Quantum Dynamics Using Lie Algebras, with Explorations. in the Chaotic Behavior of Oscillators. Ryan Thomas Sayer

Transcription

1 Quanum Dynamics Using Lie Algebras, wih Exploraions in he Chaoic Behavior of Oscillaors Ryan Thomas Sayer A hesis submied o he faculy of Brigham Young Universiy in parial fulfillmen of he requiremens for he degree of Maser of Science Jean-François S. Van Huele, Chair Manuel Berrondo Sco D. Bergeson Deparmen of Physics and Asronomy Brigham Young Universiy December 22 Copyrigh 22 Ryan Thomas Sayer All Righs Reserved

2 ABSTRACT Quanum Dynamics Using Lie Algebras, wih Exploraions in he Chaoic Behavior of Oscillaors Ryan Thomas Sayer Deparmen of Physics and Asronomy, BYU Maser of Science We sudy he ime evoluion of driven quanum sysems using analyic, algebraic, and numerical mehods. Firs, we obain analyic soluions for driven free and oscillaor sysems by shifing he coordinae and phase of he undriven wave funcion. We also facorize he quanum evoluion operaor using he generaors of he Lie algebra comprising he Hamilonian. We obain coupled ODE s for he ime evoluion of he Lie algebra parameers. These parameers allow us o find physical properies of oscillaor dynamics. In paricular we find phase-space rajecories and ransiion probabiliies. We hen search for chaoic behavior in he Lie algebra parameers as a signaure for dynamical chaos in he quanum sysem. We plo he rajecories, ransiion probabiliies, and Lyapunov exponens for a wide range of he following physical parameers: srengh and duraion of he driving force, frequency difference, and anharmoniciy of he oscillaor. We idenify condiions for he appearance of chaos in he sysem. Keywords: quanum dynamics, oscillaor, driving force, dynamical chaos, Lie algebra, mean field, anharmoniciy, Lyapunov exponen, ransiion probabiliy

3 ACKNOWLEDGMENTS I would like o hank Dr. Van Huele for all of he hours of help and guidance he gave me in preparing his documen. I would also like acknowledge Dr. Berrondo, whose pas work wih Lie algebras provided a saring poin for my own research.

4 Conens Table of Conens Lis of Figures iv vi Inroducion. Background Objecives Ouline Mehods 5 2. The Time-Dependen Hamilonian A Semi-Classical Ansaz Ehrenfes s Theorem The Free Paricle The Simple Harmonic Oscillaor Exensions And Limis Algebraic Srucures Properies of Algebras Lie Algebras Solving for Time Evoluion wih Lie Algebras The Wei-Norman Ansaz Choosing an Algebra Basis Closure Under Commuaion Normal Ordering Uniariy The Free Paricle The Simple Harmonic Oscillaor Mean Field Approximaions o Ensure Closure Some Useful Properies of he Lie Algebra Parameers Chaos Condiions for Chaos Lyapunov Exponens iv

5 CONTENTS v Phase Space Behavior Examples of Chaoic Sysems Quanum Chaos Resuls The Driven Free Paricle The Driven Simple Harmonic Oscillaor Srengh of Driving Force Duraion of Driving Force Frequency of Driving Force The Driven Quaric Poenial λx The Driven Morse and Pöschl-Teller Oscillaors Srengh of Driving Force Duraion of Driving Force Frequency of Driving Force Anharmoniciy Parameer Coninuous (Sinusoidal) Driving Force The Caldirola-Kanai Damped SHO Conclusion 8 4. Summary of Resuls Discussion Fuure Research A Wei-Norman Facorizaions 2 A. Driven Free Paricle A.2 Driven Simple Harmonic Oscillaor Bibliography 7 Index 2

6 Lis of Figures 3. Driving force, λ-varying FP Lyapunov, E o -varying, λ =.5, = ω FP Lyapunov, E o =, λ-varying, = ω FP Lyapunov, E o =, λ-varying, = ω SHO phase, small E o, λ =.5, = ω SHO ransiions, small E o, λ =.5, = ω SHO Lyapunov, small E o, λ =.5, = ω SHO Lyapunov closer look, E o =, λ =.5, = ω SHO phase, large E o, λ =.5, = ω SHO ransiions, large E o, λ =.5, = ω SHO Lyapunov, large E o, λ =.5, = ω SHO phase, E o =, λ-varying, = ω SHO ransiions, E o =, λ-varying, = ω SHO Lyapunov, E o =, λ-varying, = ω SHO phase, E o =, λ-varying, = ω SHO ransiions, E o =, λ-varying, = ω SHO Lyapunov, E o =, λ-varying, = ω SHO phase, E o =, λ =.5, -varying vi

7 LIST OF FIGURES vii 3.9 SHO ransiions, E o =, λ =.5, -varying SHO Lyapunov, E o =, λ =.5, -varying SHO phase, E o =, λ =.5, -varying SHO ransiions, E o =, λ =.5, -varying SHO Lyapunov, E o =, λ =.5, -varying Classical Duffing Phase/Lyapunov, E o -varying, = AO phase, small E o, λ =.5, = ω, χ = AO ransiions, small E o, λ =.5, = ω, χ = AO Lyapunov, small E o, λ =.5, = ω, χ = AO phase, large E o, λ =.5, = ω, χ = AO ransiions, large E o, λ =.5, = ω, χ = AO Lyapunov, large E o, λ =.5, = ω, χ = AO phase, E o =, λ-varying, = ω, χ = AO ransiions, E o =, λ-varying, = ω, χ = AO Lyapunov, E o =, λ-varying, = ω, χ = AO phase, E o =, λ-varying, = ω, χ = AO ransiions, E o =, λ-varying, = ω, χ = AO Lyapunov, E o =, λ-varying, = ω, χ = AO phase, E o =, λ =.5, -varying, χ = AO ransiions, E o =, λ =.5, -varying, χ = AO Lyapunov, E o =, λ =.5, -varying, χ = AO phase, E o =, λ =.5, -varying, χ = AO ransiions, E o =, λ =.5, -varying, χ = AO Lyapunov, E o =, λ =.5, -varying, χ = AO phase, E o =, λ =.5, = ω, χ-varying

8 LIST OF FIGURES viii 3.44 AO ransiions, E o =, λ =.5, = ω, χ-varying AO Lyapunov, E o =, λ =.5, = ω, χ-varying AO phase, E o =, λ =.5, = ω, χ-varying AO ransiions, E o =, λ =.5, = ω, χ-varying AO Lyapunov, E o =, λ =.5, = ω, χ-varying AO phase, E o -varying, λ =, = ω, χ = AO ransiions, E o -varying, λ =, = ω, χ = AO Lyapunov, E o -varying, λ =, = ω, χ = AO Lyapunov, E o -varying near chaos poins, λ =, = ω, χ = CK Lyapunov, E o =, λ =.5, = ω, γ-varying

9 Chaper Inroducion. Background This work is abou quanum dynamics, he sudy of how quanum sysems evolve in ime. Time evoluion of quanum sysems is a opic of coninuing imporance in physics research. Scieniss look for ways o map ou how iny sysems will behave over ime when aced upon by exernal forces []. There are many differen mehods ha can be employed o rea quanum dynamical sysems; some of hese mehods includes Fourier ransforms, numerical solvers, and sudden, adiabaic or perurbaive approximaions [2]. Ehrenfes s heorem can be used o ake a semiclassical approach o solving for quanum dynamics (see Sec. 2.2). Also, algebraic srucures of he dynamics can be used o simplify he calculaions, and in paricular Lie algebras can be used o reduce he parial-differenial Schrödinger equaion ino a sysem of coupled ordinary differenial equaions (see Sec. 2.4). The opic of quanum chaos, or he manifesaion of classical chaos in quanum sysems, is also a maer of grea ineres in physics research [3] [4]. The correspondence beween classical and quanum chaos has proven problemaic because of he seemingly incompaible definiions of

10 .2 Objecives 2 quanum uncerainy and loss of predicabiliy. Chaos in classical sysems is well undersood. I would be of grea value o have a way of reaing quanum chaos using classical approaches, which ofen involve he measuremen of exac rajecories. By combining he properies of quanum dynamics wih he condiions and approaches of he reamen of classical chaos, we will be able o answer quesions abou he ime evoluion of quanum sysems ha would oherwise be difficul o address. The Lie algebra mehod in paricular provides a useful bridge beween quanum dynamics and chaos. Alhough references o chaos and Lie algebra can be found [5] [6], he approaches aken in hese sudies are differen from he mehod developed here, and heir relevance o my work is no immediaely apparen. My conribuion answers quesions abou chaos and quanum dynamics in a specific way ha I have no seen elsewhere..2 Objecives My objecive is o solve for he ime evoluion of quanum dynamical sysems. I will use a semiclassical ansaz o reduce he dynamics of a quanum sysem o a ime-dependen shif in coordinae and a ime-dependen quanum phase. I will also employ an ansaz for he ime evoluion operaor in erms of a Lie algebra basis, and hen I will use he commuaion relaions of ha basis o reduce he PDE o a sysem of coupled ODE s. My second objecive is o sudy he properies of a given quanum sysem and look for signs of chaos using classical crieria. I will accomplish his by calculaing phase-space rajecories and Lyapunov exponens for he soluions of he coupled ODE s. I will also calculae energy level ransiion probabiliies in order o characerize how he sysem responds over ime o a given driving force.

11 .3 Ouline 3.3 Ouline Sec. 2.2 gives a descripion of a semiclassical analyic mehod for solving quanum dynamics, which was recenly proposed by Andrews [7]. I will apply his mehod o wo simple bu exemplary sysems: he driven free paricle (FP), and he driven simple harmonic oscillaor (SHO). Finally I will discuss he exen and limis of he efficacy of his mehod. Sec. 2.3 gives he definiion of an "algebra" and explains some of is properies. I hen specifically discusses Lie algebras and heir properies. Sec. 2.4 shows how a Lie algebra can be used o solve for he ime evoluion of a quanum dynamical sysem. This is done for he driven FP and he driven SHO. This Lie algebra mehod gives direcly useful informaion abou a quanum sysem, such as expecaion values and ransiion probabiliies. The meaning of "chaos" and "quanum chaos" is briefly reviewed in Sec I will discuss how o calculae and inerpre Lyapunov exponens as a crierion for chaos. Chaper 3 discusses specific quanum sysems, all driven by a Gaussian pulse driving force. Sec. 3. sudies he FP and is Lyapunov exponens. Nex, Sec. 3.2 calculaes he phase-space rajecories, ransiion probabiliies, and Lyapunov exponens of a driven SHO. Sec. 3.3 shows how o approach a quaric anharmonic poenial problem and gives some example plos for he equivalen classical sysem. Sec. 3.4 will hen repea wha was done for he SHO bu wih a driven anharmonic oscillaor. Sec. 3.5 will ake a brief look a he Caldirola-Kanai damped SHO and is Lyapunov exponens. In he concluding Chaper 4 we will argue ha hese mehods are powerful and allow for he deailed sudy of he properies of dynamical sysems in ime. In paricular, he Lie algebra mehod allows us o disinguish beween sysems wih and wihou chaoic behavior. I will hen briefly remark on some of he difficulies I had while compleing his research, and will finally sae some fuure goals and direcions o which his research may lead. An appendix conains he explici facorizaion of he free paricle and simple harmonic oscil-

12 .3 Ouline 4 laor ime derivaives of he Wei-Norman Lie algebra parameers.

13 Chaper 2 Mehods 2. The Time-Dependen Hamilonian The Schrödinger equaion is used o deermine how quanum sysems evolve in ime: i h ψ(x,) = H (x, p,)ψ(x,), (2.) where H (x, p,) is he Hamilonian of he sysem and ψ(x,) is he wave funcion. Once Eq. (2.) has been solved, ψ(x,) can be used o deermine imporan properies of he sysem, such as is energy, posiion, and he energy ransiions or posiion evoluion as ime progresses. These characerisics in urn inform us abou he naure of he sysem. For insance, in his work we ll be ineresed in discovering wha he ransiions and rajecories can each us abou he chaoic naure of some anharmonic oscillaors. For a sysem wih a ime-independen Hamilonian H (x, p) (wih p = h i x ), Eq. (2.) can be easily solved by he separaion of variables x and, as is ofen demonsraed in quanum exbooks [8]. However, for a sysem wih a ime-dependen Hamilonian H (x, p,), separaion of variables usually doesn work. Some oher mehod mus be employed o deal wih his general case. 5

14 2.2 A Semi-Classical Ansaz A Semi-Classical Ansaz One ineresing mehod for reaing forced quanum dynamical sysems has been proposed by Andrews [7]. In his paper i is shown ha a spaially-uniform, ime-dependen driving force f () acing on a quanum sysem resuls in wo simple effecs: a ime-dependen shif in he coordinae, x(), such ha x ξ = x x, (2.2) and a ime-dependen phase shif of he wavefuncion θ(x,). The ansaz is ha he wave funcion will have he form ψ(x,) = exp[iθ(x,)]ψ(x x(),), (2.3) where Ψ(x,) is he wave funcion of he sysem when no forcing occurs. In wha follows, I will refer o his mehod as he "Andrews mehod" for he sake of clariy. I will no repea he derivaion of his mehod, bu will refer he reader o he paper [7]. In Sec. 2.4 I will direcly compare he Andrews mehod resuls wih hose of he Lie algebra mehod, which I develop in Sec Ehrenfes s Theorem According o Ehrenfes s heorem [9], he expecaion values of posiion and momenum correspond o he rajecories of a classical paricle in a similar poenial. A paricle s shif in posiion due o a driving force will herefore be given by he difference of he semiclassical rajecories, i.e., he rajecories of he expecaion values obained using quanum mechanics. The erm x() in Eq. (2.3) is found o be he difference in posiion expecaion values of he forced sysem x ψ () and unforced sysem x Ψ (), x() = x ψ () x Ψ (), (2.4) where x is he posiion operaor, muliplicaive in he coordinae represenaion, as can be verified by explici calculaion.

15 2.2 A Semi-Classical Ansaz 7 The quanum phase shif from he force will depend on he shif in momenum p(), which is necessarily he difference in momenum expecaion values of he forced and unforced sysems, p() = p ψ p Ψ. (2.5) This shif in momenum is relaed o he quanum phase shif by he equaion p() = h θ(x,) x, which can be inegraed wih respec o x o give θ(x,) = ( p()x β()). (2.6) h The erm β() is a consan of inegraion deermined by subsiuing our ansaz Eq. (2.3) ino Eq. (2.). As i urns ou, β() is equal o he acion of he unforced sysem, found by aking he ime inegral of he classical Lagrangian. The values of x() and p() are deermined by he classical Hamilonian equaions of moion, subjec o iniial condiions x = H p, (2.7) p = H x, (2.8) x() = p() =. (2.9) I will now show he Andrews mehod in acion for a couple of cases wih general driving force: a forced free paricle and a forced simple harmonic oscillaor The Free Paricle Le s firs ake a look a he free paricle (FP) sysem. The Hamilonian of a driven FP is given by H = ˆp2 f () ˆx (2.) 2m for some arbirary ime-dependen force f (). The classical equaions of moion for x() and p(), as deermined wih he coordinae and momenum operaors by Eq. (2.), are x() = m p() and

16 2.2 A Semi-Classical Ansaz 8 p() = f (). These equaions, aken wih he iniial condiions as given in Eq. (2.9), can be solved o give x() = m p() = f ( )d d = F () m, (2.a) f ( )d = F(), (2.b) where F() is he ime inegral of he force and F () is he ime inegral of F(). We plug hese soluions ino Eq. (2.3) o ge he following explici expression for our ansaz: ψ(x,) = exp[ ī h (F()x β())]ψ(x F (),). (2.2) m To solve for β() we subsiue Eq. (2.2) ino Eq. (2.), make a change of variables ξ = x F () m (as seen in Eq. (2.2)), and simplify o ge i h h2 2 [ Ψ(ξ,) F() 2 Ψ(ξ,) = 2m ξ 2 + 2m β() ] Ψ(ξ,). (2.3) The firs wo erms correspond o he undriven FP Schrödinger equaion for Ψ(ξ,), and hey cancel ou. When we solve he remainder for β, we ge β() = F( ) 2 d = F 2 (). (2.4) 2m We can now express he ime-dependen wave funcion of he forced FP sysem in erms of F(), F (), and F 2 (), which can be easily deermined for any given ime-dependen driving force f () The Simple Harmonic Oscillaor To apply he Andrews mehod o he simple harmonic oscillaor (SHO), we follow he same procedure as wih he FP. I jus akes a lile more algebra work o ge he soluions. The SHO Hamilonian is given by H = ˆp2 2m + 2 mω2 ˆx 2 f () ˆx, (2.5)

17 2.2 A Semi-Classical Ansaz 9 where ω is he angular frequency of he oscillaor. The classical Hamilonian equaions of moion, defined by Eq. (2.7) and Eq. (2.8) are x = p m, (2.6) p = mω 2 x + f (), (2.7) and he iniial condiions are given by Eq. (2.9). These equaions can be solved wih a Fourier ransform or a Green s funcion o ge he following inegral expressions: x() = mω p() = f ( )sin[ω( )]d S() mω, f ( )cos[ω( )]d C(). (2.8a) (2.8b) We now solve for β() as we did for he FP sysem, by subsiuing he ansaz ino Eq. (2.), performing a change of variables ξ = x S() mω, and simplifying. Afer some algebra we are lef wih i h h2 2 Ψ(ξ,) Ψ(ξ,) = 2m ξ 2 + [ C() 2 mω2 x 2 2 Ψ(ξ,) + 2m S()2 2m dβ() d ] Ψ(ξ,). (2.9) When we cancel ou he erms for he undriven SHO Schrödinger equaion and simplify we recover he following: dβ d = 2m [C2 S 2 ] = p2 2m 2 mω2 x 2. (2.2) To solve his, we can ake advanage of a useful ideniy. By muliplying Eq. (2.6) by p, Eq. (2.7) by x, and hen adding hem ogeher, we ge he following: d( x p) d We subsiue his ino Eq. (2.24) o ge = p2 m mω2 x 2 + f () x. (2.2) dβ d = x p) [d( f () x], (2.22) 2 d which can hen be inegraed wih respec o ime o give us β() = 2 x() p() 2 f ( ) x( )d, (2.23)

18 2.2 A Semi-Classical Ansaz or, in erms of C() and S(), β() = C()S() 2mω f ( )S( )d. (2.24) 2mω We can now express he ime-dependen wave funcion of he forced sysem in erms of a couple of funcions, S() and C(), which can be deermined for an arbirary driving force f (). As in he FP case, his approach works equally well for any spaially-uniform force ha is urned on a =. Also, he soluions we ge are exac, as no approximaions were required Exensions And Limis The Andrews mehod is valid for any single or muli-dimensional isoropic sysem wih a Hamilonian ha is a mos quadraic in coordinae and momenum. Given a Hamilonian of he form ˆ H = 2 a()ˆp2 + 2 b()(ˆp ˆx + ˆx ˆp) + 2 c()ˆx2 + f() ˆp + g() ˆx, (2.25) where a(), b(), c(), and f() are coefficiens of arbirary ime dependence and g() is he driving force of he sysem, he equaions of moion o deermine x and p are he following: x() = a() p() + b() x() + f(), p() = b() p() + c() x() + g(). (2.26a) (2.26b) Finally, β() is deermined by aking he ime inegral of he Lagrangian of he classical sysem, i.e. he acion of he sysem: β() = 2 a()ˆp2 2 c()ˆx2 g() ˆx. (2.27) The ansaz soluion for he wavefuncion of he forced sysem will ake he same form as in Eq. (2.3) and (2.6), bu for x and p as vecors insead of scalars [7]. For sysems wih Hamilonians ha are higher order han quadraic in posiion and momenum, some approximaion mus be made o resore lineariy o he classical equaions of moion. This

19 2.3 Algebraic Srucures would apply o anharmonic oscillaors of he ype we sudy in Chaper 3. A similar problem is encounered when employing he Lie algebra mehod, as will be discussed in Sec As an example of such a sysem, consider Schrödinger s equaion wih a quaric poenial of he form i h ψ = h2 2m 2 x 2 ψ + 2 mω2 x 2 ψ + λx 4 ψ f ()xψ. (2.28) To solve his sysem using he Andrews ansaz we mus solve for he "classical" rajecory using he Hamilon equaions of moion. The Hamilonian of he sysem is given by H = ˆp2 2m + 2 mω2 ˆx 2 + λ ˆx 4 f () ˆx. (2.29) From he Hamilonian we can find he classical equaions of moion used o solve x and p: x = H p = p m, (2.3) p = H x = mω2 x 4λ x 3 + f (). (2.3) The coupled equaions can be rewrien as a single, second-order differenial equaion: m d2 x d 2 = mω2 x 4λ x 3 + f (). (2.32) For sinusoidal f (), Eq. (2.32) is known as he Duffing equaion, and in general canno be solved analyically. When solved using numerical mehods, he soluion exhibis chaos depending on he srengh of f (), as will be discussed in Sec and demonsraed in Sec Algebraic Srucures 2.3. Properies of Algebras The parial differenial Schrödinger equaion, Eq. (2.), can also be solved by using Lie algebras. In his way, as will be shown in Sec. 2.4, a second-order parial differenial equaion can be urned

20 2.3 Algebraic Srucures 2 ino a sysem of coupled firs-order ordinary differenial equaions. Lie algebras appear quie ofen in physics, hough hey re no always referred o as such []. Lie algebras are a ype of algebraic srucure. I now will give a brief explanaion of wha ha means. In general, an algebra is a se of elemens for which one or more binary operaions are defined. Binary operaions combine wo elemens of he se and give one elemen back; some common examples include he operaions of addiion, subracion, muliplicaion, and division ha define he "elemenary algebra" of he real number se, which is probably he mos familiar ype of algebra o non-mahemaicians. An ideniy elemen of a binary operaion, when combined wih any elemen, reurns he original elemen; some examples of ideniy elemens are for addiion (A + = A for any A) and for muliplicaion (A = A). An elemen A of an algebra may have an inverse, which when combined wih he elemen will yield he ideniy: A + ( A) =, A (A ) =, ec. An operaion is said o be associaive when he order in which wo or more operaions are carried ou doesn maer, as long as he order of he elemens says he same. Some examples of associaive operaions are addiion ((A + B) +C = A + (B +C)) and muliplicaion ((A B) C = A (B C)). By conras, he propery of commuaiviy says ha he order of he elemens hemselves doesn maer. In elemenary algebra he operaions of addiion and muliplicaion are commuaive, or in oher words hey "commue," (i.e. A+B = B+A, A B = B A). For some ses of elemens, commuiviy of muliplicaion is no required, as will be discussed in Sec One more imporan propery of algebraic srucures is closure. A "closed" se is one such ha a binary operaion of any wo elemens of he se will yield anoher elemen of he se. A se mus be closed in order o form an algebra. As an example, consider he se of vecors confined o he xy-plane, r = (x, y): his se is closed under any operaion ha reurns anoher vecor confined o he xy-plane, which is rue for he operaion of addiion: r + r 2 = (x,y )+(x 2,y 2 ) = (x +x 2,y +y 2 ). An operaion such as a cross produc ( ), however, would no be closed for his se: r r 2 =

21 2.3 Algebraic Srucures 3 (x,y ) (x 2,y 2 ) = (,,x x 2 y y 2 ). before he cross produc operaion can be defined, he se mus be exended beyond he xy-plane o include z componens. These properies or axioms are aken ogeher o define a group. In general, a group is a se of elemens and an operaion wih he following properies: (a) an ideniy, (b) an inverse, (c) associaiviy, and (d) closure. An Abelian group has he addiional propery of commuaiviy. Lie groups, which may or may no be Abelian, will be discussed in Sec Lie Algebras A Lie algebra is a vecor space wih an anisymmeric binary operaion, such as a commuaor. A commuaor is wrien as square brackes wih wo elemens separaed by a comma [, ] and is aken o be he difference of producs of wo elemens wih he order of he elemens in he second produc reversed: [A,B] = A B B A. (2.33) Two elemens are said o commue if heir commuaor is zero. Lie algebras and Lie groups appear quie ofen in he field of physics [] [2]. Some examples of Lie algebras include he hree-dimensional Euclidian space R 3, wih he commuaor defined as he cross produc of vecors. Anoher example of a Lie algebra is he se of n n ani-hermiian marices. In quanum physics, he angular momenum operaors and heir commuaors form a Lie algebra. In Sec. 2.4 we will make use of he propery ha a Hamilonian someimes can be expressed as a sum of elemens of a Lie algebra. In his way, he ime evoluion of he sysem can be deermined.

22 2.4 Solving for Time Evoluion wih Lie Algebras Solving for Time Evoluion wih Lie Algebras The equaion which is used o deermine he ime evoluion of a quanum sysem has he Schrödinger form i h U(). = H U(), (2.34) where U() is he ime evoluion operaor and H is he Hamilonian. The doed equal sign. = will be used o designae an operaor equaion. For an evoluion operaor wih inverse U (), Eq. (2.34) can be wrien as i h U() U () =. H, (2.35) and he equaion can be solved for specific Hamilonians in he righ-hand side by manipulaing he srucure of he evoluion operaor U() in he lef-hand side The Wei-Norman Ansaz J. Wei and E. Norman proposed a produc form [3] for he operaor U, U(). = N i= e α i()h i, (2.36) where α i () are a se of ime-dependen, complex, scalar parameers o be deermined, and H i are he basis elemens of a Lie algebra A, chosen o include all he erms of he Hamilonian H A. = {H,..,H N }. (2.37) The elemens H i in Eq. (2.36) do no necessarily commue, so care mus be aken when applying he parial ime derivaive and he inverse operaor in Eq. (2.35), as will be menioned in Sec These basis elemens are chosen based on a Hamilonian of he form H. = N i= b i ()H i, (2.38)

23 2.4 Solving for Time Evoluion wih Lie Algebras 5 wih coefficiens b i (). Some of he b i () may be zero wihou necessarily implying ha he corresponding α i () vanish. The oal ime evoluion operaor U(), which is an exponenial mapping of he algebra A, is a represenaion of he Lie group corresponding o ha algebra. The nex sep is o facorize he U() U () erm in he lef-hand side of (2.35). This is done by aking he ime derivaive of U(), one exponenial erm a a ime, and hen using he Baker- Campbell-Hausdorff ideniy o simplify (see Eq. (A.6)). A sep-by-sep explanaion of his process can be found in Appendix A. In he end we are lef wih an equaion of he form i h N i N. g i ( α (),.., α N (),α (),..,α N ())H i = i b i ()H i, (2.39) where g i ( α (),.., α N (),α (),..,α n ()) are funcions of he α i parameers and heir ime derivaives. The nex sep is he equae he coefficiens of each algebra elemen, which yields a se of coupled firs-order ordinary differenial equaions g i ( α (),.., α N (),α (),..,α n ()) = b i (), one for each H i. These can be solved o deermine he values of he α i () Choosing an Algebra Basis To minimize he work, he chosen Lie algebra basis should be he smalles algebra which includes every erm in he Hamilonian. This means ha a smaller or more workable Lie algebra can someimes be chosen by rewriing he Hamilonian. A given Hamilonian may be expressed in several differen forms. For example, oscillaor dynamics can be wrien in erms of he coordinae operaor x and is derivaives x and 2 x 2, or in erms of he creaion and annihilaion (ladder) operaors (a and a, respecively) and he number operaor N = a a. Consider a simple harmonic oscillaor whose Hamilonian can be wrien in erms of he number operaor, H =. hω(n + ). (2.4) 2 The corresponding algebra for his Hamilonian, wrien as in Eq. (2.37), would be A. = {N,}.

24 2.4 Solving for Time Evoluion wih Lie Algebras 6 Now, consider a simple harmonic oscillaor wih an added driving force: H. = hω(n + 2 ) f () h 2mω (a + a ). (2.4) The Lie algebra basis as chosen before no longer includes all he erms in he Hamilonian. The basis mus be exended o include a and a : A. = { } a,a,n,. (2.42) This new closed se forms an algebra which covers every erm in he Hamilonian defined in Eq. (2.4) Closure Under Commuaion As menioned in Sec. 2.3, an operaion beween any wo elemens of a se mus be closed for he se o consiue an algebra. For Lie algebras his means ha any non-vanishing commuaor of any wo algebra elemens mus also be an elemen of he algebra. To deermine if a chosen se mainains closure, ake he commuaor of every possible wo-elemen combinaion in he se. As an example, consider he commuaors for he algebra given in Eq. (2.42): [,a] =.,[,a ] =.,[,N] =., (2.43) [a,a ] =.,[a,n] =. a,[a,n] =. a. (2.44) The resul of every commuaor is iself an elemen of he se, which means ha he se is closed. I may be necessary o add erms o he basis ha aren found in he Hamilonian o ensure closure Normal Ordering The Wei-Norman ansaz gives flexibiliy in choosing he order of he exponenials, bu some choices lead o considerable simplificaion. When working wih producs of creaion and annihilaion operaors, he mos convenien order is wih he creaion operaors a o he lef of he

25 2.4 Solving for Time Evoluion wih Lie Algebras 7 annihilaion operaors a. This arrangemen is known as "normal ordered," and is convenience comes from he fac ha annihilaion operaors acing on ground-sae kes and creaion operaors acing on ground-sae bras will boh equal zero: a =, (2.45) a =. (2.46) Uniariy The ime-evoluion operaor defined in Eq. (2.36) is assumed o be a uniary operaor for sysems of "closed" quanum Hamilonians (here closed is mean in he physical and no algebraic sense). When a quanum sysem is no longer closed, i.e. when i is in conac wih an exernal hea bah or includes facors for inernal losses, he ime-evoluion operaor will no longer be uniary. This can be seen when looking a quanum sysems wih damping facors, such as he Caldirola-Kanai Hamilonian ha will be encounered in Sec As long as he Hamilonian under consideraion is closed o is environmen and real-valued, he uniariy of he evoluion operaor is assured. For a uniary U() is inverse mus equal is Hermiian conjugae, U. = U. Relaionships beween he α i parameers of he Lie algebra basis can be deermined from his condiion. As an example, for he algebra defined in Eq. (2.42) we ge U(). = e α ()a e α 2()a e α 3()N e α 4(). (2.47) By seing U(). = U(), he following ideniies for he real and imaginary pars of he α i

26 2.4 Solving for Time Evoluion wih Lie Algebras 8 parameers can be deermined: R{α ()} = R{α 2 ()} I{α ()} = I{α 2 ()} R{α 3 ()} = R{α 4 ()} = 2 α 2 = 2 α 2 2. (2.48a) (2.48b) (2.48c) (2.48d) These relaions have been confirmed by acual daa (see Chaper 3). Sysems for which he imeevoluion operaor is no uniary are sill solved in he same way. The only requiremen o ge from Eq. (2.34) o Eq. (2.35) is ha U() have an inverse (which may or may no equal is Hermiian conjugae) The Free Paricle The ime evoluion operaor for a free paricle wih a driving force is given by solving Eq. (2.34), where he Hamilonian of he sysem is H 2 =. h2 2m x 2 f ()x =. H f ()x. (2.49) We can consruc a Lie algebra o conains his Hamilonian, A. = {,x, x, 2 }, (2.5) x2 wih which we can propose an ansaz for he evoluion operaor in erms of he Wei-Norman represenaion of he Lie group corresponding o he chosen algebra: U. = e α () e α 2()x e α 3() x e α 4() 2 x 2, (2.5) wih complex, ime-dependen parameers α i () (i =...4). We can ake he inverse of he evoluion operaor o ge i h[ U()]U(). = H (2.52)

27 2.4 Solving for Time Evoluion wih Lie Algebras 9 i h[ (eα() e α2()x e α 3() x e α 4() 2 x 2 )](e α 4() 2 x 2 e α 3() x e α2()x e α() ) =. h2 2m 2 x 2 f ()x. (2.53) The facorizaion of he lef-hand side of his expression is worked ou in Sec. A. of Appendix A. By equaing he Lie algebra elemens in Eq. (A.5) wih he Hamilonian, we recover he following se of four coupled firs-order differenial equaions, one for each basis elemen of he algebra: α () = α 3 ()α 2 () α 4 ()α 2 () 2 (2.54) α 2 () = ī f () (2.55) h α 3 () = 2α 4 ()α 2 () (2.56) α 4 () = i h 2m, (2.57) which can be solved o give α () = ī h [ 2m ( ) 2 ] f ( )d d = ī β() (2.58) h α 2 () = ī [ ] f ( )d = ī p() (2.59) h h ( ] f ( )d )d = x() (2.6) [ α 3 () = m α 4 () = i h 2m = ī h ( h 2 2m ), (2.6) where he soluions have been simplified by using he funcions defined in Sec o replace he erms in he square brackes. The oal ime evoluion operaor can now be wrien, wih θ(x,) = h ( p()x β()). U() =. e h ī β() e h ī p()x e x() x e h ī ( h2 2m 2 x 2 ) (2.62). = e iθ(x,) e x x e ī h H, (2.63)

28 2.4 Solving for Time Evoluion wih Lie Algebras 2 To show ha he Andrews mehod soluion Eq. (2.3) can be recovered, choose an iniial wave funcion Ψ(x,) for a free paricle wih no driving force a =. To evolve his wave funcion in ime, apply he evoluion operaor U() one exponen a a ime: e ī h H Ψ(x,) = Ψ(x,) (2.64) e x x Ψ(x,) = Ψ(x x,) (2.65) ψ(x,) = e iθ(x,) Ψ(x x,). (2.66) The Simple Harmonic Oscillaor The Hamilonian of a driven quanum simple harmonic oscillaor is given by H. = H f ()x, (2.67) where H. = h 2 2m 2 x mω2 x 2. This can be rewrien in erms of raising and lowering operaors,. H = hω(n + 2 ) g()(a + a ), (2.68) wih N =. a a, g() = h 2m (), and where we ve employed he relaion x = h 2mω (a + a ). In order o deermine he effecs of he driving force on he ime evoluion of he sysem we mus use his Hamilonian o solve for he evoluion operaor U(). We will use he ansaz given by Eq. (2.47): i h[ (eα ()a e α 2()a e α 3()N e α 4() )](e α 4() e α 3()N e α 2()a e α ()a ) = hω(n + 2 ) g()(a + a ). (2.69) The facorizaion of he lef-hand side of his expression is worked ou in Sec. A.2 of Appendix A. By equaing he Lie algebra elemens in Eq. (A.25) wih he Hamilonian, we recover he following se of four coupled firs-order differenial equaions, one for each basis elemen of he algebra: α () = α 3 ()α () + ī h g() = iωα () + ī g() (2.7) h

29 2.4 Solving for Time Evoluion wih Lie Algebras 2 α 2 () = α 3 ()α 2 () + ī h g() = iωα 2() + ī g() (2.7) h α 3 () = iω (2.72) α 4 () = α 3 ()α 2 ()α () + α 2 ()α () i ω 2 = ī h (g()α () hω 2 ) (2.73) These can be solved using an inegraing facor e ± iωd α () = ī h α 2 () = ī h e iω( ) g( )d (2.74) e iω( ) g( )d (2.75) α 3 () = iω (2.76) α 4 () = h ( ) 2 g( ) e iω( ) g( )d d iω 2 (2.77) As in Sec , agreemen can be shown beween he parameer soluions found using Lie algebra and he soluion via Andrews mehod found in Sec Mean Field Approximaions o Ensure Closure When using he Lie algebra mehod for he FP and SHO sysems, obaining a closed se of generaors is sraighforward. For more complicaed sysems, whose Hamilonians have erms of higher order algebra elemens (such as N 2 ), i may be difficul o find a se of elemens ha closes wihou requiring a larger (bu sill finie) number of elemens. This problem may be miigaed by making a mean field approximaion on he Hamilonian. The process is simple: For some operaor P, he value of P 2 may be approximaed as P P, where P is he expecaion value of P. Any algebra being made from he Hamilonian will only have o include P, no P 2, and P will simply be a coefficien of P. In he case where P is imedependen, his is known as a "dynamical mean field" approximaion. This ype of approximaion will be used in Sec. 3.4 o simplify he Morse and Pöschl-Teller Hamilonians [4].

30 2.4 Solving for Time Evoluion wih Lie Algebras Some Useful Properies of he Lie Algebra Parameers One of he real srenghs of he Lie algebra mehods is ha useful properies of he sysem can be easily expressed in erms of he α i parameers. By using he ladder operaor algebra o consruc a Wei-Norman ansaz for U() as seen in Eq. (2.47), he following expressions can be deermined: U au = e α 3 a + α, (2.78) U a U = e α 3 a α 2, (2.79) U NU = N α α 2. (2.8) These expressions represen he operaors a, a and N in he Heisenberg picure. When aken wih he ideniies x = 2ω (a+a ) and p = i ω2 (a a ) (in he unis of h = m = ω = ), we recover he following expressions for expecaion values in erms of he α i () s: x = n U xu n = 2ω (α () α 2 ()), (2.8) ω p = n U pu n = i 2 (α () + α 2 ()), (2.82) N = n N n = n i α ()α 2 (), (2.83) where n i is he iniial number eigenvalue of he sysem. The expecaion values of x and p can be used o find he phase-space rajecories of he sysem, and will be employed in Chaper 3. The ransiion probabiliies of he sysem can also be found in erms of Lie algebra parameers. The probabiliy of a ransiion from he n h energy level o he m h level is defined as follows: P nm () = m U() n 2. (2.84) For some elemen H i of a given algebra we use he following ideniy: e α i()h i n = k= α i () k Hi k n. (2.85) k!

31 2.5 Chaos 23 This expression can be simplified if we are using he ladder operaor algebra. From he exponenial of he lowering operaor a, and he properies of ladder operaors [], we ge e α2()a α 2 () k n = k! n k= n! n k. (2.86) (n k)! Afer acing on n wih all of he exponenial erms of U() and simplifying, we are lef wih he following expression for he ransiion probabiliies: P nm () = eα4() e α 3()n n!m! n k= α () m n+k α 2 () k 2. (2.87) k!(m n + k)!(n k)! The value P is known as he ground-sae persisence probabiliy. The expression given by Eq. (2.87) is only valid for ransiions from lower o higher saes, i.e., n m. Transiions from higher o lower saes, i.e., n m, use he following formula: P nm () = eα4() e α 3()n n!m! m k= α () k α 2 () n m+k 2. (2.88) k(m k)!(n m + k)! These expressions will be used in Chaper 3 o plo he ransiion probabiliies for he SHO and he Morse oscillaor sysems. 2.5 Chaos Classical chaos is usually described as an exreme sensiiviy o iniial condiions [5] Condiions for Chaos The sysem needs o have some level of complexiy in order o display chaos. The exac condiions for he onse of chaos is he subjec of considerable research, bu some general resuls apply. In paricular, according o he Poincaré-Bendixson heorem, a sysem of equaions will no manifes chaos unless i has hree or more degrees of freedom and is nonlinear [6].

32 2.5 Chaos Lyapunov Exponens The mos commonly used ool o measure chaos is known as a Lyapunov exponen [7]. Lyapunov exponens are deermined by measuring srengh of he exponenial divergence of wo rajecories iniially separaed by a disance ε as follows: δx() = δx o e λ (2.89) λ = lim log( δx() ), (2.9) δx o where δx o corresponds o he iniial difference of wo rajecories of variable x (i.e. δx o = ε), and δx() corresponds o he difference of he wo rajecories a some laer ime. I should be noed ha x does no necessarily refer o a coordinae variable. A sysem of coupled differenial equaions will have one Lyapunov exponen for each degree of freedom of he sysem, wih he larges-valued exponen referred o as he "maximal" Lyapunov exponen. A sysem is considered chaoic when is maximal Lyapunov exponen is posiive-valued Phase Space Behavior Chaos of a sysem is ofen deermined by sudying he phase-space behavior of he sysem using he principles of ergodic heory [7], which considers he exploraion of he phase space in ime. The phase-space rajecories of a chaoic sysem will end o spread evenly over he region of phase space ha is accessible o he sysem. This spreading over phase space can be seen in several of he figures in Chaper Examples of Chaoic Sysems Some examples of physical sysems which manifes chaos are he double pendulum [8] and he non-recangular billiard able [9]. Anoher example, already encounered in Sec , is he

33 2.5 Chaos 25 Duffing oscillaor. This sysem is known o exhibi chaos when he driving force is sufficienly srong. Some soluions of his sysem will be shown in Sec Quanum Chaos Exreme sensiiviy o iniial condiions is difficul o ascribe o quanum sysems because he uncerainy principle makes i impossible o exacly define he iniial condiions and rajecories [2]. Therefore, oher crieria and mehods have been developed in he sudy of quanum chaos. In paricular, he quesion of quanum chaos is ofen addressed by sudying he saisical disribuion of energy levels [2]. Wha we will show in Chaper 3 is ha exac rajecories can be deermined for he Lie algebra parameers in Sec. 2.4, making i possible o deermine he values of he Lyapunov exponens of he corresponding sysems. This is done by solving Eq. (2.39) using wo differen ses of iniial condiions: firs wih α i () =, hen wih α i () = ±ε. A minus sign may be necessary o preserve uniariy when dealing wih closely coupled α s (such as he α s corresponding o a and a ). Throughou Chaper 3 I will use his mehod o calculae Lyapunov exponens for each α i of he Lie algebra soluion in order o deermine if chaos is presen. For chaoic sysems, a leas one of he Lyapunov exponens will have a posiive, nonzero value in he limi ha.

34 Chaper 3 Resuls In his chaper I presen resuls obained by numerical soluions of he dynamical equaions for he Lie parameers derived in Sec I consider successively he driven free paricle, he driven simple harmonic oscillaor, he quaric poenial oscillaor, he driven anharmonic Morse and Pöschl- Teller oscillaors, and he Caldirola-Kanai damped SHO. For all of his chaper I will use a Gaussian pulse as he driving force, of he form f () = E o sin( )e λ( o) 2, (3.) where E o is he ampliude or srengh of he force, is he frequency, λ is he inverse envelope widh (i.e., he inverse duraion) of he force, and o defines he ime he cener of he pulse arrives a he sysem. The pulse will always be cenered a o = 5 so ha a = he force will be eiher zero-valued or very small, depending on he chosen value of λ (see Fig. 3.). This driving force can be used o model how a shor laser pulse migh inerac wih a given aomic sysem. 26

35 27 λ =.5 λ = λ =.5 λ = λ =.5 λ = Figure 3. Gaussian pulse force, λ-varying

36 3. The Driven Free Paricle The Driven Free Paricle The firs sysem I wish o look a is he driven free paricle, whose Hamilonian is given in Eq. (2.49). This sysem was already solved for a generic driving force f () using he Andrews mehod and he Lie algebra mehod (see Sec and Sec , respecively). Using he driving force given in Eq. (3.), he Andrews mehod defines following useful funcions: F () = F() = F 2 () = 2m [ E o sin( )e λ( o ) 2 d, (3.2) E o sin( )e λ( o) 2 d d, (3.3) E o sin( )e λ( o) 2 d ] 2 d. (3.4) These hree funcions are used o deermine he ime evoluion, via a shif in coordinae and a quanum phase, of a given iniial wavefuncion. The sysem can also be solved wih Lie algebras by replacing he force f () in Eqs. (2.58) hrough (2.6) wih he paricular force given in Eq. (3.). I will use he soluions o he αs o deermine Lyapunov exponens. In Fig. 3.2 I plo he Lyapunov exponens (LE) as a funcion of ime, one exponen λ i for each degree of freedom of he sysem. In he Wei-Norman formulaion here is one degree of freedom for each of he basis elemens of he algebra, which means here is one λ i for each α i. For he sake of clariy I will label he λ i wih subscrips ha are relaed o he basis elemen hey represen (i.e., λ dx corresponds o x, λ corresponds o he uniy elemen operaor, ec.). The firs hing o noice is ha λ dx has he mos pronounced behavior in each of he plos. I iniially rises fas, hen slowly decays as for he res of he plo inerval. The driving force, wih an inverse widh of λ =.5, acs mosly on he inerval of = 4 o = 6, bu λ dx seems o be uninfluenced by he force urning on. Also, since he graph of λ dx is unchanged as Eo is varied, i appears ha is value is enirely independen of he driving force. The plo of λ is he nex mos ineresing of he four. I becomes nonzero around he ime he driving force begins o ac, and oscillaes as he force is acing, hen appears o "roll off" and

37 3. The Driven Free Paricle 29 Eo = Eo = Eo = Eo = λ λ x λ dx λ dx Eo = Eo = Figure 3.2 Free paricle Lyapunov exponens, E o -varying, λ =.5, = ω.

38 3. The Driven Free Paricle 3 asympoically approach he y-axis once he force has ceased. The ampliude of is oscillaions as well as is maximum value before he rolling off period correspond o he srengh of he driving force. In he plo corresponding o E o =. we can see a small downward bump in λ dx, long afer he driving force has passed: While his is an ineresing feaure, i does no appear o be significan. The final hing o noe from hese plos is ha he exponens corresponding o x and 2 everywhere fla and nearly zero. Chaos wouldn be expeced in a driven free paricle sysem, which means ha hese exponens behave as migh be prediced. I will now vary he duraion of he peak λ raher han he srengh E o of he driving force. In Fig. 3.3 and Fig. 3.4 I vary λ while holding he ampliude a E o = and E o =, respecively. The behavior of λ dx in each of he plos is idenical, regardless of he value of λ. I appears ha his LE is independen of he srengh AND duraion of he driving force. Since he sysem only consiss of a free paricle and a driving force, and λ dx appears o be independen of he driving force, here s no much lef, physically, o accoun for is behavior. One possibiliy: Since his exponen corresponds o he erm e α dx x, which is he infiniesimal generaor of ranslaion for he sysem, i could mean ha he iniial separaion ε of he α dx values is physically manifesed as an iniial displacemen of he wo rajecories, and ha his iniial displacemen becomes less pronounced over ime as he wo sysems naurally evolve. However, if his were he case, he Lyapunov exponen would be expeced o immediaely begin a some high value and hen roll off, no sar a and quickly rise before rolling off as i is seen o do. Whaever he inerpreaion may be, he behavior of λ dx does no seem o indicae he presence of chaos in he sysem. As we saw in Fig. 3.2, he Lyapunov exponens λ x and λ dx2 are everywhere almos zero-valued, and so were unaffeced by changes in driving force duraion jus as hey were unaffeced by changes in is srengh. A few hings can be seen in he graphs of λ in Fig. 3.3 and Fig. 3.4 ha weren x 2 are

39 3. The Driven Free Paricle 3.3 λ =.5.3 λ = λ =.5.3 λ = λ λ x.. λ dx λ dx λ =.5.3 λ = Figure 3.3 Free paricle Lyapunov exponens, E o =, λ-varying, = ω.

40 3. The Driven Free Paricle 32 λ =.5 λ = λ =.5.5 λ = λ λ x λ dx λ dx λ =.5.5 λ = Figure 3.4 Free paricle Lyapunov exponens, E o =, λ-varying, = ω.

41 3.2 The Driven Simple Harmonic Oscillaor 33 apparen from he graphs in Fig When he driving fore is shorer in duraion, i.e. when.2, he exponen λ is kicked o a higher posiive value and hen gradually approaches zero. When he driving force is longer in duraion (corresponding o λ =.2, λ =.5, and λ =.2), here is no posiive kick in λ and no ail-end behavior. The las place we can see any kind of rolling off a he ail is for λ =.5. I appears ha he more impulsive he driving force is, he less chance he sysem has o gradually adjus. For λ =.5, he driving force appears o be oo shor in duraion o cause any response from λ. The oscillaions in λ correspond o exacly o he region where he driving force is acing and appear o have he same frequency. These ripples srech farher in he negaive values han hey do in posiive values, which is especially apparen in he λ =.5 and λ =.2 plos in Fig I doesn seem likely ha chaos is manifes in he graphs of λ. The posiive ail-end behavior is mos likely a consequence of he impulsive naure of he force, where he sysem doesn have enough ime o gradually respond o he force. The real es of chaos is in he limi of he Lyapunov exponens for asympoically large values of ; in ligh of his crierion, chaos does no appear o be presen in any of he graphs in Fig. 3.3 or Fig. 3.4, again as we would expec for a free paricle. 3.2 The Driven Simple Harmonic Oscillaor As wih he free paricle case, he driving force f () for he SHO is chosen o ake he form of a Gaussian pulse, as given in Eq. (3.). Wih his driving force, he values of S() and C() as defined in secion. (2.2.3) are he following: S() = C() = f ( )sin[ω( )]d = f ( )cos[ω( )]d = E o cos( )e λ( o ) 2 sin[ω( )]d, (3.5) E o cos( )e λ( o ) 2 cos[ω( )]d, (3.6)

42 3.2 The Driven Simple Harmonic Oscillaor 34 and β() is deermined by Eq. (2.24). The wave funcion of he iniial undriven sysem is chosen o be he ground sae of he SHO: Ψ(x,) = ( mω hπ ) mω 4 exp( 2 h x2 )exp( iω 2 ). (3.7) The final wavefuncion of he driven sysem is hen given according o he ansaz, Eq. (2.3): ψ(x,) = exp[ ī (C()x β())](mω h hπ ) mω 4 exp[ 2 h S() (x mω )2 ]exp( iω ). (3.8) 2 The sysem can also be solved using Lie algebras o deermine he ime evoluion operaor, as seen in Sec Once he ime evoluion operaor has been deermined, oher properies of he sysem, such as phase plos, ransiion probabiliies, and Lyapunov exponens can deermined as oulined in Sec and Sec I will now ake a look a each of hese properies, saring by varying he driving force srengh E o as I did in Sec. 3. for he free paricle. Afer varying E o, I will make plos of varying driving force duraion λ and driving frequency for E o = as well as E o = (again, as was done in Sec. 3. for he free paricle) Srengh of Driving Force In Fig. 3.5 I make phase-space plos (i.e., plos of p vs. x ) for wo sysems: one wih he α i () = (in red) and he oher wih he α i () iniially offse by a small value of ε =. (in blue). The plos are made for various small values of E o. For all hese plos I ve se he driving force inverse duraion λ =.5 and he driving frequency = ω (or a resonance wih he naural frequency of he oscillaor). In laer plos I will vary boh of hese quaniies while holding E o a a fixed value. In he case of no driving force E o = he sysem behaves exacly like one would expec: The red do in he middle corresponds o he sysem wih all he α i s iniially se o zero, and he blue circle wih radius jus over e 3 corresponds o he sysem wih iniial condiions α a () = ε, α a () =