Anharmonik salıcının devinim denklemlerinin oluşturulmasında dalgalanma değişkenlerinin kullanımı

Abstract

Bu çalışına, lineer olmayan katkıların yeterince küçük olarak düşünüldüğü, titreşim devinimlerinin modellenmesi için kullanılan Anharmonik salmıcı probleminin çözümü için; kısmi türevli bir yapıdaki Sclırödinger denkleminin çözü mü olan dalga fonksiyonu yerine "Taban Operatör Açılım" yönteminin kullanılması ile ilgilidir. Bu yöntem ile, taban operatörü olarak tanımlanan ve konum operatörleriyle momentum operatörlerinin çeşitli kuvvetlerinin çarpımlarım içeren terimlerden oluşan yapıların beklenen değerlerinin zamana bağlı bilinmeyen fonksiyonlar olarak gündeme getirildiği bir yaklaşıma gidilmektedir. M. Demiralp tarafından ortaya atılan bu yaklaşımda, taban operatör kümesinin ilgilenilen sistemin Hamiltoniyen'i ile komütatörünün küme içinde kalan bileşenler oluşturduğu varsayımına dayandırılmaktadır. Bu varsayımın geçerli olması durumunda oluşan sıradan diferansiyel denklem sistemi sabit katsayılı lineer sıradan bir diferansiyel denklem yapısı taşımakta ve homojen olma özeliğini de göstermek tedir. Seyrek matris yapısı taşıyan bu denklem sisteminin çözümü özel sayısal önlemlerin almmasım gerektirir. Bu çalışmada, seyrek matris yapışım gidermek, bir anlamda denklemleri tıkızlaştırmak için dalgalanma operatörü olarak nitelendirilen ve konum ope- ratörüyle konum operatörünün beklenen değeri arasındaki farkın kuvvetleriyle momentum operatörüyle momentum operatörünün beklenen değeri arasındaki farkın kuvvetlerinin çarpımının iç.erildiği terimler içeren yapılardan yararla nılmıştır. Dalgalanma değişkenleri, yalnızca zamanla değişen büyükler olup, denklemleri tıkızlaştırmak için kullanılan dalgalanma operatörlerinin beklenen değerleri olarak tanımlanmıştır. Yapılan çalışmalarda; dalgalanma değişkenle rinin gözardı edilmesiyle elde edilen denklemlerin klasik denklemlerle çakıştığı gözlenmiştir. Çalışmada ikinci ve üçüncü dereceden dalgalanma değişkenlerinin içerildiği lineer olmayan diferansiyel denklem sistemlerinin çözümleri Mathematica paket programı ile elde edilmiştir. Çözümlerde kaotik görünümlü davranışlar gözlen- miştir.Ayrıca momentum, komün ve dalgalanma değişkenlerinin çözümlerinin grafikleri ve çözümlerin yorumlarını da içermektedir.

This work deals with tlıe use of tlıe metlıod of Bases Operatör Expaıısion for the solutioıı of the aıılıarnıoııic oscillator tlıat is tısed for modelling the oscillatory motions witlı suıall eııouglı ııoıı-linear parts, instead of the wave function which is soltttion of a Sclırdinger's eqtıation in partial differential form. The poteııtial function with the least order of noıı-linear term, named as The Forth Order Anharmonic Oscillator is as follows V(x) = ^k,x2 + İfc2x4 here fe] and &2 are ııaıned as harınonic and amıarmonic force constants respectively.Our model is valid for the cases for which the non-linear part can be considered small enough and this is generally valid for the solid state phases. The quantunı nıotioııs of the systenı can be represented by the Schrdinger's equation.But in this work ııo attention is paid for the solution of the equation which has partial differentioııs, instead xısing the metlıod of Bases Operatör Expansions, systenı is respected by a systenı of ordiııary difFereııtial equations. As tlıe Bases Operatör Expansion nıethod, an approach tlıat uses the ex- pected values of tlıe forıııs tlıat constitııte multiplications of ternıs of various powers of the location and tlıe nıomentum operators, as uııkııown functions of tlıe time is enıployed.This approach is proposed by Prof.Dr.Metiıı Demiralp and it relies oıı tlıe assumptioıı tlıat tlıe comnıtıtator of the set of base oper¬ ators with tlıe Hamiltonian of the systenı produces componeııts tlıat are also in the set. If the set of the base operators is coııstructed as beiııg closed under the operation of the comnmtatioıı, it is possible to write tlıe rate of change of the expected valııes of ali the base operators as liııear combinations of again tlıe base operators.By tlıis, it becomes possible to reaclı a set of denumarably iııfiııite number of unknowııs for which the expected values of ali the base op- v erators are considereci as seperate variables, aııcl to obtaiıı a system of orclinary linear differeııtial equations for these unkııowns. Tlıis system of equations is lıomogeneotıs and the coefficients matrix is as of a constant rnatrix depending only on the system parameters and of infiııite dimensions.If the operators men- tioned above are selected arbitrary, it leads that the eleınents of the coefficients matrix densely zero. Special cotıtions must be takeıı for the numerical solu- tions of the orclinary differential eqtıatioııs with such coefficients matrixes. in this respect, sonıe similarity transfoıınations can also be coıısidered. Though these and similar approaches in general help to reach to the desired accuracy they cannot wholly overcome the restrictions brought by the limitations on the memory and time demands of the numerical calculatioııs. The purpose of get- ting rid of the sparse nıatrix structure is that using as less as possible variables to reach as much accuracy as possible. Tlıis point becomes more crucial when the number of unknowns become infinite. in this work, for changiııg the sparse matrix structure and in a respect put the equations in a thight form, a structure nanıed as fluctuations operatör which consists of the terin like the uıultiplications of the powers of the differences between both the location ant the momenttim, operators and their respective expected values, is used. The fiuctuation variables depend only on time and they are defined as the expected values of the fluctuation operatör that is used for making equation tighter. Taking the degree of freedom of the system. as öne it is made posibble to discuss the evaluation of the system in the classical dynamics sense by time dependent location and momentunı variables. And this leads us to reach to the quantum dynamics of the system by the expected vahıes of the location and the momenttim operators. For this purpose the location and momenttim functions depending on the time that are closely related by the location and momentunı variables are defiııed as follows, / O \ q(t} = (x), p(t) = (-iîı-\ \ ux l and for obtaining the fluctuation variables systematically the quanturn counter- parts of the location and nıoınentum operators are defined as ç = x, p - -i%-^- then the fluctuation variables are defined as the expected values of the opera¬ tors that are created as the mtütiplications of the non-negative integer powers of the operators (q - ç(t)) and (p ~ p(t)). The first three secoııd degree fluctuation variables are defined as follows, vi VI (*) = {(«- ?(*))*> ¥*(*) = |((« - «r(*))(p - K*)) + (p - P(*))(« - «(*))> **(*) = {(P-X*))2> Similarly tlıe tlıird degree fluctuation variables are taken as ¥*(*) = {(«-S(*))8> V5(<) s İ((ç - ç(i))2(p - p(t)) + (p - p(*))(« - (?(t))2} V«(*) s |((« ~ «(*))(£ - K*))2 + (P - P(*))2(« - «(<))> ^7(t) = ((p-Ki))3> it is generated as VJI»H*(*) = |((« - «(*))n~*(p - p(*))fc + (P - ?(*))*(? - g(*))n"*> fc = l, 2,3..., n n > 2 J[n] = |(n2 + n-4) Evohıation equations of the system is obtained analytically by use of the expected valııes of the operators and tlıe results are vii ç(3)(f)=Ip(3)(<) T »i p<»)(<) = - n«W«) - 5V'"(!<3)«)vS3)M *<.>«) _JU»(«) itli tâ\t) =-v48)(*) - v"(«(s)(*))vS8)(*) - ±v'"(qM(t»<p<#="" style="margin: 0px; padding: 0px; outline: 0px; color: rgb(34, 34, 34); font-family: Verdana, Arial, sans-serif; font-size: 10px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">(*) = - 2V"(g^(t))ff\t) - İVV'MVfM *?>(«) =I^'(«) /ı w . ^)(i)=l93f(i)-y''(^)(t))^3)(i) rf)W=1^3)W-2V"(^)(t))43)W l* v ^)(t) = _2y»(ç(3)(i))9?(3)(t) 5(3)(0)=5o ; P(3)(0)=Po ; ^3)(0) = c:a2 ^3)(0)=c2«/3 ; <43)(0) = c3/32 ; ^3)(0) = c4«3 ^(0) =c5a2/3 ; ^3)(0) = c6a^2 ; ^3)(0) = c7/93 it is observecl that eqtıatioııs obtaiııed by caııceling the flactuation variables are identical with the classical equations and for reducing the error caused by this caııcelations of the fluctuation terms. it is seen that the difference between the classical and the quantunı dynanıics can be observed by considering the fluctuation terms in serial fashion. in the case if there are singularities in the Hamiltonian of the system for viii some finite vahıes of the variables tlıe ııse of tlıe fluctuation variables may not be eııough for to characterize the motioıı. Observatioııs on this points reveals that the ınethod of canceling the fiucttıation variables iıı steps may not produce convergent results if the Haıuiltonian has first and secoııd inverse powers of the location variables. Btıt in this work since the system is an anharmonic oscillator, Hamiltoniaıı does not have any singularty and it becomes sufficient to take the expected values of the operators consist of the multiplications of the natural powers of tlıe locatioıı and momeııtunı operators, as xmknowns and to define the fluctuation variables with respect to this strııcture. Sohıtions of the non-linear system of eqxıations that have üuctuation vari¬ ables of the second and tlıircl degrees are obtaiııed by use of the Mathematica package. The graphical representatioııs of the momenttim, location and the fiuctuation variables and their interpretatioııs are also given.</p