THE THEORY OF MEASUREMENT IN WAVE MECHANICS

Transcription

1 THE GREAT PROBLEMS OF SCIENCE WORKS COLLECTED BY MME. P. FÉVRIER VII THE THEORY OF MEASUREMENT IN WAVE MECHANICS (USUAL INTERPRETATION AND CAUSAL INTERPRETATION) BY Lous de BROGLIE of the French Academy Perpetual Secretary of the Academy of Scences Professor at the Sorbonne Translated by D. H. Delphench PARIS GAUTHIER-VILLARS, ÉDITEUR-IMPRIMEUR-LIBRAIRE Qua des Grands-Augustns,

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3 PREFACE. The present volume defnes a sort of complement to the boo that I recently publshed on the nterpretaton of wave mechancs by the theory of the double soluton ( 1 ). I recall, n more detal, certan questons that seem to me to necesstate a new examnaton of the role of measurement n quantum physcs, but developed n a more concrete fashon that s closer to expermental realty than what has been done up to now. The plan of ths boo s the followng: After recallng some well-nown prncples of wave mechancs n the frst chapter, I wll present the theory of measurement that s due to J. von Neumann n chapters II and III, whle presentng some arguments that were developed by Ensten and Schrödnger not long ago, and I wll show that ths theory, despte ts elegant character and the perfectly satsfyng appearance of ts formalsm, nonetheless leads to some consequences that are very dffcult to accept. The dffcultes that t rases derve, on the one hand, from the fact that, n accord wth presently domnant deas, t does not allow for the permanent localzaton of corpuscles n space and, on the other, that t vsualzes the processes of measurement n a very abstract manner. After summarzng the fundamental concepts of the theory of the double soluton n chapters IV and V, whle addng some complementary notons that dd not fnd ther places n my prevous treatses, I recall the study of the processes of measurement n chapter VI and VII from a concrete vewpont. I wll ntroduce the essental deas that wave trans are always bounded and that we can mae observatons or measurements on mcrophyscal realty only by the ntermedary of observable, macroscopc phenomena that are trggered by the local acton of a corpuscle. Upon addng to these fundamental remars the dea of the permanent localzaton of corpuscles n space such as would result from the theory of the double soluton, I wll show that one thus obtans a clear mage of the processes of measurement that do not rase the same obectons are the theory of von Neumann and hs hers. A last chapter s dedcated to a very rapd examnaton of von Neumann s thermodynamcs and ts nterpretaton wth the ad of the deas that were dscussed prevously. The goal of the present boo s, n summaton, to exhbt the reasons by whch t seems to me to re-establsh the noton of a permanent localzaton of mcrophyscal corpuscles, and why, once I agan became aware of that necessty, I sought n recent years to resume the attempt to nterpret wave mechancs that I setched out n 197. September 1956 Lous DE BROGLIE ( 1 ) Bblography [3].

4 TABLE OF CONTENTS Page PREFACE TABLE OF CONTENTS. Chapter I. Revew of some generaltes on wave mechancs and measurement. 1. Some nown prncples of wave mechancs 1. Reducton of the probablty pacet 4 3. Destructon of phases by measurement. Interference of probabltes Dvergence between the statstcal schema of wave mechancs and the usual schema of statstcans. 8 Chapter II. The theory of measurement, accordng to von Neumann. 1. Pure case and mxture. 11. The statstcal matrx of J. von Neumann for the pure case The statstcal matrx for a mxture of pure cases Irreducblty of the pure case The statstcal laws of wave mechancs wll be mpossble to nterpret by the ntroducton of hdden varables Crtque of the precedng concluson.. 3 Chapter III. The theory of measurement, accordng to von Neumann (cont.). 1. Generaltes on measurement.. 4. The statstcs of two nteractng systems, accordng to von Neumann The measurement of a quantty n the von Neumann formalsm 8 4. Less-admssble consequences of the theory of measurement n the present nterpretaton of wave mechancs 9

5 Table of Contents. Chapter IV. Causal nterpretaton of wave mechancs (theory of the double soluton). Page 1. Ideas at the bass for the theory of the double soluton Another way of expressng the gudance formula, and some generalzatons Proof of the gudance formula Introducton of nonlnearty and the form of the wave functon u Illustraton of the hypotheses made on u by an example The relatonshp between u and Ψ.. 45 CHAPTER V. Some complementary notons to the theory of the double soluton and gudance. 1. Exstence of sngular soluton n the exteror problem. 48. The Raylegh-Sommerfeld formula Constructon of the functon u wth the ad of the Raylegh-Sommerfeld formula n the case of statonary states Interpretaton of the statstcal sgnfcance of Ψ n the statonary states Two theorems from the theory of the double-soluton-plot wave Some words about the wave mechancs of systems n confguraton space. 60 CHAPTER VI. Poston of the causal nterpretaton n regard the problem of measurement n mcrophyscs. 1. The specal role played by the poston of the corpuscle. 6. Any measurng devce wll nvolve a separaton of wave trans n space Recoverng the usual schema of statstcans Interpretaton of the uncertanty relatons. 71 CHAPTER VII. Measurement of quanttes by the nteracton of two corpuscles. 1. The nconvenence of the measurement that was envsoned prevously for an solated corpuscle Interpretaton of measurements of the second nd by the usual theory Interpretaton by the theory of the double soluton Case of a measurement process nvolvng the nteracton of two corpuscles whose sngular regons R are not spatally dsont The dea of drectly. Examnaton of a remar by Ensten Conclusons. Pure case and mxture 85

6 v The theory of measurement n wave mechancs CHAPTER VIII. Glmpse of von Neumann s thermodynamcs. Page 1. Introducton to von Neumann s formalsm n thermodynamcs Reversble and rreversble evoluton How the theory of the double soluton must nterpret the rreversblty that results from measurement processes.. 91 APPENDIX. Study of the passage from classcal mechancs to wave mechancs n a partcular example 93 BIBLIOGRAPHY. 97

7 CHAPTER I REVIEW OF SOME GENERALITIES ON WAVE MECHANICS AND MEASUREMENT 1. Some nown prncples of wave mechancs. The present nterpretaton that wave mechancs allows supposes that one can descrbe a corpuscle or a system of corpuscles n as complete a fashon as possble wth the ad of a wave functon Ψ that s, moreover, capable of havng several components, as n the Drac theory of the electron or n that of corpuscles wth hgher spn. The functon Ψ s always assumed to be normalzed by the formula: Ψ dτ = 1. The evoluton of the wave functon n the course of tme s governed by a partal dfferental equaton vz., the wave equaton whch s the well-nown Schrödnger equaton n the smplest case of a corpuscle wthout spn n the non-relatvstc approxmaton. It wll tae on a more complcated form for partcles wth spn (the Drac electron, for example), because n these cases t wll become, n realty, a system of partal dfferental equatons that couple the varous components of Ψ. In a general fashon, the wave equatons, along wth the ntal condtons and boundary condtons, wll determne the evoluton of the functon Ψ completely. If one completely forgets the orgns of wave mechancs and the physcal ntutons upon whch t s founded then most authors wll consder the functon Ψ to be a smple mathematcal nstrument that serves to predct the probabltes of the varous results of measurements that are performed n the corpuscle or the system, snce that functon (by chance?) wll have the same form as the waves of classcal physcs. Now, here are brefly summarzed the postulates that consttute, n a way, the recpes that permt one to utlze the functon Ψ whch s assumed to be nown for the calculaton of the probabltes of the measurements that one can perform on corpuscular quanttes. One assumes that each of these quanttes wll correspond to a lnear, Hermtan operator A whose proper-value equaton: () Aϕ = αϕ wll permt one to defne a contnuous or dscontnuous (or even partally contnuous and partally dscontnuous) set of proper values α and correspondng proper functons ϕ(α). The proper functons ϕ wll form a complete system of functons wth an orthonormal bass, n such a way that one can always wrte: (3) Ψ = c(α) ϕ(α) dα, or, more smply, n the case of a dscontnuous spectrum:

8 The theory of measurement n wave mechancs. (4) Ψ = cϕ, upon enumeratng the proper values and the proper functons by an ndex. Moreover, a mathematcal formalsm le the Steltes ntegral wll permt one to combne the two cases of contnuous spectrum and dscontnuous spectrum nto ust one formula. The set of proper values of A defne the spectrum of that operator. The fundamental prncple that one taes to be the bass s then the followng one: Let Ψ be the wave functon of a corpuscle (or a system), upon whch one must perform the measurement of a quantty A, wth the ad of an approprate devce. One wll develop Ψ n proper functons ϕ of the correspondng operator A, and one can assert that the probablty for the measurement to gve a value that belongs to an nterval dα s c(α) dα. In the case of a dscontnuous spectrum, one wll say, more smply, that the probablty of the value of gven by c. The mathematcal expectaton of the value α or, f one prefers, the mean value of the result of the measurement of A that s performed upon a very large number of corpuscles that have the same functon Ψ wll be: (5) A = c α = Ψ * A Ψ dτ. When these general prncples are appled to the measurement of the poston of a corpuscle, that wll gve the followng result: The probablty for the coordnates of a corpuscle to be found nsde the nterval x x + dx, y y + dy, z z + dz.e., n order for the corpuscle to be found n the volume element dτ = dx dy dz wll be Ψ dx dy dz. An analogous statement wll be vald for the probablty of the presence of the fguratve pont of a system n the confguraton space to whch t corresponds. The statements that relate to Ψ (e.g., the prncple of nterference or ts localzaton) can be deduced from the general formalsm n such a way that, from the vewpont of that formalsm, the probablty of presence Ψ wll be seen to have the same status as any other probablty c. The set of all possble developments of Ψ n the dfferent systems of proper functons ϕ that correspond to the varous measurable quanttes wll thus appear to be entrely equvalent from the formula standpont. That dea, whch serves as the bass for the theory of transformatons, gves rse to some elegant mathematcal developments, although we shall dscuss only ts physcal sgnfcance. The general postulate that was assumed above n relaton to the statstcal sgnfcance of the c wll mply, by an argument that I wll not reproduce, the followng consequence: The same expermental devce can permt one to measure two quanttes A and B smultaneously wth any precson only f the correspondng operators commute;.e., f one has ABϕ = BAϕ for any ϕ. If that were not true.e., f ABϕ BAϕ, n general then any expermental devce that would permt one to attrbute a value to A that s affected wth a certan uncertanty and would leave behnd an uncertanty n the value of B that s greater than the measurement of A would have to be more precse, and conversely. The typcal example of two quanttes that are not smultaneously

9 Chapter I. Revew of generaltes on wave mechancs. 3 measurable wth precson s provded by any par of quanttes that are canoncally conugate, n the sense of analytcal mechancs, such as, for example, the coordnate x of a corpuscle and the correspondng component p x of the quantty of moton. In the latter h case, the correspondng operators (whch are x and ) are such that AB BA = π x h π the values of x and p x wll always satsfy the Hesenberg nequaltes:, and n turn, wll not commute. One then shows that the uncertantes that exst n (6) δx δp x h, and, n turn, can never be zero smultaneously. Moreover, there exst quanttes that, wthout beng canoncally conugate, nonetheless, do not commute; for example, the three rectangular components M X, M Y, M Z of the moment of the quantty of moton, for whch one wll fnd: h M X M Y M Y M X = M π and one wll then show that the uncertantes n the values of two of these components cannot be zero smultaneously, n general. One can translate these results nto a somewhat dfferent language by sayng that our general prncple wll mae the value of any measurable physcal measurement correspond to a probablty dstrbuton that has the form of Ψ. In the dscontnuous case, the probabltes of the values α wll be P = c, and n the contnuous case, the probablty densty wll be ρ(α) = c. Snce the state of a corpuscle (or a system) s defned by a certan functon Ψ, the set of measurable physcal quanttes wll correspond to a set of probablty dstrbutons that the theory presently consders (perhaps mstaenly, as well wll see) to be ntervenng wth exactly the same status for the corpuscle (or system) n the state Ψ. One can then defne a dsperson for every probablty dstrbuton that s equal to the square root of the mean square of the dstance from the mean value. One thus sets ths dstrbuton to be: (7) σ(a) = On can then prove that one wll have: ( α ) α = Z, α α. (8) σ(a) σ(b) 1 AB BA for two quanttes A and B. If the operators A and B commute then the rght-hand sde of (8) wll be zero, whch one can nterpret by sayng that one can get precse values (so the dspersons wll be zero) of the quanttes A and B by the same measurng devce. If the operators A and B do not commute then the rght-hand sde of (8) wll gve a non-zero lower lmt for the product of the dspersons, n such a way that no measurement operaton can provde

10 4 The theory of measurement n wave mechancs. precse values for A and B smultaneously. For two canoncally-conugate quanttes, one h wll have AB BA =, and one wll fnd that: π (8 bs) σ(a) σ(b) = 4 h π, whch wll consttute a way of statng the uncertanty relatons (6) that s more precse. Before pursung the study of the consequences of ths formalsm, I would le to nsst upon somethng that s extremely abstract: The wave functon Ψ shall be consdered to be a smple mathematcal functon that s a complex soluton to a partal dfferental equaton that wll have speang casually the form of an equaton of wave propagaton. Whle castng a pall on the physcal consderatons that guded me n the begnnng of my research and on the ones that were then developed by Schrödnger, one wll no longer see to gve any physcal pcture for the relatonshps between the wave and the corpuscle. We do not even now whether the wave Ψ s anythng but a mathematcal expresson that wll permt the calculaton of probabltes, and whether t wll reman somewhat obscured from physcal realty. On the other hand, smultaneously consderng all of the developments of the wave Ψ and gvng the same status to all of the probablty dstrbutons that one can deduce from t s somewhat strange, snce one nows that each of these dstrbutons wll be physcally sgnfcant only after one performs the correspondng measurement, and that the measurement wll, as we shall see, completely modfy the ntal state of thngs. Obvously, one can always say that any physcst that nows Ψ wll have the rght to appeal to t n order to calculate the values of a physcal quantty that represent the possble results of a measurement of that quantty and the correspondng probabltes. However, the probablty dstrbutons thus obtaned wll have only a subectve value, and can tae on an obectve value only after the effectve performance of the measurement, whch mples the nterventon of an approprate devce. Later on, we wll return to these questons, whch wll reman very obscure n the formalsm that we are presently usng, and we shall pursue the study of the consequences of that formalsm.. Reducton of the probablty pacet. Measurement plays an essental role n the nterpretaton of the formalsm that was presented above, and whch we are presently assumng, even f t does seem a lttle mysterous. It s what changes the state of our nowledge of the system under study whle gvng us new nformaton, and as a result, we are oblged to modfy brefly the form of the wave Ψ that represents our nowledge of the corpuscle (or the system). For example, f the measurement s a measurement of poston that s more or less precse then the wave tran Ψ that s ntally assocated wth the corpuscle wll be found to be reduced to a less-extended wave tran, whch can even be almost pont-le f the measurement s precse, snce the regon where the probablty of presence Ψ s non-zero wll have dmnshed n extent. One thus gets the term reducton of the probablty pacet that Hesenberg recently gave to that modfcaton of Ψ. On the contrary, f the measurement conssts of the determnaton of one of the

11 Chapter I. Revew of generaltes on wave mechancs. 5 components of the quantty of moton p x then t wll be n the space of momenta that reducton of the probablty pacet wll tae place, snce that wll then be the extent of the values of p x that effectvely appear n the Fourer representaton of Ψ that wll be dmnshed. The queston of the reducton of the probablty pacet wll then pose a dffcult problem n the present nterpretaton, namely: Is t the acton of the measurng devce that modfes the wave Ψ or s t the nowledge we acqure from the results of the measurement that mples that modfcaton? I do not now f all of the authors who have adopted the present probablty nterpretaton are n accord on the answer to that queston. Some of them (and that wll probably be the case for Bohr) wll anxous to preserve a certan character of physcal realty for the wave Ψ, and to say that t s the acton of the measurng apparatus on the wave Ψ that provoes the reducton of the probablty pacet. Others, who are perhaps beng more logcal, wll say that t s the nowledge of the result of the measurement that necesstates the modfcaton of the wave, snce, whle the result of the measurement s not nown to us, t wll be the old predctons of the probabltes that correspond to the orgnal form of Ψ that wll reman vald n order for us to mae those predctons. However, f one adopts the second opnon then the wave Ψ wll only be a purely subectve representaton of the probabltes, and cannot be a representaton of obectve realty to any degree. How then can t obey an equaton of wave propagaton and, despte everythng, provde us wth a statstcal representaton that s probably exact of phenomena whose physcal realty s not n doubt? Ths queston remans truly obscure; we shall return to t. The reducton of the wave tran Ψ wll gve rse to a new stuaton that s characterzed by a new form of Ψ, whch s a stuaton that s unpredctable n advance, snce only the probabltes of the varous possble measurements can be calculated before mang an effectve measurement. We shall have to demand whether that unpredctablty results from a real ndetermnacy, as one presently assumes, or, on the contrary, on the value of certan hdden varables, as s suggested by the theory of the double soluton, whch s a queston that has a close relatonshp wth a theory that was stated by von Neumann n hs theory of measurement n wave mechancs. The Hesenberg uncertanty relatons show that a devce that permts one to smultaneously perform varous measurements on a corpuscle cannot smultaneously tell us precsely the values of all of the quanttes that characterze the corpuscle. There wll therefore be an ncomplete maxmum nowledge of these quanttes that s compatble wth the uncertanty relatons. Once we have acqured ths maxmum nowledge, we can construct the wave functon that serves to represent our nowledge mmedately after the measurement, and upon startng wth the ntal form of Ψ, we can follow ts ultmate evoluton n the course of tme wth the ad of the wave equaton. At any nstant, we can then calculate the probabltes of the results of varous measurements that one can perform at that nstant. That wll be true up to the pont at whch we now the result of the new measurements, whch wll modfy the state of our nowledge and brefly nterrupt the regular evoluton of the wave Ψ. The regular evoluton of that wave between two measurements whch s an evoluton that s ruled by the wave equaton s tself determned entrely by the ntal form of Ψ (and possbly by the boundary condtons), snce the wave equaton s of frst order n tme. The evoluton of Ψ between two measurements wll be determned, but not the observable phenomena, snce the

12 6 The theory of measurement n wave mechancs. nowledge of the wave functon wll gve only probabltes for them. If the descrpton of physcal realty by the functon Ψ s a complete descrpton.e., f there exst no descrpton that s more complete, for example, by the ntroducton of hdden varables then physcal phenomena wll be undetermned. 3. Destructon of phases by measurements. Interference of probabltes. The A measurement wll ntroduce a dscontnuty nto the evoluton of the wave functon. The nowledge of t after the measurement does not allow one to reconstruct the form that t had before the measurement. Consder a large number of corpuscles (or systems) that are ntally found n the same state that s represented by Ψ. Measure a certan quantty A for each of them that has proper functons ϕ and proper values α. After these measurements, the proporton of corpuscles (or systems) for whch one wll have found the varous values α for A wll gve us the squares of the modul of the coeffcents c n the development Ψ = cϕ of the wave functon before the measurement. The nowledge of Ψ for all of the corpuscles (or systems) after the measurement wll then provde us wth the values of the c, but n order to now the c themselves, we would need to now ther arguments, and thus, the relatve phases of the components c ϕ of the ntal wave functon. It was that remar that led Bohr to emphasze that any measurement must have the effect of completely destroyng the phases. It s ths destructon of the phases by the act of measurement that brngs about what consttutes a brea n the evoluton of Ψ. Indeed, the dfferences n phase between the components of the development cϕ are of paramount mportance, and any nowledge that relates to the wave functon that does not nvolve nowledge of these phase dfferences wll be radcally ncomplete. The mportance of these phases s clearly manfested to us n the study of nterference phenomena for the probabltes. Consder two quanttes A and B whose operators do not commute, and whch, n turn, are not smultaneously measurable. The proper values and proper functons of A are α and ϕ, whle those of B are β and χ. One easly proves that snce A and B do not commute, the system of the ϕ cannot concde wth that of the χ. Meanwhle, snce the χ defne a complete system, each ϕ can be expressed wth the ad of the χ n the form: (9) ϕ = s χ, n whch the s are elements of a untary matrx S. More than one term n the rght-hand sde wll appear n ths development, snce the system of ϕ and that of χ do not concde. Suppose then that the state of the corpuscle (or system) beng examned s represented by the wave functon: (10) Ψ = cϕ = c s χ.,

13 Chapter I. Revew of generaltes on wave mechancs. 7 If one then measures the quantty A then one wll fnd one of the proper values α, where the probablty of fndng α wll be c a pror. After the measurement, the corpuscle (or system) wll be found n the state ϕ, and n that new state a measurement of B wll lead to the value β wth the probablty s. Therefore, the probablty of fndng the value β for B by frst measurng A and then B wll be equal to c s. However, now suppose that we have performed the measurement of B drectly on the ntal state. Then, from the form of the last expresson n (10), the general prncple that relates to the probabltes of the results of measurement wll tell us that the probablty of fndng β wll be equal to cs. That expresson wll be entrely dfferent from the precedng one, because t wll depend upon the phases (or arguments) of the c and s, whle c s obvously does not. That s what one calls the nterference of probabltes. We llustrate ths wth a smple example: Tae a one-dmensonal doman wth length L. The normalzed proper functons of the quantty of moton wll be ϕ = π 1 p x h e L n that doman. Then, let: (11) Ψ = c e L π p x h c = 1 be the wave functon of the corpuscle n ts ntal state. If one frst measures p, and then x then the probablty of the poston x = x 0 wll be: 1 π p x 0 h c e, L or smply 1 / L, whch wll mply the equal probablty of all postons on the segment of length L. However, f, on the contrary, one measures the coordnate x n the ntal state drectly then the probablty of the value x = x 0 wll be Ψ(x 0 ), and ths wll nvolve the nterference of the plane waves whose superposton wll consttute the Ψ, a result that s necessary n order to account for nterference n optcs and the dffracton of electrons. One wll then see that the nterference of probabltes, whose exstence s necessary for the nterpretaton of expermental facts, wll depend essentally upon phases, whose role s then seen to be paramount. The fact that the probablty of the value β of B, when measured drectly n the ntal state, wll be cs, and not c s, can seem, on frst glance, to be contrary to the theorem of composed probabltes, but n realty, that s not so: The probablty

14 8 The theory of measurement n wave mechancs. c s s ndeed the one that one must choose when one frst maes a determnaton of A, and then B, snce t s equal to the sum of the products of the probltes for frst gettng a value α for A tmes the probablty of then gettng the value β for B. The theorem of composed probabltes s then safe, and f one envsons the probabltes from a purely subectve vewpont then one can say that there s no reason for the probablty c s to be equal to that of drectly obtanng the value β of B by a measurement of that quantty n the ntal state. However, f one analyzes that dea closely then one wll see that all of the probablty dstrbutons that are ntroduced n the usual theory (except, wthout a doubt, Ψ ) wll exst n the ntal state only subectvely for the physcst who must mae the predctons on the result of possble measurements. These dstrbutons wll exst obectvely only after the correspondng measurement has been performed when one further gnores the result of that measurement. It s that stuaton that wll explan, later on, why the schema of the usual probablstc nterpretaton of wave mechancs s not n agreement wth the usual schema that s assumed by statstcans. 4. Dvergence between the statstcal schema of wave mechancs and the usual schema of statstcans. In the usual schema of statstcans (whch we wll present by assumng that one s dealng wth contnuous varables), one defnes a probablty densty ρ X (x) for every random varable X such that ρ X (x) dx wll be the probablty for X to have a value between x and x + dx. One wll lewse defne ρ Y (y) for another contnuous random varable Y. One then defnes a densty ρ(x, y) such that ρ(x, y) dx dy s the probablty of obtanng values for X and Y by the same measurement operaton (the statstcans often say by the same proof ) that are contaned n the ntervals x x + dx and y y + dy, respectvely. That defnton wll seem qute natural f one adopts a concrete mage of the probablty n whch ndvduals appear, for each of whch the quanttes X and Y wll have a well-defned value, so statstcs wll be ntroduced by the smultaneous consderaton of a very large number of ndvduals for whch X and Y have dfferent values. Outsde of ρ X (x), ρ Y (y), and ρ(x, y), statstcans wll also consder the probablty ( X densty ρ ) Y ( x, y) of Y, when coupled to X, whch wll correspond to the probablty of obtanng the value y of Y when one nows that X has the value x, and one lewse ( Y defnes the probablty of X, when coupled to Y, wth the ad of ρ ) X ( x, y). One must now have the followng relatons, whch one can consder to be obvous, between the fve probablty denstes that we ust defned: ρ X ( x) = ρ( x, y) dy, ρy ( y) = ρ( x, y) dx, ( Y ) ρ( x, y) ( X ) ρ( x, y) ρ X ( x, y) =, ρy ( x, y) =, ρy ( y) ρ X ( x)

15 Chapter I. Revew of generaltes on wave mechancs. 9 from whch, one wll nfer that: (13) ρ X (x) = ( Y ρ ) X ( x, y) ρy ( y) dy, ρ Y (y) = ( X ρ ) Y ( x, y) ρx ( x) dx. Now, the essental fact s that the precedng schema, whch s usually taen for granted by statstcans, s not applcable to the probablty dstrbutons that are envsoned n the present nterpretaton of wave mechancs. Indeed, t s, n general, mpossble to defne the densty ρ(x, y) for two measurable quanttes, snce t s, n general, mpossble to smultaneously measure the values of the quanttes X and Y. Formulas no longer mae sense then. Wthout a doubt, t s always possble to ( Y defne the denstes ρ X (x), ρ Y (y), ) ( X ρ X ( x, y), and ρ ) Y ( x, y), but they wll no longer be related by formulas and (13). As an example of ths, recall the precedng case that was examned of two measurable quanttes A and B that are not commutatve, and rewrte formulas (9) and (10) by passng from the dscontnuous case to the contnuous case. We wll have: (14) ϕ(α) = s(α, β) χ(β) dβ, χ(β) = s 1 (α, β) ϕ(α) dα. If Ψ s of the form: (15) Ψ = c( α) s( α, β ) dα = c( α ) s( α, β ) χ( β ) dα then one wll fnd that: (16) ρ A (α) = c(α), ρ B (β) = c ( α) s ( α, β ) d α, where the second formula expresses the nterference of probabltes, so: (17) ( ρ A) ( α, β ) = c(α) s(α, β), B ( ρ B) ( α, β ) = A 1 c( γ ) s( γ, β ) d γ s ( α, β ), ( B but here the products ρ B (β), ) ( A ρ ( α, β ), and ρ A (α), ρ ) ( α, β ) have no reason to be equal, A whch wll ndeed show that the non-exstence of the densty ρ(α, β), whch must be equal to ther common value. Where does ths very strange specal character of the statstcal dstrbutons of modern quantum mechancs come from? The answer seems to be contaned n the essental role that s played by measurement. Snce the probablty dstrbutons of modern quantum mechancs (wth the possble excepton of some of them) do not consttute obectve probabltes, they can be regarded as all correspondng to a collecton of ndvduals at the same nstant for whch the quanttes wll have welldefned values. The mplct hypothess that maes the relatons and (13) obvous for the statstcan s not realzed here. It s only after the acton of the measurng devce of a quantty for the corpuscle (or system) that the probablty dstrbuton can be consdered to be realzed obectvely. To spea more precsely, f one magnes that the measurement of a certan quantty s B

16 10 The theory of measurement n wave mechancs. performed smultaneously on an nfntude of corpuscles (or systems) that ntally have the same functon Ψ then t wll be only after performng the measurement on all of these corpuscles (or systems) that one wll really have a collecton of ndvduals that each possess a precse value of the measured value such that these values wll be dstrbuted accordng to the law of probablty n c, and t can be further remared that the law of probablty n c wll thus not be found to be realzed obectvely by a collecton, so much as for the measured quantty and the ones that commute wth t, to the excluson of the other ones. If the physcst nows the wave functon n the ntal state, when no measured has been performed, then he can calculate the varous probablty dstrbutons that he can subsequently decde to measure. However, each of these dstrbutons can be found to be thus realzed, and thus correspond to a collecton, only after performng the correspondng measurement. The dstrbutons can never be all found to be realzed smultaneously, snce one cannot smultaneously measure all of the quanttes, and one must employ two ncompatble measurng devces n order to measure two noncommutatng quanttes. Certanly, the physcst always has the rght to smultaneously consder the set of probablty dstrbutons before any measurement that can be deduced from the varous developments of the ntal Ψ, but these probabltes wll then have a subectve character, and are not obectve probabltes that are statstcally realzed by the same collecton of ndvduals. As we have seen, that s what prevents us from attrbutng the propertes and (13) whch wll be obvous for obectve dstrbutons that refer to a collecton of ndvduals wth well-defned characterstcs to the probablty dstrbutons of conventonal wave mechancs. We thn that t s for the same reason that the celebrated theorem of von Neumann, whch we wll dscuss soon, s bascally only a trusm, and does not at all prove the mpossblty of re-establshng determnsm n wave mechancs by the ntroducton of hdden varables.

17 CHAPTER II. THE THEORY OF MEASUREMENT, ACCORDING TO VON NEUMANN ( 1 ). 1. Pure case and mxture. Frst, recall some consderatons regardng the nterference of probabltes. Let there be a very large number N of corpuscles (or systems) that all have the same wave functon Ψ. If A s a measureable physcal quantty wth proper values α and proper functons ϕ then f one has Ψ = cϕ then the measurement of A must lead one to fnd the value α 1 for c 1 N systems, the value α for c N systems, etc. The mean value of A wll be c α. Now, magne that nstead of havng N systems n the same state, we have c 1 N systems n the state ϕ 1, c N systems n the state ϕ, etc. The measurement of A wll then gve us the same statstcal results as n the former case. One mght then beleve that the two cases are equvalent, but we shall see that ths s not true. Indeed, consder a measurable physcal quantty B that does not commute wth A. The proper functons of B wll not concde wth those of A, and f β and χ are the proper values and proper functons of B, resp., then one wll have ϕ = dl χl, n whch the development wll generally contan several terms. Frst, magne the prevous case, n whch we had N systems that were all n the same state: l Ψ = cϕ = cdl χl., l The measurement of B for all of these systems wll then gve N β l, and the mean value of B wll be:, l c d l tmes the value B = wth: cdl βl = Ψ BΨ dτ l ( ) B ϕ l = l = ϕ Bϕ dτ. ( ) c cl B ϕ l,, l ( 1 ) See bblography [1], [].

18 1 The theory of measurement n wave mechancs. We then place ourselves n the second case, where we had c 1 N systems n the state ϕ 1. The measurement of A on the frst c 1 N systems wll gve the value β l for a proporton of these systems that s equal to d l, etc. In total, the value of β l of B wll be obtaned: N c d tmes, and n turn, the mean value of B wll be: l () B = wth: c dl βl =, l ϕ Bϕ dτ. ( ) B ϕ l = l ( ) c B ϕ l, One then sees that the two cases that we envsoned are completely dfferent for any quantty that does not commute wth A. In the frst one there s nterference of probabltes, whle n the second one that nterference s not present. One cannot therefore consder the N systems as defnng a collectve system that s composed of N c 1 ndvdual systems that have the value α 1 for A, etc. Moreover, t s obvous that t wll therefore be entrely legtmate to consder the N systems as defnng a collectve system that s composed of N d 1 systems that have the value β 1 for B, etc., wth d 1 = c d1, and ths second collectve system wll not concde wth the frst one. We can thus not consder the set of N systems as defnng a well-defned collectve system, snce that collectve system wll vary accordng to the quantty that s envsoned. We then recover the dea that we prevously brought to lght: The probabltes that are envsoned n conventonal wave mechancs correspond to a unque collectve system that s realzed n the state Ψ. In order to dstngush the case where the probablty dstrbuton for a quantty A has only a subectve value before the measurement from the one where that dstrbuton s realzed after the measurement, von Neumann sad that the former case consttutes a pure case, whle the latter one consttutes a mxture. Wthout mang any act of measurement ntervene, one can magne N 1 systems that have a wave functon Ψ, N systems that have a wave functon Ψ, etc. The set of all N systems wll then defne a mxture of N 1 pure cases that correspond to Ψ, N pure cases that correspond to Ψ (), etc. We recover the second case that was studed at the begnnng of ths paragraph by tang N 1 = N c 1, If we set N / N = p then we wll have a mxture that s defned by the set of statstcal weghts p wth p = 1. If we set c = p e α then we wll see that the p = c are the statstcal weghts of the mxture that s equvalent to the pure case Ψ, as far as the measurement of A s concerned. However, ths mxture s realzed only after the measurement that

19 Chapter II. The theory of measurement, accordng to von Neumann. 13 transformed the ntal pure case nto ths mxture. The mxture that s equvalent to the pure case Ψ for the measurement of a quantty B that does not commute wth A wll nvolve statstcal weghts that are dfferent from the precedng ones, and wll be realzed only by a measurement that nvolves a devce of a dfferent type. That s why one cannot reduce a pure case to a well-defned mxture. We have seen that the mean value of B s gven by formula for the pure case Ψ. If one replaces ths pure case wth a mxture that s found to be realzed by the measurement of A then the mean value of B wll be gven by formula (). It s easy to specfy the manner n whch the two expressons and () dffer. Formula can be wrtten: (3) B = ( αl α ) ( ϕ) c cl e Bl., l If one assumes that the phases α (.e., the arguments of the α ) are nown completely wth equal probabltes for ther possble values then the mean value of the expresson (3) wll be obtaned by tang a mean over the values of the α, whch are all assumed to be equally probable. The terms where l wll gve zero, and we wll recover the expresson (). In other words, one passes from the pure case Ψ to the mxture that s realzed by the measurement of A by assumng that ths measurement has made one lose all nowledge of the phases α. Here, we ndeed recover the concluson that the measurement of A that s performed on the ntal state that s represented by Ψ = cϕ wll have the effect of completely destroyng the phase dfference that exst between the components ϕ of the ntal Ψ. Fnally, we have obtaned a neat dea of the dfference between a pure case that s defned by a wave functon Ψ and a mxture that s defned by a set of pure cases wth wave functons Ψ 1, Ψ, that are affected wth statstcal weghts p 1, p,. The statstcal matrx of J. von Neumann for the pure case. Frst, envson a pure case that s defned by a wave functon of a gven form. That functon can be consdered to be a vector n a Hlbert space. If ϕ 1, ϕ,, ϕ n, s a complete, orthonormal system of bass functons (for example, the proper functons of a Hermtan, lnear operator A) then the ϕ can be consdered to defne a complete system of untary vectors n Hlbert space, and the expresson Ψ = cϕ wll be analogous to the expresson of a vector wth the ad of ts components along orthogonal drectons that are defned by the untary vectors. One can say that the c are the components of Ψ n the bass system of the ϕ. The Hlbert space that we consder wll be a complex space, and the components c wll be complex, n general. Now, let: Ψ = cϕ and χ = dϕ be two vectors n Hlbert space. By defnton, ther scalar product s (D beng the doman varaton of the varables n the ϕ):

20 14 The theory of measurement n wave mechancs. (4) (Ψ χ) = Ψ χ d τ = D c dl ϕϕl dτ =, l and one wll have: (5) (χ Ψ) = (Ψ χ) * ; c dl δl =, l c d,, l one wll ndeed then have the generalzaton of the classcal expresson for the scalar product to complex vectors. The scalar product of a vector Ψ wth tself, whch s analogous to the square of the length of an ordnary vector, s called the norm of that vector, and wll have the value: (6) N(Ψ) = (Ψ, Ψ) = If the vector s normalzed then one wll have: Ψ dτ = D c. N(Ψ) = 1 and c = 1. An operator on Hlbert space wll correspond to an operator that maes one vector go to another one χ = AΨ, whch wll then defne the operaton that taes Ψ to χ, and one wll have: dlϕl = A cϕ, so, upon multplyng by ϕ and ntegratng over D, one wll get: (8) d = c ϕ A ϕ d τ l D = a c. The a, whch are elements of the matrx that s generated by A n the system of the ϕ, wll then be the coeffcents of the lnear transformaton that taes the components of Ψ to those of χ. The conservaton of norm would mpose the condton that the matrx a must be untary. If Ψ s once agan the wave functon of a pure case then magne the operaton on Hlbert space of proectng onto the vector Ψ; let P Ψ be the correspondng operator. It n s obvous that P Ψ = P Ψ, and that, more generally, P Ψ = P Ψ. Snce all of the powers of P are dentcal, one says that ths operator s dempotent. Now, let there be a complete system of orthonormal bass functons ϕ 1,, ϕ n, We have a development for Ψ: Ψ = cϕ, wth c = ϕ Ψ dτ and D c = 1.

21 Chapter II. The theory of measurement, accordng to von Neumann. 15 One can obvously fnd an nfntude of orthonormal bass systems for whch Ψ s one of the bass vectors. In one of these systems, the functon ϕ wll have a development of the form: (9) ϕ = dψ +, wth d = Ψ ϕ dτ = c. The operator P Ψ, whch s the proector onto Ψ, s defned by: (10) P Ψ ϕ = dψ = c Ψ for any ϕ. The matrx that s generated by the operator P Ψ n the bass system of ϕ has an element wth ndces m, n: (11) (P Ψ ) mn = ϕ mpψ ϕn dτ = cn ϕm D D Ψ D dτ = c m c n. Thus, the matrx P Ψ that s attached to the pure case beng consdered s expressed wth the ad of the coeffcent of the development of Ψ n the bass system that s beng utlzed. One has thus defned what von Neumann called the statstcal matrx that s attached to the pure case Ψ; formula (11) maes t obvous that ths matrx s Hermtan. The statstcal matrx possesses two fundamental propertes: 1. Its trace s equal to 1. Indeed: Tr P Ψ = ( P Ψ ) nn = c ncn = 1.. It s dempotent. Indeed, one wll have: n n (13) ( P Ψ ) mn = cmc p cpcn = c m c n = (P Ψ ) mn, p and thus, n terms of matrces, P Ψ = P Ψ, and by recurrence, n P Ψ = P Ψ. Now, let A be a quantty n the system beng consdered. If the ϕ are functons of an arbtrary orthonormal bass (whch are no longer proper functons of A, here) then we have seen that the mean value of A wll be: (14) A = ( ) c cl A ϕ l,, l where the ( ) A ϕ l are the elements of the matrx that s generated by the operator A n the system of the ϕ, and c s the component of Ψ along ϕ. One can also wrte:

22 16 The theory of measurement n wave mechancs. (15) A = ( ) ( P ) l A ϕ Ψ l = Tr(P Ψ A) = Tr(A P Ψ )., l Therefore, the nowledge of the statstcal matrx wll provde us wth a smple means of calculatng A. The statstcal matrx of a pure case s frequently called an elementary statstcal matrx (Enzelmatrx), n contrast to the more general statstcal matrces that we shall encounter later on whle studyng mxtures of pure cases. An elementary statstcal matrx can be easly put nto dagonal form. In order to ths, t wll suffce to tae the bass system to be a system where the Ψ consdered s one of the bass functons; for example, ϕ 1 = Ψ. The elementary statstcal matrx wll then tae the form: (16) All of the terms wll be zero, except for the frst dagonal term, whch s equal to 1; ths results from (11) easly. The trace of the statstcal matrx wll be an nvarant under changes of bass functons, and n turn, a nown property of untary transformatons; t must then be equal to 1, as the table (16) shows. Ths table wll also permt one to verfy mmedately that the statstcal matrx s dempotent. 3. The statstcal matrx for a mxture of pure cases. We shall now consder a mxture of pure cases. We have already defned such a mxture by consderng N systems, of whch, Np 1 are n the state Ψ, Np are n the state Ψ (),, wth p = 1. However, we can also ntroduce the dea of mxture for ust one system. Indeed, t can happen that we are gnorant of the exact form of the wave functon of a system, and that we now only that t has a probablty p 1 of beng n the state Ψ, a probablty p of beng n the state Ψ (), etc., a probablty p n of beng n the state Ψ (n), wth p = 1. The state of our nowledge about the system s then represented by a mxture of pure cases wth the statstcal weghts p. Each of the pure cases n the mxture has an elementary statstcal matrx Ψ ( ). We attrbute a Hermtan statstcal matrx: (17) P = pp Ψ ( ), wth (18) P (m) = n = 1 n = 1 p c c ( ) ( ) l m P

23 Chapter II. The theory of measurement, accordng to von Neumann. 17 to t, where the statstcal weghts p are postve numbers between 0 and 1 whose sum s ( ) equal to 1. The c are the components of the varous Ψ () n the system wth bass ϕ 1, m, ϕ n. The statstcal matrx (17) thus appears to be a superposton of elementary statstcal matrces. As an example, suppose that one has taen the bass functons to be the proper functons that relate to poston δ(q q ), where δ s the sngular Drac functon. The formula: (19) Ψ () ( (q, t) = Ψ ) ( q, t) δ ( q q ) dq ( ) wll then show that the c are equal Ψ () (q, t), and that one wll fnd that the components of the statstcal matrx are: (0) P(q, q ) = n ( ) ( ) pψ q Ψ q = 1 ( ) ( ). Ths s Drac s statstcal matrx. The mean value of a measurable quantty A of the system wll be: A = p A Ψ ( ), n = 1 where A Ψ ( ) s the mean value that A wll have when the system s n the pure state Ψ (). From (15), we wll then get: () A = p ( P ( ) A) n n Ψ = pp ( ) Ψ = 1 = 1 A. The formula wll therefore be the same as t s for the pure case. The statstcal matrx of a mxture, le that of a pure case, wll always have a trace that s equal to 1, because: (3) Tr P = Pmm = m n ( ) ( ) pcm c m = m = 1 n ( ) p cm = 1. = 1 m By contrast, whle the matrx of a pure case s always dempotent, the same thng s not true for the statstcal matrx of a mxture. Indeed, one can prove that any dempotent statstcal matrx s elementary. In order to do that, one assumes that P = P, and one wrtes P n dagonal form, whch s always possble. If p s the th dagonal element of P then the relaton P = P wll demand that one must have p = p, and the p wll then be equal to 0 or 1. The equaton Tr P = 1 that s satsfed by all statstcal matrces wll then show that one of the p s dfferent from 0, and therefore equal to 1. The system wll then have a unque Ψ that agrees wth one of the bass functons that reduces P to ts dagonal

24 18 The theory of measurement n wave mechancs. form. Therefore, the necessary and suffcent condton for a statstcal matrx to be an dempotent s that t be elementary. Now, consder the non-elementary statstcal matrx of a mxture. If the Ψ, Ψ (),, Ψ (n) that defne the pure cases that appear n the mxture are orthogonal (whch can happen only n exceptonal cases) then one can tae them to be the frst n bass functon ( ) of an orthonormal system. One wll then have c = δ m, snce Ψ () wll reduce to ϕ and P (m) wll be zero for l m, whle the P wll be equal to p for n and zero for > n. The statstcal matrx wll then tae the followng dagonal form: m (4) p p p n However, ths s the one exceptonal case. In general, the functons Ψ,, Ψ (n) wll not be orthogonal. One can nonetheless reduce the matrx P to dagonal form, even n ths case, but the dagonal elements p wll no longer be equal to p 1,, p n, 0, 0, Snce the matrx P s Hermtan, the p wll be real numbers. Moreover, snce Tr P = 1, one wll have p = 1. We shall show that the p cannot be negatve. In order to do that, f ξ are the components of a vector Ξ n Hlbert space then consder the scalar product of Ξ wth PΞ. One wll then have (5) (Ξ, PΞ) = n ( ) ( ) ξm pcm cn ξn m, n = 1 = n = 1 p ( ) ( Ξ, Ψ ) for ts value. Snce the square of a modulus s, a fortor, postve or zero, we wll see that the scalar product (5) s necessarly postve or zero. Now, f we put P nto ts dagonal form then that scalar product wll have the followng expresson: (6) (Ξ, PΞ) = p m ξm, m whch must be 0, and for any Ξ. Therefore, the p m must all be postve or zero. Snce ther sum s equal to 1, one wll have 0 p m 1. One nfers from that that p m so for any arbtrary vector Ξ n Hlbert space: (7) (Ξ (P P ) Ξ) = ( p m p m ) ξm 0. m p m 0,

25 Chapter II. The theory of measurement, accordng to von Neumann Irreducblty of the pure case. We now come to a theorem that plays a maor role n the proof by whch von Neumann wanted to establsh the mpossblty of explanng the present probablstc character of wave mechancs wth the ad of hdden varables. The mportant theorem n queston s stated as follows: It s mpossble to represent a pure case n the form of a mxture, or also: A pure case s never reducble to a superposton of pure cases. He thus establshed the ntrnsc specal character of the pure cases. Indeed, f ths theorem were not true then t would have to be possble at least, n some cases to obtan a relaton of the form: (8) P = αq, n whch, P and Q are elementary statstcal matrces.e., dempotent Hermtan matrces wth trace 1 and the α are postve numbers such that α = 1. Now, one wll then have: (9) P = = = = α Q + α α ( Q Q + Q Q ), 1 α Q + α α [ Q + Q ( Q Q ) ], 1 α + α α Q α α ( Q Q ) α Q α α ( Q Q ), >, > because α = 1 α. One wll therefore have: (30) P P = α ( Q Q ) α α ( Q Q ). > However, P = P and (31) Q = Q, so: αα ( Q Q ) = 0, > and snce all of the α are postve: (3) (Q Q ) = 0. Now, the square of a Hermtan matrx can be zero only f the matrx tself s zero. Indeed, f A s a Hermtan matrx then the elements of A wll be:

26 0 The theory of measurement n wave mechancs. (a ) = al al = al a l, and f the (a ) are zero then one must also have l l al = 0, whch wll demand that a l = 0, and n turn, that A = 0. Snce Q Q s a Hermtan matrx then the condton (3) wll mply that Q = Q. All of the Q wll be the same, and one wll have: P = α Q = Q, snce α = 1. P wll not be truly a sum of elementary statstcal matrces then, whch s contrary to hypothess. It s therefore ndeed proved that the pure cases are rreducble and can never be reduced to a mxture of pure cases. The pure cases of wave mechancs wll thus possess the followng two propertes: 1. They wll be represented by elementary (.e., dempotent) statstcal matrces, whle any mxture wll have a matrx that s not elementary (.e., not dempotent).. There wll be no way of reducng a mxture to a pure case. 5. The statstcal laws of quantum mechancs wll be mpossble to nterpret by the ntroducton of hdden varables. In classcal physcs, any tme one must ntroduce probabltes n place of rgorous laws one wll always assume that there exsts determnsm n the phenomena, but that ths determnsm s too complcated or too subtle for us to be able to follow t n detal, snce the observable manfestatons are of a statstcal character and, for that reason, they wll be expressed by probabltes. The laws of probablty and the element of chance that they seem to ntroduce wll not be the proof of a true contngency, but the result of our ncapacty to follow a determnsm that s too fne-graned or too complcated. That s the defnton of chance that one fnds n the wrtngs of all thners who predated the development of wave mechancs, and n partcular, n the wors of Henr Poncaré. The best-nown example of such a pseudo-statstcal theory n physcs s the netc theory of gases. There, one wll assume that the motons of the gas molecules, as well as ther mutual collsons, are governed by the rgorous laws of classcal mechancs, n such a way that there wll be a subordnate determnsm. However, the molecules are suffcently numerous that ther motons wll be so complcated that we cannot actually follow ths elementary determnsm n all of ts detals. Moreover, the molecular motons complete elude our senses, and we can only predct the macroscopc effects of these motons, such as pressure, temperature, local fluctuatons of densty or energy, the Brownan agtaton of a vsble granule due to ts rregular collsons wth molecules, etc. Snce these macroscopc phenomena wll result from an enormous number of complcated, elementary phenomena, we seem to be constructng a statstcal theory that wll nvolve only probabltes, but that ntroducton of chance s only apparent, and, for example, the dsorganzed motons of a granule n ts Brownan agtaton wll seem to us