Semiclassical Propagators of Wigner Function: A comparative study. Jose Mauricio Sevilla Moreno Universidad Nacional de Colombia Facultad, Departamento de F´ısica Ciudad, Colombia 2020 Semiclassical Propagators of Wigner Function: A comparative study. Jose Mauricio Sevilla Moreno Tesis presentada como requisito parcial para optar al t´ıtulode: Magister en Ciencias: F´ısica Director: Carlos Leonardo Viviescas Ram´ırez L´ıneade Investigaci´on: Caos Cu´anticoy M´etodos Semicl´asicos Grupo de Investigaci´on: Caos y complejidad Universidad Nacional de Colombia Facultad de Ciencias, Departamento de F´ısica Bogot´a,Colombia 2020 To my parents, Jos´eand Esperanza. Acknowledgments I would like to thank my thesis advisor, Dr. C. Viviescas, for welcoming me into the Caos y Complejidad group, working with me closely as a collaborator more than a superior, listening to all the comments and ideas I had al along the time I spent in the group. His support , patience and encouragement were essential on the development of this project as well as myself as a physicist. I also would like to thank all the Caos and Complejidad group members whom were always open to listen and discuss ideas on the office. But most of all, I need to thank my friends that were always there for me, listening, guiding and advising specially when I was feeling lost and without force to go on. Thanks to Alejandro Monta~na, C´esar C´elis,Daniela Garz´on,Daniel Padilla, Frank Bautista, Juli´anTriana and Nicol´as Medina for all the things that have done for me that I can't even try to describe in just a few words. Last but not least, I want to thank my family, my parents, my brother and my sister, who always have been an inspiration source for my life. ix Abstract Semiclassical approximations for the dynamics have been widely used in different represen- tations of quantum mechanics. In particular, phase space representations exhibit a clear way to implement those approximations by direct comparison with classical mechanics theory. During the last years, such approximations have been recovering interest, due to the fact that numerical applications suit the modern computational architectures. In this work different semiclassical approximations are built and compared on performance and complexity, showing that they are a suitable way to calculate the quantum dynamics. A study over the caustics is presented on a the Morse potential showing the similarities of the final Wigner function considering and not considering them. Following this, we introduced an initial and final value representations (IVR and FVR) to compare the dynamical properties with the center-center representation proposed by Dittrich et al. [Dittrich et al., 2006] showing that the later can be used as a initial value representation in numerical applications, getting better performance with less complexity than the IVR and FVR using different criteria in such comparison. Contents Acknowledgments vii Abstract ix 0 Introduction 1 1 Theoretical Background 6 1.1 Wigner Function . .8 1.1.1 Dynamics . .9 1.2 Semiclassical Propagation . 11 1.2.1 Wigner function propagator . 12 1.2.2 Initial and Final Value Representations of the Propagator . 16 2 Numerical Implementation 23 2.1 Center-Center Representation . 24 2.1.1 Algorithm . 27 2.2 *Caustics Counting . 28 2.3 Initial Value Representation . 30 CONTENTS xi 2.3.1 Algorithm . 31 2.4 Final Value Representation . 32 2.4.1 Algorithm . 33 2.5 Comparison Criteria . 34 2.5.1 Observable Calculation . 34 2.5.2 Autocorrelation Functions . 34 2.5.3 Normalized Inner Product . 35 2.5.4 Marginals . 35 3 Results 37 3.1 Model: Morse Oscillator . 38 3.2 Caustics . 39 3.2.1 Initial State . 40 3.2.2 Determinant . 41 3.2.3 Reconstruction of the Final State . 43 3.2.4 Final Wigner and Wave Functions . 43 3.3 Center-Center Representation . 47 3.3.1 Marginals . 49 3.4 Initial Value Representation . 51 3.4.1 Wigner Function . 51 3.5 Final Value Representation . 53 3.5.1 Wigner Function . 53 xii CONTENTS 3.6 Comparison . 55 3.6.1 Autocorrelation . 56 3.6.2 Observables . 57 3.6.3 Normalized Inner Product . 61 3.6.4 Complexity . 61 3.7 Parallelization . 63 3.7.1 IVR-FVR . 63 3.7.2 Center-Center . 63 4 Conclusions and Perspectives 64 A Coherent States 66 A.1 Wigner Function for Coherent States . 66 A.2 Relationship with the Husimi Function . 69 B Wigner Function Calculations 71 B.1 Marginal Probabilities and Area . 71 B.2 Weyl Correspondence . 73 C Caustics 76 D Classical Evolution 79 D.1 Dynamics . 80 D.2 Distributions . 80 CONTENTS xiii D.3 Numerical Implementation . 81 E Quantum Propagation 84 E.1 State Dynamics . 84 E.1.1 Algorithm . 87 E.2 Winger Function Dynamics . 88 E.2.1 Algorithm . 90 Bibliography 92 Chapter 0 Introduction Form the very beginning of the development of quantum mechanics an interpretation of the theory was sought. The straightforward path was comparing this new, at that time, theory with the well known results developed so far, i.e., the classical mechanics. This comparison was not clear as the quantum theory has a mathematical structure which is clearly way different than the classical theory and therefore this had to be done using particular results such as expectation values. The quantum theory resulted to have different results than the classical mechanics. This was one of the biggest issues of quantum mechanics, as the community was not accepting the theory because of the lack of interpretation or the strangeness of some already existing. As the continuous comparison of classical and quantum mechanics showed that they are indeed no compatible, the question of the relationship between them remained. it was until 1928 when after studying an statistical interpretation of works done by Dirac [Dirac and Fowler, 1927], that van Vleck [Van Vleck, 1928] built up an expression of the quantum dynamics in terms of classical quantities which is used as a basis of the so called semiclassical methods. It uses the idea of classical propagation of a set of classical particles in order to get the time evolution of wave functions. This idea of semiclassics, relies on the dynamics1, even when the dynamical rules of classical and quantum mechanics are not the same and even worse, they are not even comparable by some important reasons. First, the Schr¨odingerequation which is the quantum mechanical equation of motion is complex as well as a wave function [Landau and E.M., 1977], and the Hamilton's equations are not[Arnold et al., 2013]. Second, the representation of single particles is completely different, as in quantum mechanics a state is represented by a non localized function2, while in 1This means that these methods are not constructed to find other quantum characteristics, such as ground states. 2In fact, it is localized but not on a single point. 2 0 Introduction classical mechanics, a state is defined to be a single point on phase space3, and so on. The idea behind the van Vleck formula can be understood using several ways to proceed, for instance, from the Feynman's path integral point of view [Feynman, 1948, Feynman, 1966], which consists on the usage to trajectories to construct the dynamics of the quantum mechanical state. This trajectories do not necessary represent classically possible trajectories, but the classical trajectories are also included. We may do some comments about this way to proceed, first is that for the evolution of a single quantum mechanical particle represented via a wave function cannot be evolved using a single classical particle but a distribution of particles instead, which leads to the question Shall we use a set of classical particles to compare a single quantum mechanical particle? and the answer is yes, but this brings up more questions about what happen with the amount of quantum particles on a limit where quantum mechanics goes to classical mechanics, but we will not discuss about this concern4, a second comment is that, the evolution is described by two terms, an amplitude and the exponential of a phase. The two of them depend on the particular trajectory taken but these two quantities are calculated from an action corresponding to the chosen path. The semiclassics here enters on reducing the complete set of trajectories to just the classical ones. This result has huge implications. In particular that, the van Vleck propagator itself is not enough to describe certain quantum mechanical features such as tunnelling, due to the fact that as we are only considering the classical trajectories, there are forbidden areas so cannot be transport of probability or density classically speaking, but it occurs quantum mechanically. The van Vleck formula has another issue which was pointed out by Gutzwiller [Gutzwiller, 1967], as the amplitude term has some insights that play a very important role on the dynamics an ad- ditional treatment shall to be added to the van Vleck result. This is basically what are called caustics (For more on caustics, see appendix C), the caustics are points (lines, surfaces and so on depending the order and the number of degrees of freedom of the system) where many trajectories concentrate, this generates a divergence on the amplitude that is not real. As semiclassics is not valid on that exact point5, but it is just before and after in such a way that the only difference is that the trajectory gains a phase of π=2 on the caustic[Maslov and Fedoriuk, 1981], this lead to a big problem and it is that the caustics must be counted, so that at the end of the propagation the complete phase is added to the contribution of each particular trajectory.