Table Of ContentLecture Notes in Physics
V. M. (Nitant) Kenkre
Interplay
of Quantum
Mechanics and
Nonlinearity
Understanding Small-System
Dynamics of the Discrete Nonlinear
Schrödinger Equation
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V. M. (Nitant) Kenkre
Interplay of Quantum
Mechanics and
Nonlinearity
Understanding Small-System Dynamics
¨
of the Discrete Nonlinear Schrodinger
Equation
V.M.(Nitant)Kenkre
UniversityofNewMexico
Albuquerque,NM,USA
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-030-94810-8 ISBN978-3-030-94811-5 (eBook)
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ThisbookisdedicatedtoagroupofpeopleImet
duringthe4-yearperiod,1964-68,inPowai,India.
Itisalittlelessthan60yearsagothatIjoinedthe
IndianInstituteofTechnology,Bombay,tobeginmy
undergraduatestudies.There,Ihadthegood
fortunetostudyundertheguidanceofkindand
thoughtfulteacherswhokeptmyinterestinthe
sciencesalivethroughoutmycourseinengineering
matters.NotableamongthemwereJ.S.Murty,
R.E.Bedford,M.S.Kamath,andC.Balakrishnan.
IfeelIoweagreatdebttothemforthepersonal
helpandsupporttheyprovidedme.Evenmore
importantinmyintellectualformationthantheir
inputswastheexcitingatmospherecreatedbymy
dailyinteractionswithbrilliantcolleagues,my
classmates.Theywereanexceptionalbunch;Ihave
seldomcomeacross,anywhereintheworld,the
levelofintelligenceandoveralltalentthattheywere
blessedwith.Idebated,quarreled,discussed,and
learnedmuchfromthem.Someofthemarenomore.
EspeciallythinkingofRajanM.RanadeandAshok
C.Kulkarni,closefriendswhoseroomsflankedmy
own,Idedicatethisbook,withsinceregratitude,to
thememoriesofmyteachersandcolleagueswho
havedeparted,andtothosewhoarestillaround.
Preface
Procrastinationgivesyoutimetoconsiderdivergentideas,
tothinkinnonlinearways,tomakeunexpectedleaps.
—AdamGrant
Procrastination,mydearwifehascomplainedfor50years,istheincessantdriving
power behind my life. While I do not believe she is right, it is possible, I think,
thatmyfascinationwithnonlinearscience,alwaysfromasafedistancesoIwasnot
evercompelledtoclaimanyexpertiseinthefield,mighthavearisenthatway,asthe
modernthinker,AdamGrant,quipsinthestatementquotedabove.
The title of the book you have in your hands, dear reader, is bound to raise
your expectations and make you anticipate profound discussions concerning the
philosophy of quantum physics and the exquisite nuances of nonlinear science.
Walking a fine line between wanting to get your attention by whatever proper
means available to me and my commitment to practice honesty, I must admit,
at the outset, that this book is nothing more than a description of a casual but
pleasurable adventure I had for about a decade starting in the mid-1980s. That
adventure occurred during my efforts to assist a few of my students, who were so
inclined,togettheirPh.D.degreesdoingtheoreticalresearchinnonlinearaspectsof
condensedmatterphysics.Ilearnedjustenoughaboutthesubjectforthatpurpose,
primarily from my own escapades from what I considered to be my main field
of research (statistical mechanics), helped always by books and by colleagues
wiser than myself. There are no involved debates, deep ruminations, and sudden
epiphanies reported here. A single mathematical entity, the “discrete nonlinear
Schrödingerequation,”isthecenteroffocusinthisbook,alongwithsimpleideas
and calculations based on it. What I can certainly promise the reader, however,
is the expression of a tyro’s delight at learning about pretty transitions and nifty
connectionsdiscoveredwithabeginner’sjoyinabeautifulfieldofscience.
As with a recent book I have published, Memory Functions, Projection Tech-
niques and the Defect Technique (Kenkre 2021), the intended readership here is
young starting-out theoretical physicists early in their research career, at their
postdoctoral or their advanced graduate stage. I visualize eager, fresh researchers
searchingforweaponsoftheoreticalresearch,notyethardenedbytherequirements
vii
viii Preface
and rigors of their profession, inexperienced perhaps, but at their creative peak. I
believethatthisbookwillprovidethemwithsimpleconceptsandtoolstheycanuse
toengageinusefulresearchintheoreticalphysicsevenastheyacquaintthemselves
withinterestingresultsinthefieldofnonlinearphysicsincondensedmatter.
Quantum mechanics is usually regarded as being linear in its structure. Linear
algebra is, indeed, the underlying mathematical discipline necessary to construct
it, and linear superposition of its wave functions or state vectors is a frequently
occurring phrase in discussions of the subject. Yet, the procedure to extract the
expectation value of an observable from the state vector (or wave function) surely
involvesanonlinear,tobeprecise,abilinear,operation.Itisperhapspartlywithan
expressdesiretoreturntolinearityinthisoperationthatvonNeumannintroduced
thedensitymatrixsothatobservableextractionfromthestatecouldbedonethrough
thelinear traceoperation.Arguably,themostimportantanddiscussedpropertyof
quantummechanicalsystemsissuperposition.Linearityisimplicitinthestatement
thatiftwoindependentstatevectors,amplitudes,orwavefunctionsareappropriate
to describe a process, their superposition, i.e., a sum with constant multiplying
coefficients, is also appropriate for such description. Linearity is also inherent in
the properties of the operators that represent all quantum mechanical observables.
The fact that, typically, the Hamiltonian of the system that operates on the state
vector to yield the time derivative of the latter is independent of the state vector
leadstolinearityinthetimeevolution.
However, there is an obvious problem with this way of thinking. While it is
true that classical mechanics is often thought of as being generally nonlinear by
contrast,itiswellknownthatareformulationanalogoustovonNeumann’s,which
introduces into classical mechanics Liouville densities, at once bestows linearity
on classical mechanics as well. The place of the commutation of the Hamiltonian
with the system density matrix in quantum mechanics is now taken by Poisson
brackets involving the (classical) Hamiltonian and the Liouville density (Balescu
1975; Reichl 2009). If one (mistakenly) falls back on the nonlinearity of the
classical equations of motion for observables, e.g., the coordinates and momenta
oftheconstituentparticlesordegreesoffreedom,andarguesthatthisisspecialof
classical mechanics, one recognizes one’s error immediately on noticing that the
equations for the corresponding operators for coordinates and momenta are also
generally nonlinear. Linearity is characteristic of the usual equation of motion for
thestatevectororwavefunction,saytheSchrödingerequation,oroftheLiouville-
vonNeumannequationforthedensityordensitymatrix.Thisiscertainlynottrue
oftheHamiltonianequationsfortheobservablesortheircorrespondingoperators.
Surely, we spend much time, as we should, with the process of finding eigen-
values and eigenvectors of the Hamiltonian and other operators and rely, for the
purposes of calculation, on the mathematics of linear algebra. However, it is
necessarytobepreciseinone’smindaboutwhatonemeansbythestatementthat
quantummechanicsislinear.
Preface ix
Wecanassertwithoutargumentthat,bythelinearitystatement,weatleastmean
that,intheevolutionequation
d|(cid:2)(t)(cid:2)
ih¯ =H|(cid:2)(t)(cid:2),
dt
theHamiltonianoperatorH isnotitselfafunctionof|(cid:2)(t)(cid:2).Inthetitleofthisbook,
we consider a violation of this assumption and study the effects of the interplay
of familiar quantum features with such nonlinearity. Investigations of a violation
of linearity on a fundamental level have been carried out by several illustrious
scientists (Weinberg 1989; Leggett 2002; Jordan 2009). The last of the authors
citedhasreferredtoGeorgeSudarshan’squestionsandinsightsintotheproblem.In
thecontextofthepresentbook,theviolationdoesnotarisefromanyfundamental
source,ratherfromwhatisanapproximaterepresentationofthedynamicssupposed
to arise from a coarse-grained description through the elimination of some of the
variablesinherentinthedynamics.1
Specifically, our interest is in elucidating the consequences of the discrete
nonlinearSchrödingerequation(DNLSE).Theequationaroseoriginallyinpolaron
physics when phonons interact strongly with moving electrons, or with quasipar-
ticles such as other phonons, from the creative arguments of several investigators
(Landau 1933; Pekar 1954; Holstein 1959a,b). Much later it also arose under the
guise of the Gross-Pitaevskii equation (Pitaevskii 1961; Gross 1961, 1963) in the
dynamics of Bose-Einstein condensates, and even formally in optical waveguides
and in the dynamics of classical anharmonic oscillators. My interest in this book
is primarily in the first area and to some extent in the second. I shall refrain from
touching the field of optical waveguides or anharmonic oscillators given that fine
expositions(Eilbecketal.1985;HennigandTsironis1999)alreadycontaindetailed
referencestothoserespectiveareas.
The subject of nonlinear Schrödinger equations is teeming with activity and
encompasses an enormous community of physicists and mathematicians. My
purpose will be to focus only on the discrete form of the nonlinear Schrödinger
equationandthattooinspatiallyverysmallsystems.Fromthemultitudeofexpert
treatments in the area of nonlinear Schrödinger equations, even restricting to the
discrete variety, I especially mention two, Ablowitz et al. (2004) and Kevrikidis
(2009).Bothareexcellentandcanteachthereadervaluabletechniquesandtoolsin
thegeneralfield.Therearealsoother,similar,booksonthesubjectwithspecialized
slants, for instance (Christiansen and Scott 1990). For the general field of soliton
physics, an eminently readable presentation is Physics of Solitons (Dauxois and
Peyrard2006).Givenallthiswealthofmaterialalreadyavailableintheliterature,it
isimportanttounderstandwhyIhaveundertakentowritethisone.Ibelieveithas
a special pedagogical element. The book is characterized by the fact that it offers
alimitedtreatmentoftheDNLSEinsystemsofsmallspatialextent,alwaystaking
advantageofanalyticalsolutionswhereverpossible.Thedetailsofthisstatementare
1Thus,inoneinstance,thenonlinearityarises,orissupposedtoarise,fromtheremovalofphonon
degreesoffreedomfromastronglyinteractingelectron-phononsystem.
x Preface
explainedinChap.1.Therestrictionthatweimposeonourownsphereofanalysis
willallowustofocusonsimplematterswithsimpletools.
Theworkdescribedinthepagesofthisbookstemsfromabout50publications
primarily within my own research group but also in those of closely connected
colleagues, which were produced largely during about a decade starting around
1985.Interestintheso-calledDavydovsoliton(Scott1992)wasgreatlyresponsible
fortheactivityinthesubjectatthattime.2 Ibelieve thelessonslearnedaswellas
issues left unresolved during that relatively brief spurt in research on the part of
the community focused on that topic are as timely today as they were then. My
own activities were not motivated by the Davydov problem except for using it as
a backdrop. Despite the elementary nature of the questions raised and treatments
offered,Ihopethatthereaderwillfindsomethingofvalueinthesepages.
ConsideringthatIfeelthecontentofthisbookmightbemostusefultostarting-
out researchers, let me venture to express my opinion to them that research in
theoretical physics comes in three flavors. One has to do with the explanation of
experiment.Thecentralimportancefortheadvancementofscienceofthiskindof
activityisobvious.Einsteinunravellingtheessenceofthetemperaturedependence
of the specific heat of insulators provides a clear example as do Bardeen, Cooper,
and Schrieffer presenting their explanation for superconductivity. A second flavor
isthatofseekingrelationsbetweenformalismswithaviewtoestablishingbridges
across theories: an excellent instance is Dyson’s building a lexicon that facilitated
theunderstandingoftheconnectionsbetweenFeynman’sformalismontheonehand
and Schwinger’s on the other in their investigations of quantum electrodynamics.
A third type of activity of the theoretical physicist occurs when, starting with
an equation or similar mathematical object, one barges forward and discovers
something interesting, unexpected, and surprising, and then presents the results
to the experimentalist. The discovery of waves of electromagnetic radiation from
manipulations of Maxwell’s equations happened, as we all know, in this manner.
Thislastofthethreeflavorsiswhatpermeateswhateverisdescribedinthepresent
book, a single exception being the analysis of excimers in Chap.11. Experiments
areanalyzedseriouslybutonlyinthespiritofdesigningthemtoseeneweffects;set
puzzlesarenotsolved.3
2GlimpsesoftheactivityaroundtheDavydovsolitonproposalcanbehadintheDauxois-Peyrard
bookmentionedabove.MuchmoreattentionisfocusedonitintheChristiansen-Scottbookonthe
DNLSE.
3Each of these kinds of theoretical research is, no doubt, important to the advancement of our
branchofscience.Mostofushavetriedallthree,tovariousextents.Thefirstisperhapsthemost
difficult.Someamongmycolleaguesconsiderittheonlykindworthpursuing,insistingtheorists
tobenomorethanhiredhandsofexperimentalists.Attheotherextremeliesometheoristswho
confess that they have the ability only to work in the last of the modes mentioned. That mode
is,perhaps,theeasiestasthereisnogoalsetbyotherstomeet.Ofallthree,this"forward”kind
of research activity has less of puzzle solving and has, I tend to think, more kinship to artistic
endeavors.Mymusicianfriendshaveoftenconfidedinmethatthesolvingofsetproblemsthat
theybelievetobecharacteristicofscientificworkturnsthemoff.Dancingtoone’sowntunesis
howtheydescribetheirownactivity.Thatiswhy,amongthetheorist’sthreeavenues,thelastone
mentionedaboveistheoneIconsidermostakintotheartist’smannerofworking.
