adiabatic perturbation theory in quantum dynamics

Lecture Notes in Mathematics 1821 Editors: J M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Stefan Teufel Adiabatic Perturbation Theory in Quantum Dynamics 13 Author Stefan Teufel Zentrum Mathematik Technische Universit ¨ at M ¨ unchen Boltzmannstr. 3 85747 Garching bei M ¨ unchen, Germany e-mail: teufel@ma.tum.de Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 81-02, 81Q15, 47G30, 35Q40 ISSN 0075-8434 ISBN 3-540-40723-5 Springer-Verlag Berlin Heidelb erg New York This work is subject to copyright. 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Typesetting: Camera-ready T E Xoutputbytheauthor SPIN: 10951750 41/3142/du-543210 - Printed on acid-free paper Table of Contents 1 Introduction 1 1.1 The time-adiabatic theorem of quantum mechanics . . . . . . . . . 6 1.2 Space-adiabatic decoupling: examples from physics . . . . . . . . . . 15 1.2.1 Moleculardynamics 15 1.2.2 The Dirac equation with slowly varying potentials . . . . 21 1.3 Outlineofcontentsandsome left outtopics 27 2 First order adiabatic theory 33 2.1 Theclassicaltime-adiabaticresult 33 2.2 Perturbationsof fibered Hamiltonians 39 2.3 Time-dependent Born-Oppenheimer theory: Part I . . . . . . . . . . 44 2.3.1 Aglobalresult 46 2.3.2 Localresults andeffective dynamics 50 2.3.3 Thesemiclassicallimit: firstremarks 57 2.3.4 Born-Oppenheimer approximation in a magnetic field andBerry’sconnection 61 2.4 Constrainedquantummotion 62 2.4.1 Theclassicalproblem 62 2.4.2 Aquantummechanicalresult 65 2.4.3 Comparison 67 3 Space-adiabatic perturbation theory 71 3.1 Almostinvariantsubspaces 75 3.2 Mapping tothereferencespace 83 3.3 Effectivedynamics 89 3.3.1 ExpandingtheeffectiveHamiltonian 92 3.4 Semiclassical limit for effective Hamiltonians . . . . . . . . . . . . . . . 95 3.4.1 Semiclassical analysis for matrix-valued symbols . . . . . . 96 3.4.2 Geometrical interpretation: the generalized Berry connection 101 3.4.3 Semiclassical observables and an Egorov theorem . . . . . 102 4 Applications and extensions 105 4.1 The Dirac equation with slowly varying potentials . . . . . . . . . . 105 4.1.1 Decoupling electronsandpositrons 106 VI Table of Contents 4.1.2 Semiclassical limit for electrons: the T-BMT equation . 111 4.1.3 Back-reaction of spin onto the translational motion . . . 115 4.2 Time-dependent Born-Oppenheimer theory: Part II . . . . . . . . . 124 4.3 Thetime-adiabatictheoremrevisited 127 4.4 How goodistheadiabaticapproximation? 131 4.5 The B O. approximation near a conical eigenvalue crossing . . 136 5 Quantum dynamics in periodic media 141 5.1 TheperiodicHamiltonian 145 5.2 AdiabaticperturbationtheoryforBlochbands 151 5.2.1 Thealmostinvariantsubspace 155 5.2.2 Theintertwiningunitaries 159 5.2.3 TheeffectiveHamiltonian 161 5.3 Semiclassicaldynamicsfor Blochelectrons 163 6 Adiabatic decoupling without spectral gap 173 6.1 Time-adiabatictheorywithoutgapcondition 174 6.2 Space-adiabatic theory without gap condition . . . . . . . . . . . . . . 178 6.3 Effective N -body dynamics in the massless Nelson model . . . . 185 6.3.1 Formulationoftheproblem 185 6.3.2 Mathematicalresults 193 A Pseudodifferential operators 203 A.1 Weylquantizationandsymbolclasses 203 A.2 Composition of symbols: the Weyl-Moyal product . . . . . . . . . . 208 B Operator-valued Weyl calculus for τ -equivariant symbols . 215 C Related approaches 221 C.1 LocallyisospectraleffectiveHamiltonians 221 C.2 Simultaneous adiabatic and semiclassical limit . . . . . . . . . . . . . 223 C.3 TheworkofBlountandofLittlejohn etal. 224 List of symbols 225 References 227 Index 235 1 Introduction Separation of scales plays a fundamental role in the understanding of the dynamical behavior of complex systems in physics and other natural sciences. It is often possible to derive simple laws for certain slow variables from the underlying fast dynamics whenever the scales are well separated. Clearly the manifestations of this basic idea and the precise meaning of slow and fast may differ widely. A spinning top may serve as a simple example for the kind of situation we shall consider. While it is spinning at a high frequency, the rotation axis is usually precessing much slower. The orientation of the rotation axis is thus the slow degree of freedom, while the angle of rotation with respect to the axis is the fast degree of freedom. The earth is an example of a top where these scales are well separated. It turns once a day, but the frequency of precession is about once in 25700 years. In this monograph we consider quantum mechanical systems which display such a separation of scales. The prototypic example are molecules, i.e. systems consisting of two types of particles with very different masses. Electrons are lighter than nuclei by at least a factor of 2 · 10 3 , depending on the type of nucleus. Therefore, assuming equal distribution of kinetic energies inside a molecule, the electrons are moving at least 50 times faster than the nuclei. The effective dynamics for the slow degrees of freedom, i.e. for the nuclei, is known as the Born-Oppenheimer approximation and it is of extraordinary importance for understanding molecular dynamics. Roughly speaking, in the Born-Oppenheimer approximation the nuclei evolve in an effective potential generated by one energy level of the electrons, while the state of the electrons instantaneously adjusts to an eigenstate corresponding to the momentary configuration of the nuclei. The phenomenon that fast degrees of freedom become slaved by slow degrees of freedom which in turn evolve autonomously is called adiabatic decoupling. We will find that there is a variety of physical systems which have the same mathematical structure as molecular dynamics and for which similar mathematical methods can be applied in order to derive effective equations of motion for the slow degrees of freedom. The unifying characteristic, which is reflected in the common mathematical structure described below, is that the fast scale is always also the quantum mechanical time scale defined through Planck’s constant  and the relevant energies. The slow scale is “slow” with S. Teufel: LNM 1821, pp. 1–31, 2003. c  Springer-Verlag Berlin Heidelberg 2003 2 1 Introduction respect to the fast quantum scale. However, the underlying physical mech- anisms responsible for scale separation and the qualitative features of the arising effective dynamics may differ widely. The abstract mathematical question we are led to when considering the problem of adiabatic decoupling in quantum dynamics, is the singular limit ε → 0inSchr¨odinger’s equation i ε ∂ ∂t ψ ε (t, x)=H(x, −iε∇ x ) ψ ε (t, x) (1.1) with a special type of Hamiltonian H.Forfixedtimet ∈ R the wave function ψ(t, ·) of the system is an element of the Hilbert space H = L 2 (R d ) ⊗H f , where L 2 (R d ) is the state space for the slow degrees of freedom and H f is the state space for the fast degrees of freedom. The Hamiltonian H(x, −iε∇ x )isa linear operator acting on this Hilbert space and generates the time-evolution of states in H. As indicated by the notation, the Hamiltonian is a pseudodif- ferential operator. More precisely, H(x, −iε∇ x ) is the Weyl quantization of a function H : R 2d →L sa (H f ) with values in the self-adjoint operators on H f . As needs to be explained, the parameter 0 <ε 1 controls the separation of scales: the smaller ε the better is the slow time scale separated from the fixed fast time scale. Equation (1.1) provides a complete description of the quantum dynamics of the entire system. However, in many interesting situations the complexity of the full system makes a numerical treatment of (1.1) impossible, today and in the foreseeable future. Even a qualitative understanding of the dynamics can often not be based on the full equations of motion (1.1) alone. It is therefore of major interest to find simpler effective equations of motion that yield at least approximate solutions to (1.1) whenever ε is sufficiently small. This monograph reviews and extends a quite recent approach to adiabatic perturbation theory in quantum dynamics. Roughly speaking the goal of this approach is to find asymptotic solutions to the initial value problem (1.1) as solutions of an effective Schr¨odinger equation for the slow degrees of freedom alone. It turns out that in many situations this effective Schr¨odinger equation is not only simpler than (1.1), but can be further analyzed using methods of semiclassical approximation. Indeed, in other approaches the limit ε → 0 in (1.1) is understood as a partial semiclassical limit for certain degrees of freedom only, namely for the slow degrees of freedom. We believe that one main insight of our approach is the clear separation of the adiabatic limit from the semiclassical limit. Indeed, it turns out that adiabatic decoupling is a necessary condition for semiclassical behavior of the slow degrees of freedom. Semiclassical behavior is, however, not a necessary consequence of adiabatic decoupling. This is exemplified by the double slit experiment for electrons as Dirac particles. While the coupling to the positrons can be neglected in very good approximation, because of interference effects the electronic part behaves by no means semiclassical. 1 Introduction 3 A closely related feature of our approach – worth stressing – is the clear emphasis on effective equations of motion throughout all stages of the con- struction. As opposed to the direct construction of approximate solutions to (1.1) based on the WKB Ansatz or on semiclassical wave packets, this has two advantages. The obvious point is that effective equations of motion allow one to prove results for general states, not only for those within some class of nice Ansatz functions. More important is, however, that the higher order corrections in the effective equations of motion allow for a straightforward physical interpretation. In contrast it is not obvious how to gain the same physical picture from the higher order corrections to the special solutions. This last point is illustrated e.g. by the derivation of corrections to the semi- classical model of solid state physics based on coherent states in [SuNi]. There it is not obvious how to conclude from the corrections to the solution on the corrections to the dynamical equations. As a consequence in [SuNi] one ε- dependent force term was missed in the semiclassical equations of motion, cf. Sections 5.1 and 5.3. Adiabatic perturbation theory constitutes an example where techniques of mathematical physics yield more than just a rigorous confirmation of results well known to physicists. To the contrary, the results provide new physical insights into adiabatic problems and yield novel effective equations, as wit- nessed, for example, by the corrections to the semiclassical model of solid state physics as derived in Section 5.3 or by the non-perturbative formula for the g-factor in non-relativistic QED as presented in [PST 2 ]. However, the physics literature on adiabatic problems is extensive and we mention at this point the work of Blount [Bl 1 ,Bl 2 ,Bl 3 ] and of Littlejohn et. al. [LiFl 1 ,LiFl 2 ,LiWe 1 ,LiWe 2 ], since their ideas are in part quite close to ours. A very recent survey of adiabatic problems in physics is the book of Bohm, Mostafazadeh, Koizumi, Niu and Zwanziger [BMKNZ]. Apart from this introductory chapter the book at hand contains three main parts. First order adiabatic theory for a certain type of problems, namely for perturbations of fibered Hamiltonians, is discussed and applied in Chapter 2. Here and in the following “order” refers to the order of ap- proximation with respect to the parameter ε. The mathematical tools used in Chapter 2 are those contained in any standard course dealing with un- bounded self-adjoint operators on Hilbert spaces, e.g. [ReSi 1 ]. The proofs are motivated by strategies developed in the context of the time-adiabatic theo- rem of quantum mechanics by Kato [Ka 2 ], Nenciu [Nen 4 ] and Avron, Seiler and Yaffe [ASY 1 ]. Several results presented in Chapter 2 emerged from joint work of the author with H. Spohn [SpTe, TeSp]. In Chapter 3 we attack the general problem in the form of Equation (1.1) on an abstract level and develop a theory, which allows for approxi- mations to arbitrary order. Chapter 4 and Chapter 5 contain applications and extensions of this general scheme, which we term adiabatic perturbation theory. As can be seen already from the formulation of the problem in (1.1), 4 1 Introduction the main mathematical tool of Chapters 3–5 are pseudodifferential operators with operator-valued symbols. For the convenience of the reader, we collect in Appendix A the necessary definitions and results and give references to the literature. In our context pseudodifferential operators with operator-valued symbols were first considered by Balazard-Konlein [Ba] and applied many times to related problems, most prominently by Helffer and Sj¨ostrand [HeSj], by Klein, Martinez, Seiler and Wang [KMSW] and by G´erard, Martinez and Sj¨ostrand [GMS]. While more detailed references are given within the text, we mention that the basal construction of Section 3.1 appeared already sev- eral times in the literature. Special cases were considered by Emmrich and Weinstein [EmWe], Brummelhuis and Nourrigat [BrNo] and by Martinez and Sordoni [MaSo], while the general case is due to Nenciu and Sordoni [NeSo]. Many of the original results presented in Chapters 3–5 stem from a collabo- ration of the author with G. Panati and H. Spohn [PST 1 ,PST 2 ,PST 3 ]. The first five chapters deal with adiabatic decoupling in the presence of a gap in the spectrum of the symbol H(q, p) ∈L sa (H f ) of the Hamiltonian. Chapter 6 is concerned with adiabatic theory without spectral gap, which was started, in a general setting, only recently by Avron and Elgart [AvEl 1 ] and by Bornemann [Bor]. Most results presented in Chapter 6 appeared in [Te 1 ,Te 2 ]. The reader might know that adiabatic theory is well developed also for classical mechanics, see e.g. [LoMe]. Although a careful comparison of the quantum mechanical results with those of classical adiabatic theory would seem an interesting enterprize, this is beyond the scope of this monograph. We will remain entirely in the framework of quantum mechanics with the ex- ception of Section 2.4, where some aspects of such a comparison are discussed in a special example. Since it requires considerable preparation to enter into more details, we postpone a detailed outline and discussion of the contents of this book to the end of the introductory chapter. In order to get a feeling for adiabatic problems in quantum mechanics and for the concepts involved in their solution, we recall in Section 1.1 the “adiabatic theorem of quantum mechanics” which can be found in many textbooks on theoretical physics. For reasons that become clear later on we shall refer to it as the time-adiabatic theorem. Afterwards in Section 1.2 two examples from physics are discussed, where instead of a time-adiabatic theo- rem a space-adiabatic theorem can be formulated. While molecular dynam- ics and the Born-Oppenheimer approximation motivate the investigations of Chapter 2, adiabatic decoupling for the Dirac equation with slowly varying external fields will lead us directly to the general formulation of the problem as in (1.1). [...]... often the slowly varying parameters come from an idealization of the coupling to another quantum system The idealization consists in prescribing the time-dependent configurations of the other system in the Hamiltonian of the full quantum system It is the content of space -adiabatic theory to understand adiabatic decoupling without relying on this idealization, as to be explained in detail in the next section... compute the evolution of its spin Such a simple minded adiabatic approximation thus will never explain why an electron beam is split in a Stern-Gerlach magnet because of spin This problem, as we will see in Section 4.1.2, is solved once we switch to the space -adiabatic setting Thus one of our main interests in this monograph will be a general spaceadiabatic perturbation theory which allows us to systematically... subspaces P∗ (t)H are not only adiabatically decoupled from the remainder of the Hilbert space, but that the dynamics inside of them can be formulated in terms of a much simpler Schr¨dinger equation as (1.19), turns out to produce many interesting results o In Section 1.2 of the introduction, we obtain, e.g., the famous Thomas-BMT 1 equation for the spin -dynamics of a relativistic spin- 2 particle from (1.20)... the time-axis and in the space -adiabatic case one considers Hamiltonians fibered over the configuration space of the slow degrees of freedom The terminology space -adiabatic partly originates in the latter observation Also in the space -adiabatic setting effective dynamics on the decoupled subspace P∗ H are of primary interest As in the time -adiabatic case one can map the subspace P∗ H in a natural way to... for sharing their insights on adiabatic problems with me and for their interest in and support of my work Important parts of the research contained in Chapter 3 and Chapter 4 were initiated during visits of Andr´ Martinez and Gheorghe e Nenciu in Munich in the first half of 2001, whose role is herewith thankfully acknowledged There are many more scientists to whom I am grateful for their interest in and/or... gap condition in this section, but refer to Chapter 6, which is devoted to adiabatic decoupling without spectral gap Space -adiabatic theory in the general form to be presented in this monograph is quite recent and will be motivated and set up in Section 1.2 We close this introductory section on the time -adiabatic theorem with some remarks on higher order estimates Going back to the beginning of this... to any finite order ♦ 1.2 Space -adiabatic decoupling: examples from physics 15 1.2 Space -adiabatic decoupling: examples from physics Applications of the time -adiabatic theorem of quantum mechanics can be found in many different fields of physics Indeed, the importance of a good understanding of adiabatic theory is founded in the fact that whenever a physical system contains degrees of freedom with well... decoupling without gap-condition is considered Chapter 3: Space -adiabatic perturbation theory This chapter contains the general scheme of space -adiabatic perturbation theory dealing with the abstract problem formulated in (1.1) The theory uses in its formulation and its proofs pseudodifferential operators with operatorvalued symbols and a short presentation of the relevant material can be found in Appendix... matrix-representation (1.20) 12 1 Introduction Generalizations In summary Theorem 1.2 and Theorem 1.4 contain what we will call first order time -adiabatic theory with gap condition The terminology suggests already that there are several ways of generalizing this theory (i) Adiabatic theorems with higher order error estimates and higher order asymptotic expansions in the adiabatic parameter ε (ii) Adiabatic theorems... concrete problem ?” and in Section 4.5 we study the effective Born-Oppenheimer Hamiltonian near a conical eigenvalue crossing Chapter 5: Dynamics in periodic structures As a not so obvious application of space -adiabatic perturbation theory we discuss the dynamics of an electron in a periodic potential based on Panati, Spohn, Teufel [PST3 ] Indeed it requires considerable insight into the problem and some . approximately transports the time-dependent spectral subspaces of H(t)whichvarysuffi- ciently smoothly as t changes. In the classical result one considers spectral subspaces associated with parts of the spectrum. ♦ Effective dynamics In many situations one is interested only in the dynamics inside the subspaces P ∗ (t)H, which might be of particular interest for physical reasons or just be selected by the initial. BertelsmannSpringer Science + Business Media GmbH springer.de c  Springer-Verlag Berlin He idelberg 2003 PrintedinGermany The use of general descriptive names, registered names, trademarks, etc. in