Philosophic foundations of quantum mechanics

Bornrecognized theprobability interpretation ofthe waves.Heisenberp;sawthat them^P^niliipflil mnfrTFiisniofthetheoryinvolves anunsurmountableuncertainty of predictions and a disturbance

TEXT FLY WITHIN

gj<OU_166132

Philosophic Foundations of Quantum Mechanics

Philosophic Foundations

PROFESSOROF PHILOSOPHYINTHEUNIVERSITYOF CALIFORNIA

UNIVERSITY OF CALIFORNIA PRESS

BERKELEY AND LOS ANGELES 1944

BERKELEY ANDLOSANGELESCALIFORNIA

CAVJRIDGE UNIVERSITY PRESS

nf'ONDON, ENGLAND

COPYRIGHT, 1Q44, BY

THE REGENTS Ofr THE UNIVERSITY OF CALIFORNIA

PRINTED IN THE UNITED STATES OF AMERICA

BY THE UNIVERSITY OP CALIFORNIA PRESS

Two

GREAT theoretical constructions have shaped the face of modern

physics:the theoryof relativityandthe theoryofquanta Thefirsthasbeen, onthe whole, the discoveryofone man, sincetheworkofAlbertEinstein has remained unparalleled bythe contributions of others who, like

Hendrik Anton Lorentz, came very close to the foundations of special tivity, or, like HermannMinkowski, determined the geometricalformof thetheory It is different with the theory of quanta This theory has been de-velopedbythecollaboration ofanumberofmeneachofwhomhas contributed

rela-an essential part, andeachof whom, inhiswork, hasmadeuse ofthe results

of others

^'The necessity of such teamworkseems to be deeply rooted in the subject

matterof quantumtheory In thefirst place, the developmentof thistheoryhasbeengreatlydependent onthe productionofobservationalresultsandon

the exactnessofthe numerical valutanhtfl.inftfi. Withoutthe helpof thearmy

ofexperimenterswhophotographedspectrallinesor watchedthe behaviorof

elementary particles by means of ingenious devices, it would have been possibleevertocarrythroughthe theoryofthe quanta evenafteritsfounda-tionshadbeenlaid.Inthe secondplace,thesefoundationsareverydifferent in

im-logical form fromthose ofthe theory ofrelativity. Theyhave never hadthe

formofoneunifyingprinciple, notevenafterthe theory has beencompleted.

Theyconsist ofaset of principleswhich, despite theirmathematicalelegance,

donotpossessthe suggestive characterofaprinciplewhichconvinces usatfirst

sight,asdoes theprinciple ofrelativity.And,finally,they departmuchfurther

fromtheprinciples ofclassicalphysicsthanthe theoryof relativityeverdidinits criticism of space and time; their implications include, in addition to a

transition from causal laws to probability laws, a revision of philosophicalideasabouttheexistence ofunobservedobjects,evenoftheprinciples oflogic,

andreachdowntothe deepestfundamentalsofthe theoryofknowledge

Inthedevelopmentofthetheoreticalformofquantumphysics, wecan

dis-tinguish four phases The first phase is associated with the names of Max

Planck, AlbertEinstein,andNilsBohr.Planfik'p intrndnnf.jop nff.h^qnonfnin

1900wasfollowedbyEinstein'sextensionofthequantumconr^pttowardthat

ofa needleradiation (1905) Thedecisive step, however, wasmade IP Rnhr'aapplication (1913^ ofthelmanfomidea.to theanalvsHofthestructure oftheatgin^whichledtoanewworldofphysicaldiscoveries

Thesecond phase, which began in 1925, representsthe work of ayounger

generation which had been trained in the physics of Planck, Einstein, and

Bohr, andstartedwherethe olderoneshadstopped. It, isamostastonishingfactthatthisphase,whichleduptoquantumnr>eftlmrnfg

Tbepanwithouta clear

insight intowhat was actuallybein^ done LouisdeBroglie introduced waves

a!T"companions of particles; Erwin Schrodinger, guided by mathematical

analogieswithwaveoptics,discoveredthe^twofundamentaldifferential

equa-tions ofquantum mechanicsTMaxBorn, WernerHeisenberg, Pascual Jordan,and, independently nf thfo grnnp, f^ul A M. Dirac constructed the matrix

mechanics which seemedto defyany wave interpretation

Tj^isperiodsentsanamazing triumphofmathematicaltechnique which, masterly applied

repre-andguidedbya physicalinstinctmorethan,byIngipaJprinciplesldetermined

thepathtothe discovery of q

^ifif^iVw*"fh wggQHftt-flembraceallobservable

3afta.All thiswasdoneinaveryshorttime;by1926 themathematical shape

ofthenewtheoryhad becomeclear.

Thethirdphase followedimmediately:itconsisted inthe physicaltation of theresults obtained Schrodinger showed the identity of wq.ve me-

interpre-chanicsandmatrix mechanics Bornrecognized theprobability interpretation

ofthe waves.Heisenberp;sawthat them^P^niliipflil mnfrTFiisniofthetheoryinvolves anunsurmountableuncertainty of predictions and a disturbance ofthe object by themeasurement And here oncemore Bohr^ntervenc^M.ntlie

\flork of theyoungergeneration anJTshoweJthat the"description of naturegivenbythe theory was toleave opena specific ambiguity whichJiejormu-

lated in his principle ofcomplementarity.

Thefourth phase continuesuptothe presentday;it is filled withconstantextensions of the results obtained toward further and further applications,including the application to new experimental results. These extensions are

combinedwithmathematicalrefinements;in particular,the adaptationofthe

mathematicalmethodtothepostulates of relativityisinthe foregroundoftheinvestigations.Weshallnotspeakoftheseproblemshere, sinceour inquiryisconcerned with thelogicalfoundationsofthetheory.

Itwaswiththephaseofthe physicalinterpretationsthat the noveltyofthe

logicalformofquantummechanicswasrealized.SomethinghadBeenachieved

in thisnew theorywhich was

Contrary to traditional coTipppts nf fcn

9wledge

and reality It was not easy, however, to say what had happened, i.e., toproceedtothephilosophical interpretation ofthetheory. Basedonthe physicalinterpretations given, a philosophy for common use was developed by thephysicists which spoke of the relation of subject and object, of pictures of

reality whichmustremain vagueandunsatisfactory, ofoperationalismwhich

is satisfiedwhenobservationalpredictions are correctlymade, andrenouncesinterpretations as unnecessary ballast. Such concepts may appear useful forthepurpose ofcarryingon themerely technicalworkofthephysicist. Butit

seems tous that the physicist, whenever hetriedto be consciousofwhat he

did,could not helpfeelingalittleuneasy withthisphilosophy.Hethenbecame

awarethathewaswalking,so tospeak,onthe thiniceofasuperficiallyfrozen

lake,andherealizedthathe mightslipandbreak throughatany moment.

It was this feeling of uneasiness which led theauthor to attempt a sophical analysis ofthe foundations ofquantummechanics Fullyawarethatphilosophy should not try to construct physical results, nor try to prevent

physicists from finding such results, he nonetheless believed that a logical

analysis ofphysics which did not use vague conceptsandunfairexcuseswas

possible The philosophy of physics should be as neat and clear as physicsitself; itshouldnottake refugeinconceptionsof speculativephilosophywhich

mustappearoutmodedinthe ageofempiricism,noruse the operationalform

ofempiricismasawaytoevade problems ofthelogicof interpretations rectedbythis principletheauthorhastried in the presentbookto developa

Di-philosophical interpretation of quantum physics which is free from

meta-physics, andyet allows us to consider quantum mechanical results as

state-ments about an atomic worldasrealasthe ordinary physical world

It scarcelywillappearnecessarytoemphasizethatthisphilosophical analysis

iscarriedthroughindeepest admirationoftheworkofthephysicists,andthat

itdoes not pretend tointerferewiththemethodofphysicalinquiry Allthatisjntendedinthisbook iff fllflj-ifWfinn nf

pnrmfipf.fi ;nowherein this

presenta-tion, therefore, isany contribution toward the solution of physical problems

tobeexpected.Whereasphysicsconsists intheanalysis ofthe physical world,philosophy consists in the analysis of our knowledge of the physical world

Thepresentbookismeanttobephilosophical inthissense

Thedivision ofthebookissoplannedthatthefirstpart presents thejgeneralideasonwhichquantummechanicsisbased;thispart, therefore, outlinesour

and siiT^ni^lJ^^s^tg.zesviltfi. The presentation is

suchthatitdoesnot presupposemathematicalknowledge,noranacquaintancewiththemethodsofquantumphysics In the second partwepresent the out-

lines of the mathematical[

methods_of quantum mechanics: this isso writtenthat a knowledge of thecalculus should enable the readerto understand theexposition.Sincewepossesstoday anumberof excellenttextbookson quantum

mechanics,suchanexpositionmayappearunnecessary;wegiveit,however,inorder to open ashort cuttoward themathematical foundations of quantummechanicsforall thosewhodo not havethetime forthoroughstudies of thesubject, or who would like to see in a short review themethods which theyhaveappliedinmanyindividualproblems. Ourpresentation, of course,makes

noclaimtobecomplete. Thethirdpartdealswith the variousinterpretations

of quantummechanics;it ishere thatwe makeuseof boththe philosophicalideasot thetirstpart andthemathematicalformulations ofthe second The

properties ofthedifferent interpretations are discussed, andaninterpretation

intermsofathree-valuedlogicisconstructedwhichappearsasasatisfactory

logicalformofquantummechanics

I am greatly indebted to Dr Valentin Bargmann of the Institute of

Ad-vanced Studies in Princeton for his advice in mathematical and physical

questions;numerousimprovementsinthepresentation, inPartII in particular,aredueto his suggestions IwishtothankDr NormanC DalkeyoftheUni-

versity of California,LosAngeles, andDr.ErnestH Hutten, formerlyatLos

Angeles, nowinthe University of Chicago, for the opportunityof discussing

with themquestionsofalogicalnature, andfor theirassistance inmattersofstyle andterminology Finally Iwish to thankthe staff ofthe University ofCalifornia Press for the care and consideration with which they have edited

my book and for their liberality in following my wishes concerning some

deviationsfromestablishedusageinpunctuation

Apresentationofthe viewsdevelopedin this book, includinganexposition

ofthesystemofthree-valuedlogicintroducedin 32,wasgivenbytheauthor

at the Unity of Science Meetingin the Universityof Chicago onSeptember

5,1941

HANS REICHENBACH

DepartmentofPhilosophy,University ofCalifornia,LosAngeles

June, 1942

PAGE

1. Causallawsandprobabilityla*ws 1

2.Theprobability distributions 5

3. Theprinciple ofindeterminacy 9

4. Thedisturbance ofthe objectbytheobservation 14

5.The determinationofunobservedobjects 17

6. Waves and corpuscles 20

7.Analysisofaninterferenceexperiment 24

8. Exhaustiveand restrictive interpretations 32

PART II: OUTLINES OF THE MATHEMATICS OF

QUANTUM MECHANICS

9. Expansionofa functioninterms ofanorthogonal set 45

10. Geometricalinterpretation in the function space 52

11. Reversionand iteration oftransformations 58

12. Functions of severalvariablesandthe configuration space 64

13. DerivationofSchrodinger'sequationfrom deBroglie's principle 66

14. Operators, eigen-functions, andeigen-valuesofphysicalentities 72

15. The commutation rule 76

16. Operatormatrices 78

17. Determinationofthe probability distributions 81

18. Time dependenceofthe ^-function 85

19. Transformation tootherstatefunctions 90

20 Observational determinationofthe ^-function 91

21 Mathematicaltheoryofmeasurement 95

22 Therulesofprobabilityandthe disturbancebythemeasurement. 100

23 Thenatureof probabilitiesandofstatisticalassemblagesin

quantummechanics 105

x CONTENTS

PART III: INTERPRETATIONS

24 Comparisonof classical and quantum mechanical statistics Ill

25. Thecorpuscleinterpretation 118

26. Theimpossibility ofa chainstructure 122

27. The wave interpretation . 129

28. Observational language and quantummechanical language ' . 136

29. Interpretationbyarestrictedmeaning 139

30 Interpretationthrough a three-valuedlogic 144

31 Therulesof two-valuedlogic 148

32 Therules ofthree-valuedlogic 150

33 Suppression ofcausalanomalies throughathree-valued logic 160

34 Indeterminacyin the objectlanguage 166

35 Thelimitation of measurability 169

Parti GENERAL CONSIDERATIONS

1. Causal Laws and Probability Laws

The philosophical problems ofquantum mechanics are centered aroundtwo mainissues. Thefirstconcerns the transitionfrom causal lawsto probability

laws^Jhesecond concerns thejnterpretationofunobserved objects.Webegin

with thediscussion ofthe first issue, andshall enterinto the analysis of the

secondinlater sections.

Thequestionofreplacingcausallawsbystatisticallawsmadeitsappearance

inthehistory ofphysics long beforethe times ofthe theoryofquanta Sincethe time of Boltzmann'sgreat discovery which revealedthe secondprinciple

ofthermodynamicsjbobeastatisticalinstead ofacausallaw,the opinion hasbe"enTFepeatedlyutteredthata similar latemay meetallother physicallaws

Theidea of determinism, i.e., ofjtrifit^causal laws governing the elementary

phflfinnnftim nfnfl/hirp, was

recognized as an extrapolation inferred from thecausal regularities of the macrocosm The validity of this extrapolation was

questioned as soon as it turned out tha/Lmacrocosmic regularity is equallycompatible withirregularity inthemicrocosmic domain,sii^cethelawofgreat

numberswilltransform theprobabilitycharacteroftheelementaryphenomena

into the_practical certainty ofstafcfttJcallawft. Observationsinthemacrocosmic

domainwillneverfurnishanyevidencefor causality ofatomicoccurrences solongas onlyeffectsof greatnumbersof atomicparticle^"are considered This

wasthe result ofunprejudicedphilosophical analysis of the physics of

Boltz-mann.1

Withthisresultadecision ofthe questionwaspostponeduntilitwaspossible

to observe macrocosmic effects of individual atomic phenomena Even with

the use of observations of this kind, however, the question is not easilyanswered, but requires the development ofamore profound logical analysis.Whenever wespeakofstrictlycausallawsweassumethemtoholdbetween

idealizedphysical states; and we know that actual physicalstatesnever

cor-1 It is scarcely possible tosaywhowasthefirst toformulatethis philosophical idea.

\Vehaveno publishedutterances ofBoltzmannindicatingthat he thoughtofthebility ofabandoningtheprinciple ofcausality. Inthedecade preceding the formulation

possi-ofquantummechanics the ideawasoften discussed F. Exner, in hisbook, Vorlesungen

uber die physikalischenGrundlagender Naturwissenschaften (Vienna, 1919), isperhapsthe first tohaveclearlystatedthe criticismwhichwegave above: "Letusnotforgetthat

the principle of causalityandtheneedfor causalityhas been suggestedto us exclusively

byexperienceswithmacrocosmic phenomena andthatatransference oftheprinciple to

,

microcosmicphenomena,viz theassumption that everyindividual occurrencebe

strictly

i, hasnolongeranyjustificationbased onexperience." p 691. Witn

; causallydetermined, has nolongeranyjustificationbased onexperience,

reference toExner, E Schrodinger has expressedsimilar ide

' ~ ' '

inZurich,1922,publishedin Naturwissenschaften, 17:9 (1929).

CO

2 PARTI. GENERAL CONSIDERATIONS

respondexactlytothe conditions assumedforthelaws.Thisdiscrepancy hasoften beendisregarded as irrelevant, as befogdue tothe imperfection oftheexperimenterandtherefore negligible inastatementaboutcausalityasapropiertyof nature With suchanattitude, however, thewaytoa solution of the

problem of causality is barred Statements about the physical world have

meaningonlysofar a. theyarfi CQiffleqteflwfohverifiableresults;anda

state-mentaboutstrictcausalitymustbetranslatable intostatementsaboutablerelationsif it is tohave autilizablemeaning Followingthis principlewe

observ-caninterpretthestatementof causality inthe followingway

If we characterize physical states in observational terms, i.e., in terms ofobservationsastheyare actuallymade, we knowthatwecanconstruct prob-ability relationsbetweenthesestates.Forinstance,ifwe knowtheinclination

ofthebarrel ofa gun, thepowdercharge, andthe weightoftheshell, we can

predictthe pointofimpact withacertain probability LetAbetheso-defined

initial conditionsand Badescription of the pointofimpact; then wehave a

probabilityimplication A

-=>- B (V\

whichstatesthatifAisgiven,Bwillhappenwithadeterminatejprnbabilityp.From thisempirically verifiablerelationwe pass to an idealrelationbycon-sidering ideal statesAf

and Br

andstatingalogicalimplication

A' D Bf

(2)

between them, whichrepresentsalawofmechanics Sinceweknow, however,

thatfromthe observational state A we can inferonly with someprobability

theexistence oftheidealstate A',andthatsimilarlywehaveonlyaprobability

relation between B and Br

the logical implication (2) cannot be utilized. Itderivesitsphysicalmeaningonlyfromthefactthatinallcases of applications

toobservablephenomenaitcan bereplacedbythe probability implication (1).

Whatthenisthe meaningofa statement saying thatif we

powdercharge,andtheweightoftheshell,wecanconsiderfurtherparameters,

suchastheresistance oftheair,the rotationoftheearth,etc.Asa consequence,the predicted value will change;butwe know that withsuch amore precise

characterization also the probability ofthe prediction increases. From riences of this kind we haveinferred that the probability p caq be madeto

expe-tihft Ygjue 1 as closely as we want bv the introductionjoL^ctEer

' nfrk

+-hfi analysis ofphysicalstates It isin thisformthatwe must

statetheprinciple ofcausalityif it istohavephysicalmeaning.Thestatementthat natureisgoverned by strict causal lawsmeansthatwe can predictthefuturewithadeterminate probabilityandthatwecan pushthisprobabilityas

1. CAUSALITY AND PROBABILITY 3

closeto certainty aswe want byusinga sufficientlyelaborateanalysis ofthe

phenomenaunderconsideration

Withthisformulation theprinciple ofcausalityisstrippedofitsdisguiseas

aprincipleapriori,inwhichithasbeenpresented withinmanyaphilosophicalsystem If causalityisstated as alimitofprobabilityimplications, it isclearthatthis principlecan be maintainedonlyinthe senseofanempiricalhypoth-

esig. There is, logically,no needforsaying that the probability of predictions

can be made to approach certainty by the introduction of more and more

parameters Inthisformthepossibility ofalimitof predictabilitywas

recog-nized evenbefore quantummechanics ledto the assertion ofsucha limit.2

The objection has been raised that we can know only a finite number ofparameters, andthatthereforewe mustleaveopenthepossibility of discover-

ing,atalater time,newparameters whichleadto better predictions.Although,

of course, wehaveno meansofexcludingwithcertaintysuchapossibility, we

must answer that there may be strong inductive evidence against such an

assumption, and that such evidence will be regarded as given if continued

attempts at findingnew para/mp^mfravefailed. Physicallaws, likethelawofconservation of energy, have been based on evidence derived from repeated

failures ofattempts to prove the contrary. If the existence of causallaws isdenied, thisassertion willalways begroundedonlyininductiveevidence.The

criticsofthebeliefin causalitywillnotcommitthemistakeof their adversaries,

and willnot try toadducea supposedevidence apriori for their contentions.The quantummechanicalcriticismof causalitymustthereforebeconsidered

asthelogicalcontinuationofalineofdevelopment whichbegan withtheductionofstatistical lawsintophysics within thekinetictheory of gases,and was continued in the empiricist analysis of the concept of causality The

intro-specific form, however, in which this criticism finally was presented through

Heisenberg's principle of indeterminacy was different from the form of thecriticismso far explained

In the precedinganalysiswehaveassumedthatit ispossible tomeasurethe

independent parametersofphysical occurrencesasexactlyaswewish; ormore

precisely, to measure the simultaneousvalues of these parameters as exactly

as we wish The breakdown of causality then consists in the fact that thesevaluesdonotstrictlydetermine the valuesofdependententities,including thevaluesofthesame parameters atlatertimes Ouranalysis thereforecontains

an assumption of the measurement of simultaneous values of independent

parameters It is this assumption which Heisenberg has shown to be wrong.The laws of classical physics are throughout temporally directed laws, i.e.,

lawsstatingdependencesofentitiesat differenttimesandwhichthusestablishcausallinesextendinginthedirection of time Ifsimultaneous valuesofdiffer-

2 Cf. the author's "Die Kausalstruktur der Welt/' Ber d Bayer Akad., Math. Kl.

(Munich, 1925), p 138:andhis paper, "Die Kausalbehauptung unddie

Mpglichkeitihrerempirischen Nachprinung," which was written in 1923 and published in Erkenntnis 3

(1932), p 32.

4 PART GENERAL CONSIDERATIONS

ent entities are regarded as dependent on one another, this dependence is

always construed as derivable from temporally^directed laws Thus the

cor-respondenceofvariousindicators ofa physicalstateisreducedtotheinfluence

ofthesamephysical cause actingonthe instruments If,for instance,

barom-eters in differentroomsofa house alwaysshowthesameindication,weexplain

thiscorrespondenceasduetotheeffectofthesame massofaironthe ments, i.e.,asduetotheeffectofacommoncause Itispossible, however, to

instru-assume the existence of cross-section laws, i.e., laws which directly connectsimultaneous valuesofphysicalentitieswithoutbeing reducibletothe-effects

ofcommon causes Itissucha cross-sectionlaw which Heisenberghas stated

inhisrelation ofindeterminacy

Thiscross-sectionlawhas theformofalimitation of measurability It statesthat the simultaneous valuesoftheindependent parameters cannot be meas-uredasexactlyaswewish.Wecanmeasureonlyone half ofalltheparameters

to a desired degree of exactness; the other half then must remain inexactlyknown Thereexistsa couplingofsimultaneouslymeasurablevaluessuchthatgreaterexactnessinthe determinationofone half ofthe totalityinvolvesless

exactnessinthe determinationofthe otherhalf, andvice versa Thislawdoes

notmakehalf oftheparametersfunctionsofthe others;ifone halfisknown,

the other half remains entirely unknown unless it is measured We know,

however, thatthismeasurementisrestrictedtoa certain exactness

Thiscross-sectionlawleads toaspecificversionofthecriticismof causality

Ifthe values oftheindependent parameters are inexactlyknown, we cannot

expect tobe able to make strict predictions offuture observations Wethencan establish'onlystatisticallaws for these observations Theidea that thereare causallaws "behind" these statistical laws, which determine exactly the

resultsoffutureobservations, isthendestinedtoremainanunverifiable

state-ment; its verification is excluded by a physical law, the cross-section law

mentioned Accordingto the verifiability theory ofmeaning, which has been

generallyacceptedfortheinterpretation of physics, the statement that therearecausal lawsthereforemustbeconsideredasphysicallymeaningless.Itisan empty assertion which cannot be converted into relations between observa-tional data

Thereisonlyonewayleftinwhichaphysicallymeaningful statement about

causality can be made If statements of causal relations between the exactvalues of certainentitiescannot be verified, we can try to introducethematleast intheformof conventionsordefinitions; thatis, we may tryto establisharbitrarilycausalrelationsbetweenthestrictvalues Thismeansthat wecanattemptto assign definitevalues totheunmeasured, ornotexactlymeasured,

entities insuch a waythat the observed results appear as the causal

conse-quencesofthe valuesintroducedbyourassumption.Ifthiswerepossible,thecausal relations introduced could not be used for an improvementof predic-tions;they couldbe used only afterobservationshadbeenmade inthe sense

2. PROBABILITY DISTRIBUTIONS 5

ofa causalconstructionpost hoc. Evenif wewishtofollow such a procedure,however, we mustanswerthe questionofwhether suchacausalsupplementa-tion of observabledata by interpolation ofunobservedvaluescan be consistently

done Although theinterpolation isbased onconventions, the answerto the

latterquestionisnot amatterofconvention, but dependsonthestructure ofthe physical world Heisenberg's principle of indeterminacy, therefore, leads

toarevision ofthestatementof causality;if thisstatementistobephysically

meaningful, itmust be made as anassertion about apossible causal

supple-mentationoftheobservational world

Withthese considerations the plan of the following inquiry is made clear.

We shall first explain Hcisenberg's principle, showing its nature as a section law, and discuss the reasons why it mustbe regarded as being well

cross-foundedonempirical evidence.Wethenshallturntothe questionofthepolationofunobservedvaluesbydefinitions.We shallshowthatthe questionstated above is to be answered negatively; that the relations of quantum

inter-mechanicsare soconstructed thatthey do not admitofacausal

supplementa-tionbyinterpolation Withtheseresultstheprinciple ofcausalityisshownto

be in no sense compatible with quantum physics; causal determinism holdsneither intheformofa verifiablestatement, norin theformof aconvention

directingapossibleinterpolation ofunobservedvaluesbetweenverifiabledata

2 The Probability Distributions

Let us analyze more closely the structure of causal laws by means of an

example taken fromclassical mechanicsand thenturnto the modificationofthisstructureproducedbythe introductionof probability considerations

In classicalphysics the physical state ofa freemass particlewhich has no

rotation, or whose rotation can be neglected, is determined if we know thepositionq,thevelocityv, andthemassmoftheparticle. Thevaluesq andv, ofcourse,mustbecorrespondingvalues, i.e.,theymustbeobservedatthesame

time Instead of the velocity v, the momentum p = m v can be used The

futurestates of themass particle, if it isnot submittedto anyforces, isthen

determined; the velocity, and with it, themomentum, willremain constant,

and the positionqcan be calculated for everytimet If external forces

inter-vene, wecanalsodeterminethe futurestates oftheparticleiftheseforces are

mathematicallyknown.

Ifweconsiderthefactthatp andqcannot beexactly determined, we must

replace strict statements about p and qby probability statements We then

introduceprobability distributions

whichcoordinatetoeveryvalue qandtoevery valuepaprobabilitythatthisvaluewilloccur The symbold() isusedhereinthe generalmeaningofdistri-

PARTI. GENERAL CONSIDERATIONS

buttonof;the expressionsd(q)andd(p) denote,therefore, different

mathemati-cal functions Asusual, theprobability given by the functioniscoordinated,nottoasharp valueqorp, buttoasmallintervaldqordpsuchthat only theexpressions

(2)

represent probabilities, whereas the functions (1) are probability densities.

Thiscanalsobestatedintheformthat theintegrals

measurement we therefore mean, more precisely, the exactness of a type of

measurement madeina certaintypeofphysical system Inthissensewe can

say that everymeasurementends with the determinationofprobability tions d. Usually dis a Gauss function, i.e., a bell-shaped curve following an

func-exponential law (cf figure 1) ;the steeper this curve, the more precise is the

measurement Inclassicalphysicswe maketheassumption thateachofthesecurves can bemadeassteep aswewant, ifonlywetake sufficientcare intheelaboration of the measurement In quantum mechanics this assumption isdiscardedforthe followingreasons

Whereas, inclassical physics, we consider thetwo curves d(q) and d(p) as

independentof eachother, quantummechanicsintroduces the rulethatthey

are not This isthe cross-sectionlawmentionedin 1 Theideais expressed

through a mathematicalprinciplewhichdeterminesbothcurvesd(q)andd(p),

ata giventimet, asderivablefromamathematicalfunction\f/(q);thetion is so given that a certain logical connection between the shapes of thecurves d(q) and d(p) follows. This contraction of the two probability distri-

deriva-butionsintoonefunction^isoneofthebasic principles ofquantummechanics

Itturns out that the connectionbetween thedistributions establishedby theprinciplehassuchastructurethatifoneofthe curvesisverysteep, the other

mustbe ratherflat. Physically speaking, this means thatmeasurements ofp andqcannotbemadeindependently andthat an arrangement whichpermits

a precise determination of q must make any determination of p unprecise,

and vice versa

Thefunction \[/(q) has the character of a wave; it is even a complex wave,

i.e.,awavedeterminedbycomplexnumbers^.Historicallyspeaking, theductionofthiswave by L deBroglie andSchrodinger goesbacktothestrug-gle betweenthe wave interpretation and the corpuscle interpretation in thetheory of light. The ^-function is the last offspring of generations of wave

intro-conceptsstemmingfrom Huygens'swavetheoryoflight;butHuygens would

2. PROBABILITY 7

scarcely recognize his ideas in the form which they have assumed today inBorn's probability interpretation of the ^-function. Letus put aside for thepresent the discussion of the physical nature of this wave; weshall be con-cerned with this important question in later sections of our inquiry. In thepresent sectionweshallconsiderthe^-wavesmerelyasa mathematicalinstru-

ment used to determine probability distributions; i.e., we shall restrict ourpresentation toshowthe wayinwhichthe probability distributionsd(q) and

d(p)canbe derivedfrom\fs(q).

Thfcderivationwhichwearegoingto explain coordinates toa curve$(q) at

a given time the curvesd(q) andd(p); this is thereason thattdoes notenterintothefollowing equations. If,atalatertime,

\l/(q)shouldhaveadifferentshape,different

func-tions d(q) and d(p) wouldensue Thus, in

gen-eral,wehavefunctions ^(g,)> d(q,f),andd(p,t).

We omit the t for the sake of convenience

Thederivationwillbe formulatedintworules,

the first determining d(q), and the second

de-terminingd(p). Weshall statethese rules here

onlyforthe simple case of free particles. The

extension tomore complicated mechanical sys- d(q)

terns willbe givenlater (17) Wepresentfirst t

theruleforthe determinationof d(q).

Ruleofthesquared\f/-function:Theprobability

of observingavalue qisdetermined bythesquare C 1 ^

-.,

Fig 1.Curve1-1-1 represents

ofthe"({/-functionaccordingto the relation a precise measurement, curve

2-2-2 a less precise

Theexplanationoftherule forthe

determina-tion of d(p) requiressome introductory mathematical remarks According toFouriera waveof anyshape can be considered as the superpositionofmany

individual waves having the form of sine curves This is well known fromsound waves, where the individual waves are called fundamental tone and

overtoneSj or harmonics In optics the individual waves are called matic waves, and their totality is called the spectrum The individual wave

monochro-ischaracterized by its frequency v, or its wave length X, these two

charac-teristics being connected by the relation v X = w, where w is the velocity

of the waves In addition, every individual wavehas an amplitude a- which

does not depend onq, but isa constantfor the whole individual wave The

general mathematical form of the Fourier expansion is explained in 9; for

the purposes of the present partit is not necessary to introduce the

mathe-maticalwayof writing

TheFouriersuperpositioncan be appliedtothewave^,althoughthiswave

isconsideredbyus,at present, notasaphysicalentity,butmerelyasa

mathe-8 PART I. GENERAL CONSIDERATIONS

matical instrument In case the wave \l/ consists of periodic oscillations

ex-tended over a certain time, such as inthe case of sound waves produced by

musical instruments, the spectrum furnished by the Fourier expansion is discrete. Thus the individual waves of musical instruments have the wave

lengthsA,-,-,-,- where A is the wave length of the fundamental tone

andthe other valuesrepresent the harmonics In case the wave ^ consists ofonly onesimpleimpact movingalong theg-axis, i.e., incase the function\f/ is

notperiodic,the Fourier expansionfurnishesa continuous spectrum, i.e., thefrequencies oftheindividualwavesconstitute,not adiscrete,buta continuousset.Asbefore, eachofthese individualwavespossessesan amplitudecr, whichcan be written<r(X), sinceitdepends onthe wavelength Xbutisindependent

ofq.

It is the amplitudes o-(X) which are connected with the momentum We

shallnottry to explainhere the trendofthought whichled to thisconnection

andwhichisassociatedwith thenamesofPlanck,Einstein,andL.deBroglie.

Suchanexpositionmaybe postponedto alatersection (13). Letus suppresstherefore any question of why this connection holds true, and let us rely,instead, upon the authority of the physicist who says that this is the case Suffice itto say, therefore, that everywaveofthe length Xiscoordinatedtoa

momentumoftheamount ,

P =

-(5)

A

where hisPlanck's constant The probability of findingamomentum pthen

isconnected with theamplitudecrbelongingtothe coordinatedwaveX.Thisis

expressedinthefollowingrule 1

Ruleofspectraldecomposition: Theprobability of observingavalue pis

deter-minedbythesquare oftheamplitude o-(X) occurringwithinthe spectral

decompo-sitionof\l/(q),intheform ^

(6)

Thefactor resultsfromtherelationbetweenp andX expressedin(5).2

A3

Thetwo rulesshowclearlytheconnection whichthe ^-functionestablishes

between the two distributions d(q) and d(p), so far as it reduces these two

distributions tooneroot.Weshall latershowthatthiskindofconnectionisnot

1The name"principleofspectraldecomposition" has been introducedbyL.deBroglie,

IntroductionaI'Etudede laMecanique ondulatoire (Paris,1930), p 151. Inhis laterbook,

LaMecaniqueondulatoire (Paris, 1939), p 47, heuses also thename "principle ofBorn,"

since thisprinciple wasintroducedby Born Forthe rule of the squared^-function he

uses thename "principle of interference" andin his laterbook thename "principle of

localization".

2

Mathematicallyspeaking, this factorcorrespondsto a density function r asintroduced

in (22), 9. The third powerin h originates from the fact thatwe assume thewaves

tobethree-dimensional.

3- PRINCIPLE OF INDETERMINACY 9

restrictedto the simple case ofone massparticle, and that the same logical

pattern isestablished by quantum mechanicsfor theanalysis of all physicalsituations.Forevery physicalsituationthereexistsa^-function,andthe prob-ability distributions oftheentitiesinvolvedaredeterminedby tworules ofthe

kinddescribed Thisisoneofthebasic principles ofquantummechanics We

shallnowconstruct the implications ofthis principle, returning once moretothe simplecase ofthemassparticle.

3 The Principle of Indeterminacy

It canbe shown that the derivation of the two distributions d(q) and d(p)

froma function\l/leadsimmediatelytothe principle ofindeterminacy Let usconsideraparticlemovinginastraightline, andletusassumethat the func-

Fig 2

respectively^v or p

Fig 3 Fig 2 Distribution of the position q, in theformof aGausscurve.

Fig 3.The dottedline indicates the directFourier expansion of the

curveoffig 2.Thesolid line isconstructed throughtheFourier

expan-sion of a ^-functionfromwhichthecurved(q) offig.2 is derivable, and

represents the distribution d(p) ofthemomentum, coordinatedtod(q).

tion\f/ ispracticallyequal to zeroexcept for acertain intervalalong the line.

Thefunction \$(q)

|

2

, i.e.,the functiond(q),thenwillhavethesameproperty;

letusassumethatit isa Gausscurve suchasis shownin figure2. Theshape

ofthe curvemeans thatwe donot knowthe location ofthe particle exactly;withpracticalcertaintyit iswithin theintervalwherethe curveisnoticeablydifferent from zero, but for a given place within this interval we know only

with a determinate probability that the particle is there Our diagram, ofcourse, representsthesituationonlyfora given time t;foralatertime, when

theparticlehasmovedtotheright,weshallhaveasimilar curve, butit willbe

shifted totheright.1

1Thecurvewill also graduallychange its[form This, however, is irrelevant for the present discussion.

1O PART GENERAL CONSIDERATIONS

Nowletusapply theprinciple of spectral decomposition. This

decomposi-tion, it is true, istobeappliedtothecomplexfunction^(g),andnottothereal

function \\l/(q)

|

2

ofour diagram Forthestudy of themathematicalrelations

inthe decomposition, however, we shall first apply itto the real function ofthediagram;theresultscanthenbetransferred tothecomplexcase.

The Fourier expansion constructs the curveof figure2 outofan infinite set

ofindividualwaves Eachoftheseindividual wavesis a pure harmonic wave

ofinfinite length; i.e., its oscillations have a sine form andextend alofigthe

whole infinite line. Their amplitudes differ, however The maximalamplitude

will be associated with a certain mean frequency j>

; for frequencies greater

andsmallerthanVQtheamplitudewillbesmaller, andoutsideacertainrange

on each side of VQ the amplitude of the individual waves will be practically

zero. Let uscallthe range withinwhichthe amplitudeshaveanoticeablesize,the practicalrange. A bunchof harmonic waves of the kind described isalsocalledawavepacket, sincethe superpositionofalltheseharmonicwavesresults

inonepacketoftheformgiveninfigure2.

Nowit isoneofthetheoremsofFourieranalysisthat thepractical rangeof

apacketofharmonic wavesisgreatwhenthe curveof figure2israthersteep,

butissmallwhenthis curve is flat. We canillustrate thisby a diagramwhen

we draw as abscissae the frequencies v, and as ordinates the correspondingamplitudes s(v) ofthe harmonic analysis, such as is done for the dottedline

in figure 3. We choose the notation s(v) for these amplitudes of the Fourierexpansion ofthe real functiond(q), or \^(q)|

2

, in order to distinguish them

from the amplitudes a(v) of the Fourier expansion of the complex function

\fs(q). In our case, -sincewe had assumedthe curved(q) tobe a Gauss curve,

thecurve s(v) isalso a Gausscurve,butofamuch flattershape,2asshown by

thedotted-linecurveof figure3.Thisillustratesthetheorem mentioned; itcan

be provedgenerallythat thesteeperthecurveof figure2, theflatter willbethedotted-line curve of figure 3, and vice versa We therefore have an inversecorrelationbetweentheshapeoftheoriginalcurve andthe shapeofthecurveexpressing its harmonicanalysis. We call this the law of inversecorrelationof

harmonicanalysis.

An instructive illustration of this law is found in some problems of radiotransmission Ifthewaveofa radio transmitter does not carryanysound, it is

apuresine waveof a sharply defined frequency If it is modulated, however,

i.e., if its amplitude varies according to the intensity of impressed sound

waves, it no longer represents one sharp frequency, but a spectrum of

fre-quencies varying continuously withina certain range This range isgivenby

the highest pitchof the sound frequencies. Consequently, a receiver with a

2Themaximumof thes(v)is at the place v=0,andthecurveissymmetricalforpositiveandnegativefrequencies v.Thisresultsfromthe fact thatwe have assumed thecurved(q) in theform ofaGausscurve. Forthe curved(p) thereexistsno such restriction,becausethe function\fs(q)isnot determinedbyd(q),butleftwidelyarbitrary; themaxi-

mumof d(p) maythereforebesituated at anyvaluev = j>, orcorrespondingly, atany

valuep.

3- PRINCIPLE OF INDETERMINACY H

sharp resonancesystemwillpick outonly anarrowdomainofthe transmitted

waves;it willthereforedropthe highersoundfrequenciesandreproducemittedmusicinadistortedform.Onthe other hand, ifthe resonance curveofthe receiver is sufficiently flat for a high fidelity reproduction of music, thereceiverwillnotsufficientlyseparatetworadiostationstransmitting adjacent

trans-wavelengths Theprinciple of inverse correlationishere expressedinthefactthat it is impossible to unite high fidelity and high selectivity in the same

adjustmentofthereceiver

Thpapplication of these considerations to the determinationof the

proba-bility distribution ofthe momentumof the particle involves some

complica-tions, which, however, in principle, do not change the result. As explained

above, the spectral decomposition which furnishes the momenta must be

applied,nottotheprobabilitycurved(q) of figure2,buttoacomplex^-curve

from whichthiscurveisderivable bythe rule ofthesquared^-function. Thecomplexamplitudesa(v)oftheresultingharmonicwavesthenmustbe squared

according to the rule of spectral decomposition It is only by means of this

detourthroughthecomplexdomainthatwearrive atthe probabilitytion d(p) of the momentum as shown in the solid line of figure 3. Like thedotted line, this curve is also a rather flat Gauss distribution, although not

distribu-quite soflat. Butitcan beshown that thelaw ofinverse correlationholds aswell forthetwo curves d(q) andd(p). We therefore shall speak ofthe law ofinversecorrelationoftheprobability distributions ofmomentumandposition.Moreprecisely,this inverse correlationistobeunderstoodas follows.Ifonlythe curved(q) isgiven, thecurved(p) isnot determined;it canhavevarious

forms dependingontheshape ofthe function\l/(q) from whichd(q) isderived

But thereis a limitto the steepness of d(p), representedby the solid line offigure 3. Shouldd(q) be derived fromanother ^-function than that assumed

for the diagram, the resulting curve d(p) can only be flatter. This generaltheorem can bestatedwithout afurther discussion ofthe question concerningthepracticaldeterminationofthe function\l/(q) in a given physicalsituation

Theanswertothelatterquestion,whichrequiresthemathematical apparatus

ofquantummechanics,mustbe postponedtoalatersection( 20).

Wesaidthat theresults obtainedforthemass particlecan beextended toall physical situations Now the difference between q and p is transferred tosituations ingeneralasadifferencebetweenkinematicanddynamicparameters.

Wethereforeshallspeakgenerally ofthe lawof inverse correlationofkinematic

and dynamicparameters It isintheformofthislawofinverse correlationthat

we express the principle of indeterminacy. The universality of this principle

follows from the fact that, whatever be the physical situation, our tionalknowledgeofit issummarizedina^-function

observa-Thelaw of inverse correlationcan be extended to thetwo parameters lime

and energy This extension can be made clear as follows A measurement of

timeisanalogoustoa determinationof position.When wespeakoftheposition

12 PART GENERAL CONSIDERATIONS

qofaparticlewe meantheposition ata giventime t;inverselywecanaskforthetime tatwhich theparticlewillbeat a given space point q. This value t

will be determinable only with a certain probability, and we can thereforeintroduce aprobability distributiond(f) analogous to d(q). Similarly we can

introduce aprobabilityfunctiond(H) statingtheprobabilitythat theparticle

will have a certain energyH. His connectedwith the frequency of the

har-monicsbythePlanckrelation IT _ /, n\

corresponding to (5), 2, and the probability d(H) is therefore determined

bytheprinciple of spectraldecomposition. Hencethetwocurvesd(f) andd(H)

are subjecttotheprinciple of inverse correlation inthesame wayasthe curvesd(q) and d(p). We shall therefore include time inthe category of kinematicparameters,andenergyinthatofthedynamicparameters.Thegenerallawofinverse correlation of kinematic and dynamic parameters then includes theinverse correlation oftimeandenergy.

This general law can be formulated somewhatdifferentlywhen we use theconceptofstandarddeviation. Let g represent themeanvalueofg,thatis,thevalueoftheabscissa forwhichthecurved(q) reachesitsmaximum; andletAg

be the standarddeviation Then the area between the curve and the axis ofabscissaeisdividedbythe ordinatesg Ag andg + Ag(drawn in figure 2)

2

insuchawaythatapproximately - ofthewholeareaissituatedbetweenthese

o

twoordinates Itisshownin the theoryof probabilitythat this ratiois

inde-pendentofthe shapeofthe Gausscurve Theprobabilityof findingavalueq

2within theinterval'g Agistherefore approximately = -. Because of these

oproperties,the quantityAgrepresentsameasureofthe steepnessoftheGauss

curve, and istherefore used as an expression characterizing the exactness ofthedistribution ofmeasurements Ifthestandard deviationissmall, themeas-urementsare exact;if it isgreat, themeasurementsare inexact Infigure2and

figure 3 the standard deviations Ag and Ap belonging, respectively, to thecurvesd(q) and d(p) areindicatedon theaxis. Now itcan be shown thatforthe case of curves of this kind, which arederived from the same^-function,

Theinequalities (2) and (3) represent theformin whichthe relation of

inde-terminacyhas beenestablishedby Heisenberg (2) expresses the inverse

cor-g. PRINCIPLE OF INDETERMINACY ig

relation ofmeasurements ofposition and momentum bystating thata smallstandard deviationin q impliesagreatstandard deviationinp,andvice versa.(3) statesthe correspondingrelationbetweenAtand A#.Theserelationsshow

atthe sametime the significance of the constanth. Sinceh hasavery small

value,the indeterminacywillbevisibleonlyinobservations withinthecosmic domain;there, however, the indeterminacycannot be neglected.The

micro-case ofclassicalphysicscorrespondstotheassumptionthath = 0.

Relation(2) canbeinterpreted inthe form:Whentheposition ofaparticle

iswelldetermined, themomentumisnot sharply determined, andvice versa

(3) canbeinterpreted inasimilarway.Thisformmakesit clearthat the

cross-sectionlawof inverse correlationbetweenkinematicand dynamicparametersstatesalimitation ofmeasurability.

We now are in a position to answer the question raised above about thelegitimacy of this cross-section law If the basic principles of quantum me-

chanics are correct, the principle of indeterminacy musthold because it is a

logicalconsequenceof thesebasicprinciples. Furthermore, itmustholdfor allphysicalsituations,becauseit isderivable directlyfromtherulesofthe squared

^-function and of spectral decomposition, without reference to any special

formofthe^-function.Theissueoflegitimacyisreducedwiththistothe

valid-ityof the basic principles ofquantummechanics Nowthese principlesare,ofcourse, empirical principles, and nophysicist claims absolute truthfor them

But whatcan be claimed for them is the truth of a well-established theory.Since it is a consequence ofthe limitation of measurability that all relations

betweenobservationaldataare restricted tostatistical relations,wecanfore say: With thesameright withwhich the physicistmaintainsanyone ofhis

there-fundamental theorems, he is entitled to assert a limitation of predictability. We

may addthat thesamelimitationfollowsforthedeterminationofpast datainterms of given observations, and that we therefore must also speak of alimitationofpostdictability.

It has sometimes been said that quantum mechanics possesses a

mathe-matical proofofthelimitation ofpredictability. Suchastatementcanablybemeantonly in thesensethat thereisa mathematical proofderiving

reason-the statement of the limitation from the basic principles of quantum

me-chanics.Theprinciple ofindeterminacyisanempiricalstatement;allthatcan

besaidmathematicallyinitsfavoristhatit issupportedbythe very evidence

on which the basic principles of quantum mechanics are founded This is,however,very strongevidence

We occasionally meet with the objection that the laws of quantum

me-chanics, perhaps, hold only for a certain kind of parameter; that at a later

stage of scienceother parametersmaybe foundforwhichtherelation oftaintydoes not hold; andthat the newparameters may enable us to make

uncer-strictpredictions Logically speaking, sucha possibility cannotbe denied It

then might be possible, for instance, to combine a measurement ofthe new

14 PARTI, GENERAL CONSIDERATIONS

parameters with ameasurement of the kinematic parameters insuch away

that the results of measurements of the dynamic parameters could be

pre-dicted.Thelawof inverse correlationbetweenkinematicand dynamic

param-etersthenstillwouldholdfortheoldparameterssolongasthenewoneswerenotused;butwhenthenewobservablesweretobeappliedfortheselection oftypesofphysicalsystems, thelawofinverse correlationwouldno longer hold

withinthe assemblagesso constructed,evenfortheoldparameters,andfore theirvalues couldbestrictlypredicted Thiswouldmean,inother words,thatwe could empirically define typesofphysical systemsfor which thesta-

there-tistical relations controlling theirparameters were notexpressible intermsof

^-functions.3

In sucha casequantummechanicswould be consideredasastatisticalpart

of scienceimbeddedinauniversal science ofcausalcharacter.Although,aswe

said, we cannot adducelogical reasons excludingsuch afurther development

of physics,and, althoughsomeeminentphysicists believe insuchapossibility,

wecannotfindmuchempirical evidenceforsuchan assumption. If a physical

principleembracingallknownentitieshasbeenestablished, itseemsplausible

to assume that it holds universally, and that there is no unknown class ofphysical entities which do not conform to this principle. Such an inductiveinference from all known entities to all entities has always been consideredlegitimate The principle of describing all physical situations in terms of

^-functions is a well-established principle, and though, certainly, quantummechanicsis still confrontedby manyunsolved problemsand mayexperience

important improvements, nothing indicates that the principle ofthe tionwillbeabandoned Since therelation ofuncertaintyandthelimitation ofpredictability follow directly fromtheprinciple ofthe^-function, these theo-

^-func-remsmustberegardedasbeingas wellfoundedin theiruniversalclaimsasallother generaltheoremsof physics

4 The Disturbance of the Object by the Observation

We now turn to considerations involving the second main issue confrontingthe philosophy ofquantummechanics theissue oftheinterpretation ofun-observedobjects. This questionfindsafirstanswer in thestatementthat theobjectis disturbedby the means ofobservation Heisenberg, who recognized

this feature in combination with his discovery ofthe principle of

indetermi-8Weuse theterm"expressibleintermsof ^-functions" inorderto includeboththe pure

caseandthemixture;cf 23.Inhisbook, Mathematische GrundlagenderQuantenmechanik(Berlin, 1932), p 160-173, J v.Neumannhas given a proof that no "hidden parameters''

canexist.Butthis proof isbasedontheassumption thatfor all kinds of statistical

assem-blages thelawsofquantummechanics, expressedintermsof ^-functions, are valid If

theindeterminism ofquantum mechanicsis criticized, this assumptionwill be equally

questioned.J. v. Neumann'sprooftherefore cannot exclude the case towhichwerefer

inthetext Itshows only that the assumptionofhidden parametersisnot compatiblewith auniversal validity ofquantummechanics

4- DISTURBANCE OF THE OBJECT 15

nacy,useditasanexplanation ofthe latterprinciple;he maintainedthat theindeterminacyofallmeasurementsisa consequenceofthe disturbancebythe

meansof observation

This statement has aroused a wave of philosophical speculation. Some

philosophers, and some physicists as well, have interpreted Heisenberg's

statementastheconfirmation, intermsof physics, of traditional philosophical*

ideasconcerning theinfluence ofthe perceiving subject onitspercepts They

haveiteratedthisideabyseeing inHeisenberg'sprinciplea statement that thesubjectcannotbe strictlyseparatedfromtheexternalworldandthat theline

of demarcationbetweensubject andobjectcan onlybearbitrarily setup; orthat thesubject createsthe objectin theact ofperception; or that theobjectseenis only a thingofappearance, whereas the thinginitselfforever escapes

humanknowledge;orthatthe thingsofnaturemustbetransformed according

to certain conditions beforethey canenterintohumanconsciousness,etc. We

cannot admit thatanyversion ofsuch a philosophicalmysticismhas a basis

in quantum mechanics Like all other parts of physics, quantum mechanics

deals with nothing but relations between physical things; all its statements

can be madewithoutreference toanobserver The disturbancebythe means

ofobservation whichiscertainlyoneofthebasic facts asserted in quantum

mechanics isanentirelyphysicalaffairwhichdoes notincludeanyreference

toeffectsemanating fromhumanbeingsas observers

Thisismadeclearbythefollowing consideration.Wecanreplacethe ing personbyphysical devices,suchas photoelectriccells, etc., whichregister

observ-the observationsandpresentthemasdatawrittenonastripof paper Theact

of observationthenconsists in reading thenumbers andsignswrittenon thepaper.Since theinteractionbetweenthe reading eyeandthepaperisamacro-

cosmicoccurrence,the disturbancebythe observationcanbeneglectedfor this

process.Itfollowsthatallthatcan besaidaboutthedisturbancebythemeans

of observationmust beinferable fromthe linguisticexpressionsonthe paper

strip, and must therefore be statable in terms of physical devices and their

interrelations. Quantum mechanics should not be misused for attempts torevivephilosophical speculationswhicharenotonalevelwiththe clarityand

precision ofthe languageof physics.Thesolution ofitsphilosophicalproblemscanonlybegiven within ascientificphilosophy suchashasbeendevelopedintheanalysis of scienceandinsymboliclogic.

Therewasasimilarperiodinthediscussion of Einstein'stheoryof relativity

inwhichtherelativity oftimeandmotionwasascribedtothesubjectivity oftheobserver.Lateranalysishasshownthat thedependenceofstatementsabout

spaceand time on the system of reference is in no way connected with theprivacyofeveryperson'ssensedata,butrepresentsthe expressionof arbitrarydefinitions involved in everydescription of the physical world We shall seethat a similarsolution can begivento the problems ofquantummechanics,although thesituationthereisevenmorecomplicatedthan inthe caseofthe

l6 PARTI. GENERAL CONSIDERATIONS

theory ofrelativity. Thedifference is that onthe arbitrariness of definitions

is superimposed, in quantummechanics, anuncertainty in theprediction ofobservableresults, afeaturewhichhasnoanalogueinthe theoryofrelativity.

We mustbegin our analysiswitharevision ofHeisenberg'sstatementthatthe uncertaintyof predictionsisaconsequenceofthe disturbancebythemeans

of observation We do not think that the statement is correct in this form,

althoughit istrue that thereisa disturbancebythe observationandthatthere

isa logical connectionbetween this principle and theprinciple of nacy Thisconnectionshould rather bestated inversely, namely, in the form

indetermi-that the principle ofindeterminacyimplies the statementofa disturbance ofobjectsbythemeansofobservation

Tosay that the indeterminacyof predictions originatesfromthe disturbance

by the instruments of observation means that whenever there is a

non-negligible disturbancebyobservation therewillalways bealimitation of

pre-dictability. A consideration of classical physics shows that this is not true.

Therearemanycases inclassicalphysicswheretheinfluence oftheinstrument

of measurementcannot be neglected, and where, nevertheless, exact tions arepossible.Suchcasesare dealtwithbytheestablishmentofaphysical

predic-theory whichincludes a theoryof the instrumentofmeasurement When weputathermometerintoaglass ofwaterwe knowthatthe temperatureofthe

water will be changed bythe introduction ofthe thermometer; therefore wecannotinterpretthe readingtakenfromthethermometerasgiving the watertemperature before the measurement, but must consider this reading as an

observation from which we can determine the original temperature of the

water only by means of inferences These inferences can be made when we

include inthema theoryofthethermometer

Whyis itnotpossible toapplythis logicalproceduretothecase ofquantummechanics? Heisenberg has shown that for a precise determination of theposition ofaparticleweneedlightwavesofaveryshortwavelength, thatis,

waves whichcarry ratherlargequantaofenergyandwhich changethevelocity

oftheparticlebytheirimpacts, with theconsequencethatthisvelocitycannot

bemeasuredbythesameexperiment If, onthe other hand,we wishto

deter-minethevelocity ofaparticle, we mustuse rather longwavelengthsinorder

notto change thevelocity to be measured; but thenwe shall notbe able toascertain preciselytheposition oftheparticle. If, however, the observationof

aparticlebyilluminatingitwithalightrayproducesan impact which throws

theparticleoff itspath, why canwenot construct a theory whichtellsusby means of inferences starting from the result of the observation what theoriginal velocity ofthe particle was? It is here that the cross-section law of

Heisenbergintervenes.Thisprinciple statesthatwhatever bethe observational

results, the corresponding distributions of position and momentum must be

derivablefroma^-function, andthereforemustbeinversely correlated. Thus,

ameasurementof positioninvolves physicalprocesses ofsucha kindthat,

rela-5. UNOBSERVED OBJECTS 17

tivetothe observationaleffectsof these processes, thevelocity distributionis

a ratherflat curve Thisis the reason that we cannot determine exactly thevelocity of the particle in the experiment mentioned The relation between

disturbance throughobservationandindeterminacymusttherefore bestated

as follows: Thedisturbance by the observation is the reason that the

deter-minationofthe physical entityconsidered isnot immediatelygiven with the

measurement, but requires inferences using physical laws; since these

infer-encesareboundtothe useofa^-function,theyare limitedbytheprinciple ofindeterminacy, andthereforeit isimpossible tocometo an exact determina-

tion.This formulationmakesclearthat the disturbancebytheobservation, initself,doesnot leadtothe indeterminacyoftheobservation.Itdoessoonlyin

combinationwith theprinciple ofindeterminacy.1

InviewofsuchobjectionsHeisenberg'sprinciplehassometimes been

formu-lated asmeaning: Wehavenoexactknowledgeofphysicalstatesbecause theobservation disturbs in an unpredictable way In this form the statement iscorrect;butthenitcan nolongerbeinterpreted assubstantiating theprinciple

ofindeterminacy It statesthisprinciple, butdoes notgiveareasonforit.And

we recognize that the "disturbancein anunpredictable way" is buta specialcase ofageneral cross-sectionlaw ofnature statingthe inverse correlation ofall physical data available The instrument of measurement disturbs, not

becauseit isaninstrumentusedby humanobservers,butbecauseit isa

phys-ical thinglike all other physical things Instruments of measurement do not

represent exceptions to physical laws; the general limitation of inferencesleadingto simultaneous values of parameters includes the case of inferencesreferring totheeffectsof instrumentsof measurement It is inthis formthat

we must state the principle ofthe disturbance bythe meansof observation.Thisis,however, only thefirststep inouranalysis ofthe disturbancebytheobservation Wehaveso fartakenitforgranted thatwe know what we mean

bysaying that the observationdisturbstheobject.Inordertocometoadeeper

understandingoftherelationsinvolvedhere,we mustfirstconstruct apreciseformulationofthisstatement

5 The Determination of Unobserved Objects

When wesay that the objectisdisturbedbytheobservation, or that the

un-observed object is different from the observed object, we must have some

knowledge of the unobserved object; otherwise our statement would be

un-justifiable. Before wecanenter into an analysis ofthe particular situation in

1Amathematical proofthat thedisturbance of the objectbytheobservation doesnotentail the principle ofindeterminacywillbe givenlater (p. 104).Theideathatit isnotthe disturbancein itselfwhich leaas to the indeterminacywas first expressed by theauthorin "ZieleundWegeder physikalischen Erkenntnis,"HandbuchderPhysik, Vol.IV

(ed. byGeiger-Scheel, Berlin, 1929), p 78.Thesameideahasalsobeenexpressedby E.Zilsel, Erkenntnis 5(1935), p.59. Thepreciseformulationof the principle of uncertainty requiresaqualificationwhichwillbeexplainedin 30.

18 PART I. GENERAL CONSIDERATIONS

quantummechanics we musttherefore discuss in general the problem ofourknowledgeofunobservedthings Howdo things lookwhen we donotlook at

them?Thisisthe questiontowhichwe mustfindananswer

It has sometimes been said that this problem is specific for quantum

me-chanics, whereasfor classicalphysics thereis nosuch problem. Thisis,

how-ever,a misunderstandingofthenatureofthe problem.Eveninclassicalphysics

we meetwiththe problemofthe natureofunobservedthings;andonly aftergivingacorrecttreatmentofthisproblem onclassicalgroundsshallwebeable

to answer the corresponding question for quantum mechanics The logical

methods bywhichtheansweristobe formulated arethesameinbothcases

Tobeginouranalysiswithanexample, letusassumewelookatatree, and

thenturn ourhead away Howdo we knowthat the treeremains initsplace

when we do not lookatit? Itwouldnot help ustoanswerthat we caneasily

turn our head forward and thus "verify" that the tree did not disappear.

What wethus verifyisonly that the treeisalwaysthere when welook atit;

butthisdoes not exclude thepossibilitythatitalwaysdisappearswhen we do

notlook atit, ifonlyitreappearswhen weturnourhead towardit. Wecould

makean assumptionofthelatterkind.Accordingto thisassumptionthevation produces a certain change of the object in such a way that thereappearstobeno change. Wehavenomeans toprovethat thisassumptionis

obser-false If it issuggested that another personmay observe thetreewhen we do

notseeitandthus confirm thestatementthat thetreedoes not disappear, we

mayrestrictour assumptionto cases inwhich noperson looksatthetree, thusascribingthe power ofreproducing thetreeto the observationof any human

being If it is suggested thatwe mayderive theexistence ofthetreefromtaineffectsremainingobservableevenwhen wedo notseethetree, suchasthe

cer-shadowofthetree,we mayanswerthatwecanassume a changeinthelawsofoptics such that there is a shadow although there is no tree. The argument

therefore proves only that an assumption concerning the existence, or

dis-appearance, oftheunobserved objectisto be connected withan assumption

concerning the lawsofnatureinbothcases.

Itwould beamistaketosay that thereisinductive evidenceforthe

assump-tion that the tree does not disappear when we do not see it, and that this

assumption isat least highly probable There is no such inductive evidence

Wecannotsay: "Wehavesooftenfoundtheunobservedtreetobeunchanged

thatwe assumethistohold always" Thepremise ofthis inductiveinference

is not true, since, in fact, we never have seen an unobserved tree. What wehaveoften seenisthatwhen weturned ourheadtothetreeitwasthere;from

thisset of factswecaninductivelyinferthat thetreewillalways betherewhen

we look at it, but there isno inductive inference leading from these facts tostatementsabout theunobservedtree. Wetherefore cannotevensay that theunchangedexistence oftheunobservedobjectisatleastprobable

We are inclined to discard considerations ofthe given kind as "nonsense",

5- UNOBSERVED OBJECTS 1Q

because it seemsso obvious that the tree is not created bythe observation

Such an answer, however, does not meet the problem The correct answer

requiresdeeperanalysis

We must say that there is more than one true description of unobserved

objects, that there is aclass of equivalent descriptions, and that all these scriptions can be used equally well. The numberof these descriptions is notlimited Thus we can easilyintroduce an assumptionaccording to whichthetreesplitsintotwotreesevery timewedonot look atit;this ispermissibleif

de-onlyWe changethe optics ofunobservedthings ina correspondingway, such

that thetwotrees produceonlyone shadow Onthe other hand, wesee that

not alldescriptions are true. Thusit is false to say that thereare two

unob-servedtrees, andthat the lawsofordinaryopticsholdforthem.Itfollowsthatthe statementsabout unobservedthingsare tobemadeinarather complicated

way Descriptions ofunobserved things must be dividedinto admissible and

inadmissible descriptions; each admissible descriptioncan be called true, and

eachinadmissibledescriptionmustbecalledfalse.Lookingforgeneralfeatures

ofunobserved things, we must not try to find the true description, but must

consider the whole class of admissible descriptions; it is in properties of this classasawholethat thenatureofunobservedthingsisexpressed

In the case of classical physics this class contains one description which

satisfiesthe followingtwoprinciples:

1) Thelaws ofnaturearethesamewhetherornotthe objects areobserved.

2) The state ofthe objects is the samewhether ornotthe objectsareobserved.

Letuscall thisdescriptionalsystemthenormalsystem. It isthissystem which

we usually consider as the "true" system We see that this interpretationisincorrect.We may,however,makethe followingstatement Incasea classofdescriptions containsa normal system, eachof the descriptionsis equivalent

tothenormalsystem. Ifwe nowconsideroneoftheunreasonabledescriptions

ofthe class,suchasthestatementthat atreesplitsintotwotreeswheneverit

isnot observed, weseethat theseanomaliesareharmless Theyresultfromtheuseofa differentlanguage, whereasthedescription as awhole says the same

asthe normalsystem. Thisisthe reason thatwecanselectthenormal system

asthe onlydescription tobeused

Theconvention that thenormal system be usedisalwaystacitlyassumedinthe language of dailylife when we speakofinductive evidencefor or againstchanges ofunobserved objects. This convention is understood when we say

thatourhouse remainsinitsplaceaslongasweareabsent;andthesameventionisunderstoodwhen wesay that the girl isnotinthe boxofthe magi-

con-cian whileheissawingtheboxintotwopieces, althoughwehaveseen thegirl

initbefore Itisbecauseofthe useof thisconventionthat ordinary statements

about unobservedthings are testable.The sameconventionisusedinscientific

language; it simplifiesthe language considerably. Wemust, however, realize

SO PART GENERAL CONSIDERATIONS

thatthischoice oflanguage has the characterofadefinitionandthat the

sim-plicityofthenormal systemdoesnotmakethissystem"more true"thantheothers Weare concerned here onlywith whathasbeen called adifference in

descriptive simplicity,1 such as we find in the case of the metrical system as

comparedwiththe yard-inch system

Instatingthataclassof descriptions includesanormalsystem, we makeastatementaboutthewholeclass.Thiswayof statingapropertyoftheclassby meansofastatementabouttheexistence ofanormal systemmaybeillustrated

byan example fromdifferentialgeometry Propertiesofcurvatureare statable

intermsofsystemsofcoordinatesandtheir properties.Thusthesurface ofthesphere can becharacterized by the statement thatit is notpossibleto intro-

duceonitasystemoforthogonalstraight-linecoordinateswhich coverslargeareas Onlyfor aninfinitesimalarea is thispossible;i.e., for small areas it ispossible to introduceapproximatelyorthogonalstraight-line coordinates, and

the degreeof approximationincreases for smallerareas For the plane,

how-ever, such a system covering the whole plane can be introduced It is notnecessary, though, to usethis "normal system" ofcoordinates for the plane,sinceanykindofcurvedcoordinatescan be usedequallywell;butthefactthat

there is such a normal system distinguishes the class of possible systems ofcoordinates holding for the plane from the corresponding classholdingfor a

curvedsurface

Similar considerations have been developed for Einstein's theory of tivity, whichistheclassicaldomain ofapplicationforthe theoryof classes ofequivalent descriptions Every system of reference, including systems indifferentstatesofmotion,furnishesacompletedescription,and wehavethere-fore intheclassofsystemsof referenceaclassofequivalentdescriptions.Ifthe

rela-classofsuch systemsincludesoneforwhichthelawsof special relativity hold,

wesay that the considered space does notpossessa "real" gravitationalfield.

Thisis true, althoughwecanintroducein sucha worldunreasonablesystemswhich contain pseudogravitational fields; they are pseudogravitational be-causethey can be "transformedaway".2

Turning from these general considerations to quantum mechanics, we first

must clarifywhat isto be meant by observable and byunobservable rences Using the word "observable" in the strict epistemological sense, we

occur-mustsaythatnoneofthequantummechanicaloccurrencesisobservable;they

areall inferredfrom macrocosmic data which constitutethe onlybasis

acces-sible to observation by human sense organs. There is, however, a class of

1

Cf.theauthor'sExperience andPrediction(Chicago,1938), 42.Descriptivesimplicity

is distinguishedfrominductive simplicity; the latter involves predictional differences.

2

Cf.theauthor's Philosophic derRaum-Zeit-Lehre (Berlin, 1928), p 271.

6. WAVES AND CORPUSCLES 1

occurrenceswhichare so easily inferablefrom macrocosmic datathattheymay

beconsidered as observablein awider sense We mean allthose occurrences

whichconsist in coincidences, suchascoincidencesbetweenelectrons, or

elec-tronsandprotons, etc.Weshall calloccurrencesof thiskind phenomena The

phenomena are connected with macrocosmic occurrences by rather shortcausal chains; we therefore say that they can be "directly" verifiedby such

devicesas the Geigercounter, a photographicfilm, a Wilson cloudchamber,

etc.

We'thenshall considerasunobservableallthose occurrenceswhichhappenbetween the coincidences, suchas the movementofan electron, or ofa lightrayfromitssourcetoacollisionwith matter.Wecall this classofoccurrencesthe interphenomena Occurrences of this kind are introduced by inferentialchainsofamuch morecomplicatedsort;theyareconstructedintheformofan

interpolation within the world of phenomena, and we can therefore considerthe distinctionbetweenphenomena andinterphenomenaasthequantum me-

chanical analogueofthedistinctionbetweenobservedandunobservedthings.Thedeterminationofphenomenaispracticallyunambiguous Speakingmore

precisely, thismeansthatintheinferencesleadingfrom macrocosmic datato

phenomena we use only the laws of classical physics; the phenomena arethereforedeterminateinthesame sense asthe unobservedobjects ofclassical

physics Putting aside as irrelevant for our purposes the problemofthe

un-observed thingsofclassicalphysics, wetherefore canconsider thephenomena

asverifiable occurrences It is different withtheinterphenomena The ductionoftheinterphenomena canonlybegiven within theframeofquantummechanical laws; it isin this connection that the principle of indeterminacyleads to someambiguitieswhich find theirexpressioninthe duality of waves

intro-andcorpuscles

Thehistory ofthetheories oflightandmattersincethetimeofNewton and Huygens shows a continuous struggle between the interpretation by cor-puscles and the interpretation by waves Toward the end of the nineteenthcenturythis strugglehad reached aphase inwhich itseemedpractically set- tled; light and other kinds of electromagnetic radiation were regarded asconsisting of waves, whereas matter was assumed to consist of corpuscles It

wasPlanck's theoryofquantawhich, inits furtherdevelopment, conferreda

serious shock to this conception. In his theory of needle radiation Einstein

showed that light rays behave in many respects like particles; later L de

BroglieandSchrodingerdevelopedideasaccordingtowhichmaterialparticlesinversely areaccompanied bywaves The wavenature of electrons then was

demonstrated by Davisson and Germerin anexperiment of a type which, adozenor soyearsbefore,hadbeenmade by M.v. Lauewithrespect toX-rays,

and which had been considered at that time as the definitive proof that

X-rays donotconsist ofparticles. Withtheseresultsthestrugglebetweentheconceptions of waves and corpuscles seemed to be revived, and once more

22 PART I. GENERAL CONSIDERATIONS

physicsseemedtobeconfrontedbythedilemmaoftwocontradictory

concep-tionseachofwhichseemedtobeequally demonstrable. Onesort ofexperiment

seemedto require the wave interpretation, another the corpuscle tion; and in spite of the apparent inconsistency of the two interpretations,

interpreta-physicistsdisplayed acertain skillin applyingsometimestheone, sometimes

theother,with the fortunateresultthattherewasneveranydisagreementwith

factsso far asverifiabledatawereconcerned

An attempt to reconcile the two interpretations was made by Born who

introduced the assumptionthat thewaves do not representfieldsof a kindof

matter spread throughspace, but that they constitute only a mathematical

instrumentofexpressing thestatisticalbehaviorofparticles;inthisconceptionthe waves formulate the probabilities for observationsof particles It is this

interpretation which we have used in 2 It has turned out, however, that

eventhis ingeniouscombination ofthetwo interpretationscannotbe carried

throughconsistently Weshalldescribe in 7 experimentswhich donot

con-form withthe Bornconception. Onthe other hand, thelatterconception has

beenincorporatedintoquantumphysicsso farasithasbeenmadethe

defini-tiveformofthe corpuscleinterpretation Whenever wespeakof corpusclesweassume themtobecontrolledbyprobability waves,i.e., bylawsof probabilityformulatedintermsofwaves.Thedualityof interpretations, therefore,isgiven

bya waveinterpretationaccordingto which matterconsists ofwaves;and a

corpuscle interpretation, according to which matter consists of particles trolled by probability waves As to the waves the struggle between the two

con-interpretations, therefore, amounts to the question whether the waves have

thing-characteror behavior-character, i.e., whether theyconstitutethe ultimateobjects ofthe physical world or only express the statistical behavior of such

objects,thelatterbeing representedbyatomicparticles.

Thedecisiveturnintheevaluationofthisstate ofaffairswas made by Bohr

in his principle ofcomplementarity This principle states that both the wave

conceptionandthe corpuscle conceptioncan beused,andthatit isimpossibleeverto verifytheoneandtofalsifytheother.Thisindiscernabilitywas shown

to bea consequenceof theprinciple ofindeterminacy, which withthis result

appeared tobethekeyunlocking thedoor through whichanescapefromthe

dilemma of two equally demonstrable and contradictory conceptions was

possible. The contradictions disappear, since it can be shown that they arerestrictedtooccurrencessituated insidethe rangeof indeterminacy;they arethereforeexcludedfromverification.

AlthoughweshouldliketoconsiderthisBohr-Heisenberginterpretation asultimatelycorrect, itseemstous thatthisinterpretationhas not beenstated

inaform whichmakessufficientlyclearitsgroundsanditsimplications.In the

formso far presenteditleavesa feeling ofuneasinesstoeveryonewho wants

to consider physical theories as complete descriptions of nature; the pathtowards thisaimseems either tobebarredbyrigorous rules forbidding us to