Adiabatic Perturbation Theory in Quantum Dynamics

238 Pages·2003·2.148 MB·English

by  Stefan Teufel (auth.)

1821
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Stefan Teufel
Adiabatic Perturbation Theory
in Quantum Dynamics
1 3
Author
StefanTeufel
ZentrumMathematik
TechnischeUniversita¨tMu¨nchen
Boltzmannstr.3
85747GarchingbeiMu¨nchen,Germany
e-mail:[email protected]
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81-02,81Q15,47G30,35Q40
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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 Molecular dynamics .............................. 15
1.2.2 The Dirac equation with slowly varying potentials .... 21
1.3 Outline of contents and some left out topics................ 27
2 First order adiabatic theory .............................. 33
2.1 The classical time-adiabatic result ........................ 33
2.2 Perturbations of fibered Hamiltonians..................... 39
2.3 Time-dependent Born-Oppenheimer theory: Part I.......... 44
2.3.1 A global result ................................... 46
2.3.2 Local results and effective dynamics ................ 50
2.3.3 The semiclassical limit: first remarks................ 57
2.3.4 Born-Oppenheimer approximationin a magnetic field
and Berry’s connection ........................... 61
2.4 Constrained quantum motion ............................ 62
2.4.1 The classical problem ............................. 62
2.4.2 A quantum mechanical result ...................... 65
2.4.3 Comparison ..................................... 67
3 Space-adiabatic perturbation theory ...................... 71
3.1 Almost invariant subspaces .............................. 75
3.2 Mapping to the reference space........................... 83
3.3 Effective dynamics...................................... 89
3.3.1 Expanding the effective Hamiltonian................ 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 electrons and positrons ................. 106
VI Table of Contents
4.1.2 Semiclassical limit for electrons: the T-BMT equation. 111
4.1.3 Back-reactionof spin onto the translational motion ... 115
4.2 Time-dependent Born-Oppenheimer theory: Part II ......... 124
4.3 The time-adiabatic theorem revisited ..................... 127
4.4 How good is the adiabatic approximation?................. 131
4.5 The B.-O. approximation near a conical eigenvalue crossing.. 136
5 Quantum dynamics in periodic media..................... 141
5.1 The periodic Hamiltonian ............................... 145
5.2 Adiabatic perturbation theory for Bloch bands ............. 151
5.2.1 The almost invariant subspace ..................... 155
5.2.2 The intertwining unitaries ......................... 159
5.2.3 The effective Hamiltonian ......................... 161
5.3 Semiclassical dynamics for Bloch electrons................. 163
6 Adiabatic decoupling without spectral gap................ 173
6.1 Time-adiabatic theory without gap condition .............. 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 Formulation of the problem........................ 185
6.3.2 Mathematical results ............................. 193
A Pseudodifferential operators .............................. 203
A.1 Weyl quantization and symbol classes ..................... 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 Locally isospectral effective Hamiltonians.................. 221
C.2 Simultaneous adiabatic and semiclassical limit ............. 223
C.3 The work of Blount and of Littlejohn et al. ................ 224
List of symbols ............................................... 225
References.................................................... 227
Index......................................................... 235
1 Introduction
Separation of scales plays a fundamental role in the understanding of the
dynamicalbehaviorofcomplexsystemsinphysicsandothernaturalsciences.
It is often possible to derive simple laws for certain slow variables from the
underlying fastdynamics wheneverthe scalesarewellseparated.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
withrespecttotheaxisisthefastdegreeoffreedom.Theearthisanexample
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.
Inthismonographweconsiderquantummechanicalsystemswhichdisplay
suchaseparationofscales.Theprototypicexamplearemolecules,i.e.systems
consisting of two types of particles with very different masses. Electrons are
lighter than nuclei by at least a factor of 2·103, 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 forunderstandingmoleculardynamics.Roughlyspeaking,in the
Born-Oppenheimer approximation the nuclei evolve in an effective potential
generatedbyoneenergyleveloftheelectrons,whilethestateoftheelectrons
instantaneously adjusts to an eigenstate corresponding to the momentary
configuration of the nuclei. The phenomenon that fast degrees of freedom
becomeslavedbyslowdegreesoffreedomwhichinturnevolveautonomously
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
mathematicalmethodscanbeappliedinordertoderiveeffectiveequationsof
motion for the slow degrees of freedom. The unifying characteristic,which is
reflectedin the commonmathematicalstructure describedbelow, is that the
fast scale is always also the quantum mechanical time scale defined through
Planck’s constant (cid:1) and the relevant energies. The slow scale is “slow” with
S.Teufel:LNM1821,pp.1–31,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003
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→0 in Schro¨dinger’s equation
∂
iε ψε(t,x)=H(x,−iε∇x)ψε(t,x) (1.1)
∂t
withaspecialtypeofHamiltonianH.Forfixedtimet∈Rthewavefunction
ψ(t,·) of the system is an element of the Hilbert space H = L2(Rd)⊗H ,
f
whereL2(Rd)isthe statespacefortheslowdegreesoffreedomandH isthe
f
statespaceforthefastdegreesoffreedom.TheHamiltonianH(x,−iε∇x)isa
linearoperatoracting onthis Hilbertspaceandgeneratesthe time-evolution
ofstates in H. As indicated by the notation,the Hamiltonianis a pseudodif-
ferentialoperator.Moreprecisely,H(x,−iε∇x)isthe Weylquantizationofa
function H :R2d →L (H ) with values in the self-adjoint operators on H .
sa f f
As needs to be explained, the parameter 0 < ε (cid:5) 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 descriptionof the quantumdynamics
of the entire system. However,in many interesting situations the complexity
ofthefullsystemmakesanumericaltreatmentof(1.1)impossible,todayand
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.
Thismonographreviewsandextendsaquiterecentapproachtoadiabatic
perturbationtheoryinquantumdynamics.Roughlyspeakingthegoalofthis
approachis to find asymptotic solutions to the initial value problem (1.1) as
solutionsofaneffective Schro¨dingerequationforthe slowdegreesoffreedom
alone.ItturnsoutthatinmanysituationsthiseffectiveSchro¨dingerequation
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
fromthesemiclassicallimit.Indeed,itturnsoutthatadiabaticdecouplingisa
necessaryconditionforsemiclassicalbehaviorofthe slowdegreesoffreedom.
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
twoadvantages.Theobviouspointisthateffectiveequationsofmotionallow
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 lastpoint is illustratede.g.by the derivationofcorrectionsto the semi-
classicalmodelofsolidstatephysicsbasedoncoherentstatesin[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-
dependentforcetermwasmissedinthesemiclassicalequationsofmotion,cf.
Sections 5.1 and 5.3.
Adiabaticperturbationtheoryconstitutesanexamplewheretechniquesof
mathematical physics yield more than just a rigorousconfirmationof 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 ]. However,
2
the physics literature on adiabatic problems is extensive and we mention
at this point the work of Blount [Bl , Bl , Bl ] and of Littlejohn et. al.
1 2 3
[LiFl ,LiFl , LiWe , LiWe ], since their ideasare inpartquite close to ours.
1 2 1 2
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-
boundedself-adjointoperatorsonHilbertspaces,e.g.[ReSi ].The proofsare
1
motivated by strategies developed in the context of the time-adiabatic theo-
rem of quantum mechanics by Kato [Ka ], Nenciu [Nen ] and Avron, Seiler
2 4
and Yaffe [ASY ]. Severalresults presented in Chapter 2 emerged from joint
1
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 mainmathematicaltoolofChapters3–5arepseudodifferentialoperators
with operator-valued symbols. For the convenience of the reader, we collect
inAppendixAthenecessarydefinitionsandresultsandgivereferencestothe
literature. In our context pseudodifferential operators with operator-valued
symbols were first considered by Balazard-Konlein [Ba] and applied many
timestorelatedproblems,mostprominentlybyHelfferandSjo¨strand[HeSj],
by Klein, Martinez, Seiler and Wang [KMSW] and by G´erard,Martinez and
Sjo¨strand [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],BrummelhuisandNourrigat[BrNo]andbyMartinezand
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 , PST , PST ].
1 2 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 (H ) of the Hamiltonian.
sa f
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 , Te ].
1 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 willremainentirelyinthe frameworkofquantummechanicswiththeex-
ceptionofSection2.4,wheresomeaspectsofsuchacomparisonarediscussed
in a special example.
Since it requires considerable preparation to enter into more details, we
postponeadetailedoutlineanddiscussionofthe contentsofthis booktothe
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
shallrefer to it as the time-adiabatic theorem.Afterwards inSection 1.2 two
examples fromphysics arediscussed,where insteadofa time-adiabatictheo-
rem a space-adiabatic theorem can be formulated. While molecular dynam-
ics and the Born-Oppenheimer approximationmotivate the investigations of
Chapter 2, adiabatic decoupling for the Dirac equation with slowly varying
externalfields will leadus directly to the generalformulationofthe problem
as in (1.1).