PECULIARITIES PASSAGE OSCILLATOR AS STRUCTURED PARTICLE THROUGH THE POTENTIAL BARRIER

Transcription

1 PECULIAITIES PASSAGE OSCILLATO AS STUCTUED PATICLE THOUGH THE POTENTIAL BAIE V. M. Soskov, M. I. Densena epulc of Kazakhstan, Alaty, Insttute of Ionosphere MES K, 48 Е-al: The results of the nuercal soluton of the oton of the oscllator through a potental arrer otaned on the ass of the descrpton of the dynacs of systes for the two types of syetry: the syetry of the syste and the syetry of space gven are sutted. These results support the need to present the equatons of oton systes n ndependent cro-varalty defnng ther nternal energy, and acro varales that deterne the oton of the center of ass of the syste n the feld of external forces. The proposed approach to the descrpton of the dynacs of systes allowed revealng prncpal features of ther dynacs. In partcular, the passage of the oscllator s set elow the arrer, as well as the "quantzaton" of energy transsson. Explanaton of the echans ased only on the laws of classcal echancs. Introducton A key dea n defnng the approach to the descrpton of the dynacs of systes ased on the knowledge of the laws of the dynacs of ther eleents s the dea of aout two types of syetry, deternng the dynacs of the syste: the syetry of space and the syetry of the syste. On the ass of ths dea s a ased echancs structured partcle (SP), whch s an equlru syste of potentally nteractng ateral ponts (MP) [, ]. It s through the representaton of energy n accordance wth the duals of the syetry as the su of the nternal energy of the SP and the energy of ts oton n space; t was possle to otan the equatons of oton n the non-hoogeneous space, ased on Newton's laws for the MP. [7] Ths equaton, n contrast to Newton's equaton for the MP, s rreversle. Irreverslty arses n connecton wth the ncluson of a descrpton of the nonlnear oton energy transforaton nto the nternal energy. Force, whch enjoys such a transforaton, can not e expressed through the gradents of scalar functons. Ths eans that the avalalty of the systes of nternal energy dynacs akes the fundaentally dfferent fro the dynacs of unstructured odes. Here t s shown y the exaple of the dynacs of the ost sple syste - oscllator (OS), whch conssts of two potentally nteractng MP. OS s rather tradtonal oject of research. He's solved a huge nuer of proles. For exaple, he has een gven consderale attenton n the work of Mandelsta [4.]. Its splcty has helped to nvestgate the prole of xng n Haltonan systes and open deternstc chaos [5]. OS s at the heart of the theory of quantu electrodynacs [6]. Currently operatng actvely nvestgated n the study of ssues related to the echanss of the structures and clusters []. As a rule, the study of the dynacs of systes n classcal echancs s carred out ether n the fraework of the Haltonan forals, or y usng eprcal equatons that take nto account the dsspaton. The study of the dynacs of the operatng syste n an nhoogeneous feld forces fro postons of exchange etween ts nternal energy and the energy of oton s alost not satsfed. Need to take account of ths exchange have ecoe clear n the study of non-equlru systes. [6]. How t was shown n [7], t s the relatonshp etween these types of energy deternes the dynacs of systes. It turned out that, despte ts splcty, the dynacs of the OS already evdent an characterstcs of the systes, such as the dualty syetres of the syste and space. Here on the exaple of the nuercal soluton prole of the passage runnng through the potental arrer s shown that allowance for of nternal energy and the posslty of ts transforaton nto energy of oton, can qualtatvely change ts dynacs. Precsely n ths task, the ost profound and easy t s coe to the understand the need for these systes wthn the two herarchcal levels y ntroducng cro paraeters characterzng the dynacs of the syste

2 eleents and acro paraeters characterzng the oton of the syste n a heterogeneous space. [3]. Only under such a descrpton are shown the qualtatve dstngush of the dynacs of dsspatve nonequlru dynacs of unstructured odes resultng nonlnear relatonshp of the nternal energy and the oton energy. The an goal of ths work was to dentfy patterns of dynacs OS due relatonshp of the nternal energy and the energy of the syste. Stateent of the Prole To solve ths prole we take as a odel the OS fro two potentally nteractng MP. So the OS can serve as a two MP connected y the sprng. The OS oton n a non-unfor external feld s characterzed y ts total energy or Haltonan. Under the condton the hoogenety of te the energy s nvarant of the oton. Let us consder the passage of OS through a potental arrer, dependng on the rato of the oton energy, the nternal energy, the arrer heght and the relatons of the lnear densons of the arrer and the OS. For these purposes, let us represent the energy of OS as the su of the nternal energy and the oton energy. In the laoratory syste of coordnates (LCS), the energy of the OS n general can e wrtten as follows: ( ) ( ) E = K + K +U +U +U x& x& () = + +U +U( x ) +U( x) Here x and x -are coordnates of the frst and second MP accordngly; x& and x& -are veloctes of the frst and second MP accordngly; and -are asses of the frst and second U - s a heght of the MP accordngly; U( x ) - s a potental energy MP n the external feld; potental arrer; - s poston of extree arrer; a - s a half-wdth of the arrer; U -s the energy of MP nteracton. Then for OS we have: U k( x x l ) = /. Fro the condton of the hoogenety of te the condton: E= & have a place. Then fro () we have E & = x& { && x + U + F } x x + x& { && x + U F } = Here U x = U / x, U x = U / x, F = k( x x l). Let us wrte the energy of the OS n the center of ass (CM). In ths coordnate syste the energy of any structured ody s the su of the energes of the eleents of the ody relatve to the CM, whch s called the nternal energy, and the oton energy of the ody, whch s gven y the coordnates and veloctes of the CM of the ody. Varales deternng the energy of the oton of CM and the nternal energy of the ody, are ndependent. To go n to the CM, perfor the followng coordnate transforaton: = x + x, + r = x x. We can wrte n ths case for eq. ():. ( ) ( ) E =T + T + U + U + U (а) r r & & () M µr = + + U ( r) + U( + ) + U( ) Here M = + ; µ = M& µr ; T M = - s a knetc oton energy for ОС; T = & - s a knetc part of the nternal energy of ОС; U = U ( ) r - s a potental part of the nternal energy; ( ) ( ) ( ) ( ) r U + U = U + U -s a potental energy of the frst and second MP n the external feld +

3 of force; = + + ( µ / ) r, = ( µ / ) r. For eq.() we have: E & = & [ MV& + U ( ) + U ( )] where U +, = U /, v= r&. +, + If the external feld s gven y the functon U( x) forces actng on each of the MT s gven y: µ µ + r& [ µ v& + k ( r l ) + U ( ) U ( )] =, (3) r + r ( x ) / a = U e, then the correspondng x U( x) U F a ( x ) = = ( x ) e x a (4) Accordng to (4), the force s proportonal to the arrer heght and nversely proportonal to the square of ts wdth. Let us: = =. Then: µ=, U ( + ) = F( + ), Ur ( + ) = F( + )/, U ( ) ( ) = F, Ur ( ) = F( ) /. Consder the possle cases. Let us we have: a>>. Then: + =, Ur( + ) Ur( ) = Ur( ). If >> r then the varales n (3) are separated. Ths leads to the well-known equatons of oton for the OS: MV& = F( ) (5) v& / = k( r l) (6) In accordng wth the eq. () the OS oton s deterned y the next equaton: && x( t) = F ( x x) + F( x ), && x t = F x x + F x. (7) ( ) ( ) ( ) where F ( x x) U / x = k( x x l) second MP on the frst MP; F ( x x ) = U / x = k( x x l) = - s nternal force whch acted fro the sde of - s nternal force whch acted fro the sde of frst MP on the second MP: k - s a sprng stffness; l - s a wdth of the arrer. F x x = F x x The next condton for the nternal forces have a plase: ( ) ( ) Fg.. The schee to the calculaton of the OS passage through the potental arrer. The nuercal calculatons were perfored oth n the laoratory syste coordnates and the coordnates of the CM. Ovously, the nuercal results do not depend on the chosen coordnate syste, ut the advantage of the CM coordnates consst n the posslty of dentfyng the physcal nature of the dual syste of coordnates (DSC) for structured ody havng nternal energy. Here, under the DSC eans that the OS oton s deterned y the CM oton n space and MP oton are deterned relatve of the CM. In DSC the OS energy splts nto the nternal energy and the energy of oton of CM, and the equatons of oton take the for:

4 M&& ( t) = F ( ) + F ( ), F( ) F( ) ( ) = ( ) +, (8) µr && t F r µ where the arguents of the external forces and are n accordance wth µ µ x = + r, x r =. In Fg. shows the schee of the task. The calculatons were perfored for the case of lnear OS. Accordng to these paraeters of the task, such as the sprng constant, the length of the OS, the value of nternal energy and knetc energy of the CM are selected. To perfor the Hooke's law, the total energy of the syste for the kl OS s set uch lower than the axu possle nternal potental energy: E< Ur= = = Eax. In calculatons the control of perforance of necessary condtons that deterne the accuracy, ncludng the conservaton of the total energy s carred out. The nvestgaton of the OS dynacs s realzed n the followng areas of paraeters: < E <U, <U <T < E, <T <U < E.. The ntal condtons for the syste of dfferental =, & =V,r = r, r & = v. equatons of the second order can e wrtten as: ( ) ( ) ( ) ( ) r Asolute values V, v, r r were deterned fro the ntal condtons defned types of energes,.e. out of T,T r,u r, respectvely. The ntal coordnate of the CM was deterned fro the condton: >> a µ V T, where T = π - s a OS perod of oscllaton n an area where the k external forces are neglgle,.e. away fro the arrer. The latter eans that the OS anages to ake a nuer of oscllatons efore the scatterng n the arrer. The half-wdth of the potental arrer a was deterned n unts and ranged < a <<. Sprng constant and ass MP were selected on the ass of the requred values of the characterstc te scale of the task and lnear oundares nterpartcle forces. In nuercal calculatons the OS consdered to have passed through the arrer, f the CM reached dstance l ehnd the arrer, when the half-wdth of the arrer l > a and a when l < a. ESULTS OF CALCULATIONS In order to verfy the correctness of the account frst calculatons were ade for the passage of one MT at preset threshold. Calculatons confred the known fact of classcal echancs that MT only passes through a potental arrer, when the condton E / U > have a place. Then we calculated for the case k, x x = l = const - whch corresponds to an asolutely rgd duell. In ths case, the Haltonan, defned up to a constant, s as follows: ( + ) x& E = +U( x ) +U( x +l). (9) In Fg. s a graph of the OS cross the arrer n dependng fro the rato of ts sze to half the wdth of the arrer: l / a. It was assued that the OS was cross a arrer, f carred out, as a nu, the followng condtons: x( t) =, x& ( t ) =. Where t - s the te when the end of the OS on at the top of the arrer. Crteron of the cross fro the arrer s a condton: ( l / a ) / > T U + e.

5 Fg.. The dependng of the total energy transsson MP (left) OS through the arrer fro the relatve sze of the OS (rght). The ntal energy of oton changes fro zero. The arrer heght s.3. Changes the sze of the OS. OS reflecton fro the arrer wth these characterstcs s sgnfcantly dfferent fro the task of reflecton MP wth the ass M = + n the area when ts sze s less than or coparale wth the half-wdth of the arrer. If the sze of the OS s uch saller than the wdth of the arrer, t ehaves lke the MT doule weght. If the sze of OS exceed the wdth of the arrer at least twce, then to pass through the arrer of energy s requred n half. Ths s ecause the OS nteracton wth the arrer s deterned only under ts area of one MP. Now consder the general case of the dynacs of the OS. If at the te of passage over the µ arrer coordnate and velocty of the CM OS oey to the qualtes ( t) = + r( t), & ( t ) =, then OS wll e consdered to have passed through the arrer. The total energy OS s: ( r / ) ( ) a r & = r& t s a relatve velocty of MP. E = µr& / + k r l / + U ( + e ), () where r = r( t ) s a dstance etween MP, ( ) In accordng wth the eq. () the energy threshold for s not unque. It s defned as the characterstcs of the arrer and the relatve velocty r& and poston r at the oent of the passng through the arrer. So t s deterned y the nternal energy of OS also. It s ovous that r and ṙ on the one hand deterned y the ntal condtons r ( ) and r& ( ), on the other hand, ntal condtons ( ) and & ( ), and the dynacs descred y the syste of equatons of oton OS. Thus the passage of OS through a arrer n the general case should e defned y the total energy, ts redstruton etween nternal energy and the oton energy, as well as the ntal condtons or the phase. Let us consder the results of the calculaton of the passage OS at zero nternal energy. They ply that the OS goes through the arrer as soon as the energy of the oton of ts CM s equal to or hgher than the energy arrer. Nuercal crteron for cross-country, n ths case, was the poston of the CM = +3 l. In Fg. 3 shows the dependng of the passage OS fro the changes of the nternal energy at constant energy CM. It s vared the startng poston of CM, for fxed values of the half-wdth a = l / ( l =) and the hgher of the arrer U =.3.

6 а Fg. 3. Dependence OS oton fro the nternal energy when CM oton energy s constant. The ntal energy of the CM s elow the arrer heght y aout 5% for all calculatons. On the fg. 3 and n susequent slar fgures the color ntensty s the easure of the nternal energy transforaton nto the CM energy and vce versa, n the followng quanttes: ε= ( T T )/ T %. а Fg. 4. Dependence of the OS otons through the arrer fro the CM energy. Barrer wdth.5/ In Fg. 4 dsplays the results of the calculaton, when varyng the ntal energy of the CM oton wth zero ntal nternal energy. arrer heght.3, wdth.5 arrer, sze OS. It s seen that etween the regons of total reflecton and transsson, there s an nteredate regon wth dscrete trench transsson and reflecton. These are deterned y the lne of the phase condton n whch the external force actng on the OS sde of the arrer, less than the forces of the two MT. In Fg. 5 dsplays the results of the calculaton also vares the ntal energy of oton CM. But the ntal nternal energy of OS s.5; the arrer heght s.3; the arrer wdth s.5; the sze of OS s. а Fg. 5. Dependence of the OS otons through the arrer fro the CM energy when the nternal energy s constant. Barrer wdth s.5.

7 а Fg. 6. Dependence of the OS otons through the arrer fro the CM energy when the nternal energy s constant. Barrer wdth s.5. а Fg. 7. Dependence of the OS otons through the arrer fro the nternal energy when CM energy s constant. Thus, the reducton n the wdth of the arrer leads to an expanson of the reflecton OS, ut the slands areas of passage are appearng. In general, we fnd that an ncrease n the nternal energy fro.5 to., or apltude, the reflecton heght of the arrer s ncreases. In ths case, there are sall slands n the perodc passage of the reflecton. а - ntal condtons s slar as on Fg ntal condtons s slar as on Fg.7а.

8 Fg. 8. The ntal and fnal postons OS. In Fg. 8 shows the results of calculatons of passage OS dependng on the energy of the CM (Fg. 8a); nternal energy (Fg. 8 ) and the ntal poston of the CM. Total nuer of OS s N = 4,4. The nuer of OS that have passed through the arrer s Nt = The nuer of reflected OS s Nr =,37. The arrer s located at around, and dd not exceed the wdth of the.5 when the sze of the OS s. Thus, the presence of the nternal energy leads to the fact that the OS can pass through the potental arrer n the case when the heght of the arrer aove the energy of oton of the OS. Ths passage provdes a posslty to transfor the nternal energy of the OS n ts nternal energy at certan phase relatonshps. It s ovous that the passage s only possle f the total energy of OS aove the energy arrer. The perodc structure of reflecton and passng through arrer whch deterned y the wdth of the arrer, ntal condtons, the phase relatonshp etween the nternal energy of eleents, the OS frequency, s appearng. In certan cases wthn ths area there are perodc slands of the passng. CONCLUSION The an result of the nuercal analyss of the OS passng through the potental arrer s a proof of concept of the need to sut energy systes as the su of ther oton energy and nternal energy. Ths representaton s acheved y transton to an ndependent cro and acro varales. [7] Mcro-varale characterzes the oton of the eleents of the syste relatve to the center of ass, and acro varales deterne the oton of the syste as a whole. Only thanks to the dvson of energy nto these two types, t was possle to estalsh the nature of the OS passage through a potental arrer. Ths nature s connected wth the posslty of OS nternal energy transforaton to the oton energy. As a result the syste can to overcong the arrer when the oton energy s less than the heght of the arrer. Ths s due to the non-lnear transforaton of the OS nternal energy nto the energy of ts oton on the gradents of the arrer [3]. The OS reflecton fro the arrer s also possle when the energy of oton of OS aove the energy arrer. It s due to the converson of oton energy nto the nternal energy. I.e. for a gven energy of oton OS can e oserved as the passage and reflecton fro the potental arrer, as n the knetc energy of the hgher energy arrer and the reverse condton. For exaple, f the nternal energy of the frst OS s zero, then t cannot pass through the arrer, even f the energy of oton aove the potental arrer. The passage depends on the ntal phase of the oton, fro the ntal values of the nternal energy and the energy of oton, the rato etween the heght of the arrer and the energy of oton, as well as the rato of the sze of OS to the wdth of the arrer. The rato etween the energy of oton and nternal energy s change ut the OS total energy s a constant. Banded structure of transsson and reflecton of the OS through the arrer deterne the phase characterstcs of the ntal paraeters of the OS. Ths structure looks lke structure of the passage of an electron n an ato through a potental arrer, whch led N.Bohr to the deas of quantzaton of energy functon of the electron fro the ato []. But the OS pass across arrer s a fundaental dfference. Thus, n the fraework of quantu echancs, the passage of eleentary partcles over a potental arrer s due to the presence of a non-zero wave functon for passage. But the OS passage s deterned y the laws of classcal echancs. The OS features of the dynacs and the fundaental dfference etween the dynacs of the rgd ody due to the presence of nternal energy and the posslty of ts non-lnear transforaton wthn the energy arrer to the oton energy and vce versa. Nonlnearty of the OD dynacs s due to the gradent of the external forces. Ths gradent s necessary to change the nternal energy [, 7]. The vanshng of the gradent leads to the constancy of the nternal energy. Thus, the an dfference the systes fro structureless partcles s the presence of nternal energy.

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