*) φ (45) MOTION OF MICROSCOPIC PARTICLES ∂tm2 We can get that the velocity of mass centre of 2.1 The Feature of Newton’s Motion of microscopic particle can be denoted by Microscopic Particles ∞∞ v= d x' / dt ' = − 2 iφφ * dx '/ φφ * dx ' (46) Since the microscopic particle described by the nonlinear gx∫∫−∞ ' −∞ Schrödinger Equation (5) has a corpuscle feature and is We now determine the acceleration of the mass center also quite stable as mentioned above. Thus its motion of the microscopic particles and its rules of motion in an in action of a potential field in space-time should have [51-55] externally applied potential. itself rules of motion. Pang studied deeply this rule of We can obtain from the above equation motion of microscopic particles in such a case. ∞ ∞∞ ∞ ∞ dV*' *' * ' *∂∂*2 * ' * ' We know that the solution Equation (14), shown in φφdx= φφ dx + φ() φdx = i { φ [ φ' +b φφ −− V φ]]} V φ φ dx = i φ φ dx '∫∫∫−∞ x −∞ t x −∞ tx' ∫−∞ ''xx x ∫−∞ Figure 1, of the nonlinear Schrödinger Equation (5) with d∂∂ xx ∞ ∞∞ ∞ ∞ dV*' *' * ' *∂∂*2 * ' * ' ∂φ ' φφdx= φφ dx + φ() φdx = i { φ [ φ' +b φφ −− V φ]]} V φ φ dx = i φ φ dx different potentials, has'∫∫∫ −∞the behaviorx −∞of t x= 0 at −∞xx' = tx. ' ∫−∞ ''xx x ∫−∞ d∂x ' 0 ∂∂ xx ∞ ∞∞* ∞ ∞ dV*' *'Therefore, * f fdx' ' = ρ(x')dx' *∂∂ can be regarded*2 as the mass * ' * ' φφdx= φφ dx + φ() φdx = i { φ [ φ' +b φφ −− V φ]]} V φ φ dx = i φ φ dx (47) d'∫∫∫−∞ x −∞ t x in the−∞ intervaltx' of x' to ∫−∞ x' + ∂∂dx' xx''. Thusxx the position of massx ∫−∞ centre of a microscopic particle at x' in nonlinear quantum 0 Where x'= x / 2 / 2m , t'= t / . We here utilize the mechanics can be represented by following relations and the boundary conditions: ∞ * ∞ * = φ φ φ φ ∞∞ x′ = x′g = x'0 ∫−∞ ' x' dx'/∫−∞ dx' (43) * * *2 * 2 * (φφ′′′−= φ ′′ φ ′ )0,(dx′′ b φ φφ′ + φφφ′ )0dx =, ∫∫−∞ xxx xx x −∞ x x Then the velocity ofb the microscopic particle is represented by ∫φφ*'dx = constant (or a function of t’) and d ∞ ∞∞ ∞ v= v = {φ *x ' φ dx '/ φφ * dx '}= ( φ* x ' φ + φ * x ' φ ) dx '/ φφ * dx ' g∫−∞ ∫∫−∞ −∞ tt''∂∂φφ ∫−∞ * dt ' limφφ*'xx= lim '= 0, ∞ ∞∞ ∞ xx''→∞ ∂∂xx''→∞ d * v= v = {φ *x ' φ dx '/ φφ * dx '}= ( φ x ' φ + φ * x ' φ ) dx '/ φφ * dx ' (44) g∫−∞ ∫∫−∞ −∞ tt'' ∫−∞ φφ(,)xt′′= (,) xt ′′= 0 dt ' Lim lim x′ then from Equation (5) and its conjugate equation ||x′ →∞ ||x′ →∞
41 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
∂∂φφ3 Returning to the original variables, the Equation (50) Where φφ' =, ''' = . Thus, we can get becomes x ∂∂' xxx '3 xx 2 dx0 ∂V ∞∞* ∞ = − d *∂∂φφ **m 2 (51) φx′′ φ dx=+=−( x ′φ φ x ′ () dx ′ 2 iφφ′ dx ′ dt∂ x dt′∫∫−∞ −∞ ∂∂t ′′t ∫−∞ x 0 (48) Where x'0 =
2 ∞∞ ∞ derive from Equation (5) by another method. As it dd' *'*' * *' ∂V x=−2{ iφφ'' dx / φφ dx=−= 2 φ V φ dx − 2 (49) is known, the momentum of the particle depicted by dt'2 dt ' ∫∫−∞ xx−∞ ∫−∞ ∂x' Equation (5) is obtained from Equation (17) and denoted ∞ * ∞ If f is normalized, i.e., φφdx′ =1, then the above ∂L ∗∗ ∫−∞ ′ by P==−− i (φφxx′′ φφ)dx . Utilizing Equation (5) ∂φ ∫−∞ conclusions also are not changed. [18-27,51] and Equations (43)-(45) Pang obtained Where V = V(x’) in Equation (49) is the external dP ∞∞∂∂22V potential field experienced by the microscopic particles. = 22V( x′′) φφ dx = − dx ′= ′′∫∫−∞ ∂∂−∞ ′ ∂V dt x x We expand at the mass centre x' =
where Where x'0 is the position of the center of the mass of 2 ′′′ ′ the macroscopic′ particle. This is the same as Equation (51). ∆=ij ((xx ' − ' ) =(xxxij−−00 )( x ) =(xi′ − xx i ′′ )( j −xj ) Therefore, we can confirm that the microscopic particles 2 ′′′ ′ ′ ∆=ij ((xx ' − ' ) =(xxxij−−00 )( x ) =(xi′ − xx i ′′ )( j −xj ) in the nonlinear quantum mechanics satisfy the Newton- For the microscopic particle described by Equation type equation of motion for a classical particle. (5) or Equation (6), the position of the mass center of the 2.2 Lagrangian and Hamilton Equations of particle is known and determinant, which is just
Copyright © Canadian Research & Development Center of Sciences and Cultures 42 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
i ∗∗ H ' * H ' L ' =(φ ∂tt φ −∂ φφ) − i , or i (60) 2 t * t (55) 2 Equation (60) is just the complex form of Hamilton ∗∗ ∗2 (∇φ ⋅∇ φ) −Vx( ) φφ +( b / 2)( φφ ) equation in nonlinear quantum mechanics. In fact, the 2m Hamilton equation can be also represent by the canonical where L'=L. The momentum density of the particle coordinate and momentum of the particle. In this case the system is defined as P = 2L/2f. Thus, the Hamiltonian canonical coordinate and momentum are defined by ∂∂ density, H = H', of the systems is as follows 1∗∗LL'1 ' q1=+==−=(φφ), pq 12 ,,(φφ) p 2 2 22∂∂( t'qi 1 ) ∂∂( t'q 2 ) i ∗∗ ∗∗ ∗2 H= φ∂ φ −∂ φφ - L = ∇φ ⋅∇ φ + φφ − φφ ( tt1)∗∗( ∂∂LL'1) Vx( ) ( b / 2)( ) ' 2 q1=+==−=(φφ), pq 122m ,,(φφ) p 2 22∂∂qi ∂∂q 2 ( t' 1 ) ( t' 2 ) ∗∗ ∗2 =( ∇φ ⋅∇ φ) +Vx( ) φφ −( b / 2)( φφ ) (56) Thus, the Hamiltonian density of the system in 2m Equation (56) takes the form From the Lagrangian density L' = L' in Equation (55) [18-27, 51-55] H''=∑ pqLii ∂−t' Pang find out i ∂L' i = φ −V (x)φ + b(φ *φ)φ and the corresponding variation of the Lagrangian ∂φ * 2 t density L’ = L can be written as 2 δδ δ L' L' i ' LL'' L ' δδLq=∑ i + δ( ∇+ qi ) δ( ∂t' q i ) and t ( * ) .( ) t . i δδqqii(∇∂) δ( t' q i) t * 2 2m Thus we can obtain From Equation (17), the definition of pi, and the Euler- Lagrange equation, ∂∂LL'' ∂ L ' 2 −∂ −∇ =ϕ + ∇2 ϕ − ϕ + ϕ ϕϕ tt(* ) .( ) i Vx()(*) b L' L' pi ∂ϕϕ* ∂t ∂∇ ϕ *2m . 2 qi qi t ∂∂LL'' ∂ L ' 2 −∂( ) −∇ .( ) =iϕ + ∇ ϕ −Vx()(*) ϕ + b ϕ ϕϕ (57) tt* one obtains the variation of the Hamiltonian in the form ∂ϕϕ* ∂t ∂∇ ϕ *2m Through comparison with Equation (5) Pang gets = ∂δδ −∂ of δH’ ∑∫ ( t'qi p it' p ii q) dx '. L' L' L' i ( ) .( ) * t * * Thus, one pair of dynamic equations can be obtained t and expressed by or ∂qi ∂pi L' L' L' = δH’/δpi, = − δH’/δqi (61) t ( ) .( ) (58) ∂t ' ∂t ' t This is analogous to the Hamilton equation in classical Equation (58) is just the well-known Euler-Lagrange mechanics and has same physical significance with equation for this system. This show that the nonlinear Equation (60), but the latter is used often in nonlinear Schrödinger equation amount to Euler-Lagrange equation quantum mechanics. This result shows that the nonlinear in nonlinear quantum mechanics, in other word, the Schrödinger equation describing the dynamics of dynamic equation, or the nonlinear Schrödinger equation microscopic particle can be obtained from the classical can be obtained from the Euler-Lagrange equation in Hamilton equation in the case, if the Hamiltonian of the nonlinear quantum mechanics, if the Lagrangian function system is known. Obviously, such methods of finding of the system is known. This is different from quantum dynamic equations are impossible in the quantum mechanics, in which the dynamic equation is the linear mechanics. As it is known, the Euler-Lagrange equation Schrödinger equation, instead of the Euler-Lagrange and Hamilton equation are fundamental equations in equation. classical theoretical (analytic) mechanics, and were used On the other hand, Pang[18-27, 51-55] got also the Hamilton to describe laws of motions of classical particles. This equation of the microscopic particle from the Hamiltonian means that the microscopic particles possess evidently density of this system in Equation (56). In fact, we can classical features in nonlinear quantum mechanics. From this study we seek also a new way of finding the equation obtain from Equation (56) of motion of the microscopic particles in nonlinear H ' 2 2 V (x) b( *) (59) systems, i.e., only if the Lagrangian or Hamiltonian of the * 2m system is known, we can obtain the equation of motion Where H' = H'. Then from Equation (56) we can give of microscopic particles from the Euler-Lagrange or H ' 2 Hamilton equations. i 2 V (x) b( *) On the other hand, from de Broglie relation t * 2m Thus Eh=υω = and pk= , which represent the wave-
43 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems corpuscle duality of the microscopic particles in quantum dk dx ' ∂ω k=ve /2, thus = 0 and v= = =−=−2kv. theory, we see that the frequency ω and wavevector k dt ' gedt' ∂ k can play the roles as the Hamiltonian of the system and These results of the acceleration and velocity of momentum of the particle, respectively, even in the microscopic particle are same with the above data nonlinear systems and has thus the relation: obtained from Equations (15) and (51). This indicates that dωω∂ dk ∂∂ ω x′ these moved laws shown in Equations (50), (51), (53), =+=0 (60), (61), and (62) are self-consistent, correct and true in dt′∂ k dt' ∂∂ x ′′ t xk′ nonlinear quantum mechanics. as in the usual stationary media. From the above (2) For the case of V(x') = αx', the solution of Equation result we also know that the usual Hamilton equation in (5) is Equation (39) by Chen and Liu[44-45], which is also Equation (57) for the nonlinear systems remain valid for composed of a envelope and carrier wave. The mass centre the microscopic particles.dk Thus, the Hamilton equation in ' k of the particle is at x0, which is its localized position. From Equation (57) can be nowdt' representedx' by another form[25- 27, 56-57]: Equation (51) we can determined the accelerations of the mass center of the microscopic particle in this case, which dx' dk is given by k and x' (62) dt' k 2 dt' x' d x′ ∂Vx( ′ ) 0 =− =−=α ′ 2 2 2 constant (63) in the energy picture, where kx=∂∂θ is the time- dxt′′∂ dependentdx' wavenumber of the microscopic particle, On the other hand, from Equation (39) we know that x' ωθ=−∂dt' ∂kt′ is its frequency, θ is the phase of the wave 23 4α t′ 2 22 θ=2( ξ − αt′′) x+ −4αξt′′ +4( ξ − η)t + θ0 , (64) function of the microscopic particles. 3 2.3 Confirmation of Correctness of the Above where ξ is same with ξ' in Equation (39). Utilizing Conclusions again Equation (62) we can find We now use some concrete examples to confirm the k=2(ξα − t '), ω = 2 α x '4( − ξα − t ')22 + (2) η = 2 αxk ' −+ 22 (2) η correctness of the laws of motion of the microscopic =ξα − ω = α − ξα −22 + η = α −+ 22 η particles mentionedk 2(above tin '), the 2nonlinear x ' 4( quantum t ') (2 ) 2xk ' (2 ) mechanics[18-27, 51-55]:. Thus, the group velocity of the microscopic particle is (1) For the microscopic particles described by found out from Equation (5), V = 0 and constant, of which the solutions dx' ∂ω vt= = =4(ξα − ') (65) are Equation (15) or (16) and (36), respectively, we obtain g dt' ∂ k that the acceleration of the mass centre of microscopic Then its acceleration is given by 2 d ∂Vx() <> 2' particle is zero because of = − = 0 in d x dk '' mx2 = =−=2a constant, here(xx= ) (66) dt ∂
On the other hand, if the dynamic Equation (62) is used has also a mass centre, its position is at x’0, which is the we can obtain from Equation (15) that the acceleration position of the microscopic particle. When Equation and velocity of the microscopic particle are (51) is used to determine the properties of motion of dk dx ' ∂ω the particle in this case Pang[18-28] found out that the = 0 and vv= = = = −4ξ , dt ' gedt' ∂ k accelerations of the center of mass of the particle is 2 respectively, where dx0′ 2 = −4α x0′ (67) ω= −∂ θ/ ∂tk ' = 4( ξη22 − ) = 2 − 4 η 2, kx=∂∂=−θξ/' 2, dt′2 At the same time, from Equation (42) we gain that θ=−−4 ξη22tx′ −+ 2' ξ θ ( ) 0 ' ' 2' 2 ' θ=2 ξ( xx ' −0) cos[2(' α tt −+0 )]4 η (' tt −−0 ) ( ξα / )sin[4(' αtt−+00 )] θ For the solution in Equation (36) at V = constant, - - - 2 - ' '- 2 - 2' 2 ' θ = ve(x' x0')/2θ=(β2 ξv e( /4xx ' −C)0t)', cos[2(' thenα ω tt= −+ (β0 )]4v e /4 ηC), (' tt −−0 ) ( ξα / )sin[4(' αtt−+00 )] θ (68)
Copyright © Canadian Research & Development Center of Sciences and Cultures 44 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
where ξ is same with ξ' in Equation (42). From 2 α and an amplitude of ξ/α, the corresponding classical
Equations (67) and (68) we can find vibrational equation is x' = x'0 sinωt' with ω = 2a and
'' x'0 = ξ/α. The laws of motion of the center of mass of k=2ξα cos 2( t− t0 ) , microscopic particles expressed by Equation (53) and ' '' 2 '' 2 ωξ =ax4sin24cos44 α( tt−−00) ξ α(tt −−) η Equations (58), (60) and (62) in the nonlinear quantum mechanics are consistent with the equations of motion of ' 2212 2 22 =24α xkk( ξ −) −+ 24(ξη −) , the macroscopic particles. Thus, the group velocity of the microscopic particle is The correspondence between a microscopic particle and a macroscopic object shows that microscopic ∂ωαxk' v=| = −=2k 2α xctgxtt' [2 α ( '' − )] − 4 ξαparticles cos[2 ( ttdescribed'' − )] by the nonlinear quantum mechanics gx∂ ξ 2 00 k 1− k have exactly the same moved laws and properties as ξ 2 4 classical particles. These results not only verify the ∂ωαxk' v=| = −=2k 2α xctgxtt' [2 α ( '' − )] − 4 ξα cos[2 (tt'' − )] necessity of development and correctness of the nonlinear gx∂ ξ 2 00 k 1− k quantum mechanics, but also exhibit clearly the limits ξ 2 While its acceleration is 4 and approximation of the linear quantum mechanics dk ∂ω =−=−α ξ2 −=− 2 ξα α '' − ''|k 2 4k 4 sin[2 (tt0 )] and can solve these difficulties of the linear quantum dt∂ x mechanics and problems of contention in it as described in (69) Introduction. Therefore, the results mentioned above have d2 x' dk important significances in physics and nonlinear science. = '' Since ''2 , here ( xx = 0 ), we have dt dt dk d2 x ' ==−−4ξα sin[2 α (tt '' )] 3. THE COLLISION PROPERTY OF dt'' dt 2 0 and MICROSCOPIC PARTICLES DESCRIBED ξ BY NONLINEAR SCHRÖDINGER x '= sin[2α ( tt − ' )] (70) α 0 EQUATION Finally, the acceleration of the microscopic particle is As a matter of fact, the properties of collision of soliton d2 x' dk solutions of nonlinear Schrödinger Equations (5) at b = = −4'α 2 x (71) dt''2 dt = 1 > 0 and b < 0 firstly were analytically studied by [39-40] Equation (98) is also the same with Equation (67). Zakharov and Shabat using inverse scattering method Thus we confirm also the validity of Equations (48), and Zakharov and Shabat equation. They found from (51), (53), (58), (60) and (61)-(62). In such a case the calculation that the mass centre and phase of soliton occur microscopic particle moves in harmonic form. This only change after this collision at V(x) = 0 and b = 1. The th resembles also with the result of motion of classical translations of the mass centre x0m and phase θm of m particle. soliton, which moves to a positive (or final) direction after From the above studies we draw the following this collision, can be represented, respectively, by [18-27, 51-55] conclusions . (1) The motions of microscopic 1 N λλ− + −= mp< particles in the nonlinear quantum mechanics can be xx00mm Π * 0, and ηpm= +1 λλ− described by not only the nonlinear Schrödinger equation m mp but also Hamiltonian principle, Lagrangian and Hamilton N λλ− θθ+ −=−2 arg mp equations, its changes of position with changing time mm Π * (72) pm= +1 λλ− satisfy the law of motion of classical particle in both mp Where ηm and λm are some constants related to the uniform and inhomogeneous. This not only manifests that th the natures of microscopic particles described by nonlinear amplitude and eigenvalue of m soliton, respectively. The quantum mechanics differ completely from those in the Equation (72) shows that shift of position of mass centre linear quantum mechanics but displays sufficiently the of solitons and their variation of phase are constants corpuscle nature of the microscopic particles. after the collision of two solitons moving with different (2) The external potentials can change the states of velocities and amplitudes. The collision process of the two motion of the microscopic particles, although it cannot solitons can be described from Equation (72) as follows. vary its wave-corpuscle duality, for example, the particle Before the collision and in the case of t → −∞ the moves with a uniform velocity at V(x') = 0 or constant, slowest soliton is in the front while the fastest at the rear, or in an uniform acceleration at V(x') = ax', which they collide with each other at t’ = 0, after the collision corresponds to the motion of a charge particle in a uniform and t →∞, they are separated and their positions are just [39-40] electric field, but when V(x' ) = a2x'2 the microscopic reversed. Zakharov and Shabat obtained that as the particle performs localized vibration with a frequency of time t varies from −∞ to ∞ , the relative change of mass
45 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
centre of two solitons, Δx0m, and their relative change of Zakharov and Shabat equation. The result shows that phases, Δθm, can, respectively, denoted as the feature of collision of the two solitons in this case − is basically similar with the above properties. In the 1 Nmλλ−−1 λλ ∆=xxx' ''+− − =ln mp−ln mp meanwhile, Aossey et al.[25] investigated numerically 0m 00 mmη∑∑ λλ−−** λλ mkm=+=11 mpk mp the detailed structure, mechanism and rules of collision (73) of the microscopic particles described by the nonlinear and Schrödinger Equation (6) at b < 0 and obtained the rules of collision of the two solitons by a macroscopic model. mN−1 λλ−− λλ ∆=−=θθθ+−2 arg mp−2 arg mp In fact, the properties of collisions of microscopic m mm ΠΠλλ−−** λλ k=11mp km= + mp particles can be also obtained by numerically solving (74) Equation (6). Numerical simulation can reveal more − − detailed feature of collision between two microscopic Where x0m and phase θm are the mass centre and th particles. We here studied numerically the features phase of m particles at initial position, respectively. of collision of microscopic particles described by the Equation (73) can be interpreted by assuming that the nonlinear Schrödinger Equation (5) at b > 0 by fourth- solitons collide pairwise and every soliton collides with order Runge-Kutta method[58-59]. For this purpose we others. In each paired collision, the faster soliton moves began in one dimensional case by dividing Equation (5) −1* forward by an amount of ηmln ( λλλλ mk−−) mk, into the following two-equations λλ> , and the slower one shifts backwards by an ∂∂φφ22 ∂u mk += χφ i 2 , (75) −1* ∂∂t2 mx ∂ x amount of ηkln ( λλλλ mk−−) mk. The total shift is 22 equal to the algebraic sum of their shifts during the paired ∂∂uu ∂2 Mv()−=χφ . (76) collisions. So that there is not effect of multi-soliton ∂∂tx220 ∂ x( ) collisions at all. In other word, in the collision process in each time the faster soliton moves forward by an amount Equations (75)-(76) may be thought to describe the of phase shift, and the slower one shifts backwards by an features of motion of studied particle and another particle amount of phase. The total shift of the solitons is equal (such as, phonon) or background field (such as, lattice) to the algebraic sum of those of the pair during the paired with mass M and velocity v0, u is the characteristic collisions. The situation is the same with the phases. This quantity of another particle (such as, phonon) or of rule of collision of the solitons described by the nonlinear vibration (such as, displacement) of the background field. Schrödinger Equation (6) is the same as that of classical The coupling between the two modes of motion is caused particles, or speaking, meet also the collision law of by the deformation of the background field through the classical particles, i.e., during the collision these solitons studied particle – background field coupling, such as, interact and exchange their positions in the space-time dipole-dipole interaction, χ is the coupling coefficient trajectory as if they had passed through each other. After between them and represents the change of interaction the collision, the two solitons may appear to be instantly energy between the studied particle and background translated in space and/or time but otherwise unaffected field due to a unit variation of the background field. The by their interaction. The translation is called a phase relation between the two modes of motion due to their shift as mentioned above. In one dimension, this process interaction can be represented by ∂u χ 2 results from two solitons colliding head-on from opposite = φ + 22 A (77) directions, or in one direction between two particles with ∂−x Mv() v0 different amplitudes or velocities. This is possible because If inserting Equation (77) into Equation (75) yields just the velocity of a soliton depends on the amplitude. The the nonlinear Schrödinger Equation (5) at V(x) = constant, two solitons surviving a collision completely unscathed χ 2 demonstrate clearly the corpuscle feature of the solitons. where b = is a nonlinear coupling coefficient, 22− This property separates the solitons described by the Mv() v0 nonlinear Schrödinger Equation (6) from the microscopic V(x) = Aχ, A is an integral constant. This investigation 2 particles in quantum mechanical regime. Thus this shows clearly that the nonlinear interactionb φφ comes demonstrates the classical feature of the solitons. At the same time, Desem et al.[56] and Tan et al.[57] pay from the coupling interaction between two particles or attention to the features of the above solitons in collision the studied particle and background field. Very clearly, process by Zakharov and Shabat’s approach. in this model the real motion of background field and its Zakharov and Shabat[39-40] also discussed analytically interaction with the particle are completely considered, the properties of collision of two particles depicted by instead of are replacement by an average field, which the nonlinear Schrödinger Equation (6) at b < 0 using is fully different from that in quantum mechanics. Just
Copyright © Canadian Research & Development Center of Sciences and Cultures 46 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
2 so, we can say that the nonlinear interaction b|φ | φ Where arn and ain are real and imaginary parts of an. or the nonlinear Schrödinger Equation (5) represent the Equations (81)-(85) can determine states and behaviors of real motions of particles and background field and their the microscopic particle. Their solutions can be found out, interactions, which are completely different from quantum which are denoted in Appendix. There are four equations mechanics. Therefore, we can believe that the nonlinear for one structure unit. Therefore, for the quantum Schrödinger Equation (5) describe correctly the natures systems constructed by N structure units there are 4N and properties of microscopic particles in the quantum associated equations. When the fourth-order Runge-Kutta mechanical systems. method[58-59] is used to numerically calculate the solutions In order to use fourth-order Runge-Kutta method[58-59] of the above equations we should further discretize them. to solve numerically Equation (5) we must further Thus the n is replaced by j and let the time be denoted by discretize Equations (75) and (76), which can be denoted n, the step length of the space variable is denoted by h in as the above equations. An initial excitation is required in . this calculation, which is chosen as, an(o) = Asech[(n-n0) iφn () t=− εφn () t J [ φ n+−1 () t ++ φ n 1 ()] t ( χ /2 ru 01 )[n+− () t − u nn 1 ()] tφ ()2 t . (χ/2r0) /4JW] (where A is the normalization constant) at φ=− εφ φ ++ φ χ −φ i n () tn () t J [ n+−1 () t n 1 ()] t ( /2 ru 01 )[n+− () t u nn 1 ()] t () t (78) the size n, for the applied lattice, un(0) = yn(0) = 0. In the .. 22numerical simulation it is required that the total energy Mun () t= W [ un+1 () t −+ 2 u n () t u n−1 ()] t + (χφφ /2 ron )[|+−11 | − | n |] .. and the norm (or particle number) of the system must 22 Mun () t= W [ u () t −+ 2 u () t u ()] t + (χφφ /2 r )[| | − | |] (79) 0 n+1 n n−1 on+−11 n be conserved. The system of units, ev for energy, A for where the following transformation relation between length and ps for time are proven to be suitable for the continuous and discrete functions are used numerical solutions of Equations (81)-(85). The one
φφ(xt ,)→ n () t and uxt( ,)→ un () t dimensional system is fixed, N is chosen to be N = 200, and a time step size of 0.0195 is used in the simulations. ∂∂φφ()tt1 2 () φφ=±+nn2 ± Total numerical simulation is performed through data nn±1()()t tr 00 r 2 .... ∂∂xx2! parallel algorithms and MALAB language.
2 If the values of the parameters, M, ε, J, W, χ and ∂∂utnn()1 2 ut () =±+ ± r0, in Equations (78)-(79) are appropriately chosen we unn±1()() t ut r 00 r 2 .... (80) ∂∂xx2! can calculate the numerical solution of the associated 2 2 2 22 Equations (81)-(85) by using the fourth-order Runge- Hereεχ=/mr += A , J /2 mr , W = Mv / r , r 0 0 00 0 Kutta method[26-27] in the systems, thus the changes of is distance between neighbouring two lattice points. 22 φnn()t= at () , which is probability or number density If using transformation: φφnn→ exp(it ε / ) we can of the microscopic particles occurring at the nth structure εφ ()t eliminate the term n in Equation (78). Again making unit, with increasing time and position in time-place
a transformation:φnn()t→= a () t atr () nn + iati () , then can be also obtained. This result is shown in Figure 2. Equations (78)-(79) become This figure shows that the amplitude of the solution can retain constancy, i.e., the solution of Equations (78)-(79) ar =−++ J( ai ai )(/2)(χ r u − u ) ai (81) n n+1 n − 1 01n+− nn 1 or Equation (5) at V(x) = constant is very stable while - ai n=−++ J( ar n+1 ar n − 1 )(/2)(χ r 01 un+− − u nn 1 ) ar (82) in motion. In the meanwhile, we give the propagation 3 n un=y /M (83) feature of the solutions of Equations (81)-(85) in the cases of a long time period of 250ps and long spacings of 400 y= W( u −+ 2 u u )(/2)( +χ r ar22 + ai − ar 22 − ai ) n n+1 nn − 1 o n++−−1111 n nin n Figure 3, which indicates that the states of solution
22 22 are also stable in the long propagation. According to the y= W( u −+ 2 u u )(/2)( +χ r ar + ai − ar − ai ) [29-30] n n+1 nn − 1 o n++−−1111 n n n (84) soliton theory we can obtain that Equations (78)- (79) have exactly a soliton solution, thus the microscopic 2=+= 2 22φ an ar n ai nn (85) particles described by nonlinear Schrödinger Equations (5) are a soliton and have a wave-corpuscle feature. 2 | k a |
Figure 2 Motion of Soliton Solution of Equations (78)-(79)
47 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
Figure 3 State of Motion of Microscopic Particle Described by Equations (78)-(79) in the Cases of a Long Time Period and Long Spacings
In order to verify the wave-corpuscle feature of that the initial two particles with clock shapes separating microscopic particles described by nonlinear Schrödinger 50 unit spacings in the channel collide with each other Equations (5) we should also study their collision property at about 8ps and 25 units. After this collision, the two [29-30] in accordance with the soliton theory . Thus we further particles in the channel go through each other without simulated numerically the collision behaviors of two scattering and retain their clock shapes to propagate particles described by nonlinear Schrödinger Equations toward and separately along itself channels. The collision 2 (5) at V(x) =εχ= / mr0 + A = constant using the fourth- properties of microscopic particles described by the order Runge-Kutta method[58-59]. This process resulting nonlinear Schrödinger Equation (5) are same with those obtained by Zakharov and Shabat[39-40] and Asossey et from two microscopic particles colliding head-on from [60] opposite directions, which are set up from opposite ends al. as mentioned above. Meanwhile, this collision is of the channel, is shown in Figure 4, where the above same with the rules of collision of macroscopic particles. initial conditions simultaneously motivate the opposite Thus, we can conclude that microscopic particles ends of the channels. From this Figure we see clearly described by nonlinear Schrödinger Equations (5) have a corpuscle feature.
Figure 4 The Features of Collision of Microscopic Particles Described by Nonlinear Schrödinger Equation (5)
However, we see also from Figure 4 to occur a wave determine the nonlinear Schrödinger equation can describe peak of large amplitude in the colliding process, which correctly the natures and properties of microscopic appears also in Desem and Chu’s numerical result[56]. particles in quantum systems. Obviously, this is a result of complicated superposition of two solitary waves, which thus displays the wave feature of the microscopic particles. Therefore, the collision process CONCLUSION shown in Figure 4 represent clearly that the solutions of the In this paper we used a nonlinear Schrödinger equation, nonlinear Schrödinger equation have a both corpuscle and instead of Schrödinger equation in quantum mechanics, wave feature, then the microscopic particles denoted by to describe the properties of microscopic particles. In the the solutions have also a wave-corpuscle duality, which is dynamic equation the nonlinear interaction is denoted due to the nonlinear interaction b|φ |2φ . Thus we can again by b|φ |2 φ , which is caused by the interaction between
Copyright © Canadian Research & Development Center of Sciences and Cultures 48 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51 the particle and other particles or background field in Hall Englewood Cliffs, New Jersey. the case of their real motions to be considered. The [12] Potter, J. (1973). Quantum Mechanics (pp. 43-105). North- nonlinear interaction can suppress the dispersive effect Holland Publishing Co., Amsterdan. of kinetic energy in the dynamic equation and change [13] Jammer, M. (1989). The Concettual Development of also the natures of microscopic particles. Thus we Quantum Mechanics. Tomash, Los Angeles. investigated deeply the dynamic and collision features of [14] Bell, J. S. (1987). Speakable and Unspeakable in Quantum microscopic particles described by nonlinear Schrödinger Mechanics (pp. 51-132). Cambridge University Press, equation using the analytic and the Runge-Kutta method Cambridge. of numerical simulation through finding the soliton [15] Einstein, A., Podolsky, B., & Rosen, N. (1935). Phys. Rev., solutions of the equation. The results obtained show that 47, 777. the microscopic particles have a wave-corpuscle duality [16] French, A. P., & Einstein, A. (1979). Centenary Volume (pp. and are stable in propagation. When the two microscopic 64-112). Harvard University Press, Cambridge, Mass. particles are collided, they can go through each other [17] PANG, X. F. (1985). Problems of Nonlinear Quantum and retain their form after their collision of head-on Mechanics (pp. 23-167). Sichuan Normal University Press, from opposite directions, which is the same with that of Chengdu. the classical particles. However, a wave peak of large [18] PANG, Xiaofeng & FENG, Yuanping (2005). Quantum amplitude, which is a result of complicated superposition Mechanics in Nonlinear Systems. New Jersey, World of two solitary waves, occurs in the colliding process. Scientific, Publishing Co.. This displays the wave feature of microscopic particles. [19] PANG, Xiaofeng (1994). Theory of Nonlinear Quantum Therefore, the collision process shows clearly that the Mechanics. Chongqing, Chinese Chongqing Press. solutions of the nonlinear Schrödinger equation have a [20] PANG, Xiaofeng (2009). Nonlinear Quantum Mechanics both corpuscle and wave feature, then the microscopic (pp. 1-312). Beijing, Chinese Electronic Industry Press. particles represented by the solutions have a wave- [21] PANG, Xiaofeng (2008). Physica B, 403, 3571. corpuscle duality. Clearly, this nature is due to the [22] PANG, Xiaofeng (2008). Fronts of Physics in China, 3, 243. nonlinear interaction b|φ |2φ . Thus we can determine the [23] PANG, Xiaofeng (2008). Nature Sciences, 3(1), 29. nonlinear Schrödinger equation can describe correctly the [24] PANG, Xiaofeng (2007). Nature Sciences, 2(1), 42. natures and properties of microscopic particles in quantum [25] PANG, Xiaofeng, Mod. (2009). Phys.Lett., B, 23, 939. systems. This result has an important significance in [26] PANG, Xiaofeng (2010). Physica B, 405, 2317. physics. Then a new and nonlinear quantum mechanical [27] PANG, Xiaofeng (2009). Physica B, 405, 4327. theory can be established based on these results. [28] PANG, Xiaofeng (1982). Chin. Nature Journal, 4, 254. [29] PANG, Xiaofeng (2003). Soliton Physics. Chengdu: Sichuan Science and Technology Press. ACKNOWLEDGEMENT [30] GUO, Bailin & PANG, Xiaofeng (1987). Solitons. Beijing: Chinese Science Press. Author would like to acknowledge the Major State Basic [31] PANG, X. F. (2008). Physica B, 403, 4292. Research Development Program (“973” program) of China [32] PANG, X. F. (2008). Frontiers of Physics in China, 3, 413. for the financial support (Grant No. 2007CB936103). [33] PANG, X. F. (2008). Nature Sciences, 3, 29. [34] PANG, X. F. (1985). Chin. J. Potential Science, 5, 16. REFERENCES [35] PANG, X. F. (1991). The Theory of Nonlinear Quantum Mechanics. In Lui Hong (Ed.), Research of New Sciences [1] Bohr, D., & Bub, J. (1935). Phys. Rev., 48, 169. (pp. 16-20). Beijing, Science and Tech. Press. [2] Schrödinger, E. (1935). Naturwissenschaften., 23, 807. [36] PANG, X. F. (2006). Research and Development and of [3] Schrödinger, E. (1926). Phys. Rev., 28, 1049. World Science and Technology, 28, 11. [4] Heisenberg, W. Z. (1925). Zeitschrift der Physik, 33, 879. [37] PANG, X. F. (2006). Research and Development and of [5] Born, M., & Infeld, L. (1934). Proc. Roy. Soc. A, 144, 425. World Science and Technology, 24, 54. [6] Dirac, P. A. M. (1948). Phys. Rev., 73, 1092. [38] PANG, X. F. (2006). Features of Motion of Microscopic [7] Diner, S., Farque, D., Lochak, G., & Selleri, F. (1984). The Particles in Nonlinear Systems and Nonlinear Quantum Wave-Particle Dualism. Riedel, Dordrecht (pp. 34-96). Mechanics. In Sciencetific Proceding-Physics and Others [8] Ferrero, M., & Van der Merwe A. (1997). New (pp. 53-93), Atomic Energy Press, Beijing. Developments on Fundamental Problems in Quantum [39] Zakharov, V. E., & Shabat, A. B. (1972). Sov. Phys. JETP, Physics. Kluwer, Dordrecht (pp. 56-97). 34, 62. [9] Ferrero, M., & Van der Merwe A. (1995). Fundamental [40] Zakharov, V. E., & Shabat, A. B. (1973). Sov. Phys. JETP, Problems in Quantum Physics. Kluwer, Dordrecht (pp. 48- 37, 823. 91). [41] PANG X. F. (1985). J. Low Temp. Phys., 58, 334. [10] Broglie de L. (1955). Nuovo Cimento, 1, 37. [42] PANG X. F. (1989). Chinese J. Low Temp. and Supercond, [11] Bohm, D. (1951). Quantum Theory (pp. 36-98). Prentice- 10, 612.
49 Copyright © Canadian Research & Development Center of Sciences and Cultures The Dynamic and Collision Features of Microscopic Particles Described by the Nonlinear Schrödinger Equation in the Nonlinear Quantum Systems
[43] PANG, X. F. (2008). J. Electronic Science and Technology [53] PANG, X. F. (2003). Phys. Stat. Sol. (b), 236, 34. of China, 6, 205. [54] PANG, X. F. (2007). Nature Sciences, 2, 42. [44] CHEN, H. H., & LIU, C. S. (1976). Phys. Rev. Lett., 37, [55] PANG, X. F. (2000). Phys. Rev. E, 62, 6989. 693. [56] Desem, C., & Chu, P. L. (1992). Soliton-Soliton Interactions [45] CHEN, H. H. (1978). Phys. Fluids, 21, 377. in Optical Solitons. In J. R. Taylor (Ed.) (pp. 107-351). [46] PANG, X. F. (2009). Mod. Phys. Lett. B, 23, 939. Cambridge University Press, Cambridge. [47] PANG, X. F. (1993). Chin. Phys. Lett., 10, 437. [57] Tan, B., & Bord, J. P. (1998). Davydov Soliton Collisions. [48] Sulem, C., & Sulem, P. L. (1999). The Nonlinear Phys. Lett., A240, 282. Schrödinger Equation: Self-Focusing and Wave Collapse [58] Stiefel, J. (1965). Einfuhrung in Die Numerische (pp. 26-89). Springer, New York. Mathematik. Teubner Verlag, Stuttgart. [49] Makhankov, V. G., & Fedyanin, V. K. (1984). Phys. Rep., [59] Atkinson, K. E. (1987). An Introdution to Numerical 104, 1. Analysis. Wiley, New York Inc.. [50] Makhankov, V. G. (1978). Phys. Rep., 35, 1. [60]Aossey, D. W., Skinner, S. R., Cooney, J. T., Williams, J. E., [51] PANG, X. F. (2009). Physica B, 404, 3125. Gavin, M. T., Amdersen, D. R., & Lonngren, K. E. (1992). [52] ANG, X. F. (2009). Mod. Phys. Lett. B, 23, 2175. Phys. Rev., A45, 2606.
Copyright © Canadian Research & Development Center of Sciences and Cultures 50 PANG Xiaofeng (2012). Advances in Natural Science, 5(4), 36-51
APPENDIX The Solutions of Equations (81)-(85) From Equations (81)-(84) we can easily find out its solutions which are as follows h ar ar (K1 2K 2 3K3 K4 ) j, n1 jn 6 j j j j h ai=++++ ai(1 L 2 L 2 2 L 3 L 4) j, n+1 jn 6 j j j j h u=++++ u( MM 1 2 2 2 MM 3 4) j, n+1 jn 6 j j j h y + = y + (N1 + 2N2 + 2N3 + N4 ) j, n 1 jn 6 j j j j where J χ K1(j=−++ aij+−11 ai jn )( ujn +− 11 − u jn ) ai 2r0 Jh χh χhh K2j=− K 1 j (1 Lj+1 ++ L 1 j − 1 ) ( ujn +− 11 − uL jn )1 j+(M 1j+−11 −+ M 1j )( ai jn L 1 j ) 24r0 42r0 Jh χh χhh K3j=− K 1 j (2 Lj+1 ++ L 2 j − 1 ) ( ujn +−1, − uL jn 1, )2 j+(M 2j+−11 −+ M 2j )( ai jn L 2j ) 24r0 42r0 Jh χh χh K4j=− K 1 j (3 Lj+1 ++ L 3 j − 1 ) ( ujn +−1, − uL jn 1, )3 j+(M 3j+−11 −+ M 3j )( ai jn hL 3 j ) 2r0 2r0 J χ L1(j= ar j+1, n +− ar jn − 1 ) ( ujn +−1 − u jn 1 ) ar jn 2r0 Jh χh χhh L2j=+ K 1 j (1 Kj+1 +− K 1 j − 1 ) ( ujn +− 11 − uK jn )1 j−−+(M 1j+−11 M 1j )( ar jn K 1j ) 24r0 42r0 Jh χh χhh LL3j=+ 1 j (2 Kj+1 +− K 2 j −1 ) ( ujn +−1, − uK jn 1, )2 j−−+(M 2j+−11 M 2j )( ar jn K 2j ) 24r0 42r0 Jh χ1h χh L4j=+ K 1 j (3 Kj+1 +− K 3 j − 1 ) ( ujn +−1, − uK jn 1, )3 j−−+(M 3j+−11 M 3j )( ar jn hK 3 j ) 2r0 2r0 = M1 j y jn / M ; h M 2 = M1 + N1 j j 2M j ; h M 3 = M1 + N2 j j 2M j ; h M 4 = M1 + N3 j j M j ;
22 22 N1j= W ( u jn+1, −+ 2 u jnjn u −1, )(/)( +χ r0 arjn++−− 1, + ai jn 1, − ar jn 1, − ai jn 1, ) h N2= Wu [( −+ 2 u u ) + ( M 1 − 2 M 1 + M 1 )] j j+1, n jn j −+1, n 2 j 1 j j −1 h hh h +(χ /r )[( ar + K 1 )22 ++ ( ai L 1 ) −+ ( ar K 1 ) 22 −+ ( ai L 1 ) ] 0jn+ 1, 2 j ++1 jn1, 22 j +−1 jn1, j −−1 jn1, 2 j −1 h N3= Wu [( −+ 2 u u ) + ( M 2 − 2 M 2 + M 2 )] j j+−+1, n jn j1, n 2 j 1 j j −1 h22 hh 22 h +(χ /r )[( ar + K 2 ) ++ ( ai L 2 ) −+ ( ar K 2 ) −+ ( ai L 2 ) ] 0jn+ 1, 2 j ++1 jn1, 22 j +−1 jn1, j −−1 jn1, 2 j −1
N4j = W [( uj+1, n −+ 2 u jn u j −+1, n ) + hM ( 3j 1 − 2 M 3j + M 3 j −1 )]
+χ +22 ++ −+ 22 −− ( /r0 )[( arjn+ 1, hK 3 j ++1 ) ( ai jn1, hL 3 j +−1 ) ( ar jn1, hK 3 j −−1 ) ( ai jn1, hL 3 j −1 ) ]
These are just base equations we make numerical simulation by computer and Runge-Kutta method.
51 Copyright © Canadian Research & Development Center of Sciences and Cultures