DUALISMIN QUANTUM THEORYDo particles sometimes behave as waves?Can waves be treated as particles? There is still disagreementin the interpretation of the quantum theory.

MAX BORN and WALTER BIEM

IN RECENT YEARS Alfred Lande hastried to give a new foundation forquantum theory, starting with statisti-cal nonquantal principles that havenot been taken from classical physics.In his books1 and articles2 he attacksthe basic concept of quantum theorythat is generally taken for granted bymost physicists nowadays. He callsthis concept "dualistic" and maintainsthat he can replace it by another moreunified one.

We think Lande has not realizedthe historical origin of the dualisticinterpretation and does not correctlydescribe its physical meaning. More-over his fight against "dualism" inmodern quantum theory seems to bea tilt against windmills.

Lande's attempt at a new founda-tion for quantum theory was re-viewed in detail by Abner Shimony inPHYSICS TODAY3 so that we need notenter into his concept of quantumtheory as a whole.

Lande's approach

As Lande says, dualism is the habitof modern physicists to use two ap-parently contradictory "theories" sideby side, the particle theory and thewave theory, to explain the phenom-ena of the microcosm. Lande main-

tains that matter can and must betreated exclusively with a particletheory, and light on the contraryshould only be treated with a wavetheory; this is what he calls "unity inquantum physics." We think thatLande's interpretation represents amore unsatisfactory dualism that ig-nores important physical discoveriesjust for prejudice. Moreover the dual-ism of waves and particles, accordingto Lande, appears to be a late in-

vention of the theoreticians (especiallyNiels Bohr and Werner Heisenberg)who tried to interpret quantum me-chanics; this view is historically in-correct.

The founder of the dualistic viewwas Albert Einstein, that is to sayyoung Einstein, who did not speculateas he did in later years, but analyzedphysical experience with remarkablepenetration and drew irrefutable infer-ences. His first work on this subject4

Max Born studied at Breslau, Heidel-berg, Zurich, Gottingen and Cambridgebefore moving to Frankfurt in 1919 asprofessor of theoretical physics. Twoyears later he took up a similar post atGottingen. In 1933 he left Germany forCambridge, Bangalore and Edinburgh.His Nobel Prize was awarded in 1954.

Walter Biem is active in solid-state phys-ics, including molecular crystals, solidhelium and hydrogen. He studied inGottingen and Aachen, worked at theMax Planck Institute of Physics and As-trophysics in Munich, and is now em-ployed at the Institute for Neutron Phys-ics of the Kernforschungsanlage Julich.

PHYSICS TODAY • AUGUST 1968 51

appeared 60 years ago in the samevolume of Annalen der Phtjsik as hisgreat paper on electrodynamics ofmoving bodies (which includes thefoundation for the special theory ofrelativity) and his fundamental workon Brownian motion, in which the ex-istence of atoms and molecules isproved for the first time by empiricalfacts.

Einstein's derivation

Einstein considered the radiation bal-ance in a cavity and used Planck'sradiation law, known since 1900,which gives the average density ofenergy u as a function of radiationfrequency v and temperature X. Theradiation energy E(i<), at a frequencyv, in a small part V of a cavity is notalways exactly uV but is subject tofluctuations. If the radiation consistsof waves, as was assumed at that time,the mean square of fluctuation of theenergy is proportional to the squareof the mean energy density, accord-ing to Hendrik Lorentz. That is,

( ) = u1 V2. This quantity canalso be calculated from Planck's equa-tion, yielding two terms

(A£2) = uW1 + €„• uV; ^ = hv (1)

(h = Planck's constant)The new linear term has a simple

meaning. If we consider the energyto consist of quanta of magnitudee0 = hv, so that E = eon when nis the number of quanta, the meansquare of fluctuation of the quantumgas becomes exactly equal to the sec-ond term eo'uV according to ele-mentary equations of statistical me-chanics. Equation 1 can therefore bewritten as a formula for the meansquare of fluctuation, (An2), of thenumber of quanta. Calling the aver-age number of quanta (n), we haveVu = c0 (n) from equation 1

(A «2) = {nY + (n) (2)

This result (and not the photoelec-tric effect as has often been main-tained) was the starting point forEinstein's statement that the structure

of light is not completely describedby wave theory, but that there arelight quanta (or photons as we nowcall them).

The most astonishing thing is thatwe need both concepts of light, thecorpuscular and the undulatory, to un-derstand the fluctuation equation 1,which as we point out once again,is absolutely incontestable by the der-ivation from Planck's radiation law.

Thus Lande's thesis that we needonly a wave theory for light, whereaswe must exclusively use a particletheory for "material" particles, wasalready refuted by Einstein's first workon this subject. Light can be de-scribed, as Einstein's considerationshows, neither with waves alone norwith particles alone. It is not bothwaves and particles but has somethingto do with both. We can not under-stand this fact on the level of classicalphysics. Classical wave theory isfound to be only a model for light, andan imperfect one. At first Einsteinand his contemporaries (including

Electrons with momentum p Electrons with wavelength x

1Duane de Broglie

p.,,11,1 to , h . ch.m by ,„,„„„„ AP ^°* t hv f t

ein's theory of

52 « AUGUST 1968 PHYSICS TODAY

X-RAY AND ELECTRON DIFFRACTION in aluminum, showing the similarity in the diffraction pat-terns produced. X-ray photograph at left was made with rays of wavelength 7.1 nanometers; electrondiffraction photograph at right is for 600-eV electrons (equivalent to about 5 ran). Electron patternhas been enlarged X 1.6. Photos by Film Studio, Educational Development Center. —FIG. 2

one of us) did not see that this wasthe case, and even later many of themdid not understand it.

Bose statistics

Similar considerations show that thesituation is the same for matter as forlight. The statistical laws for themotion of molecules have to be alteredwith regard to the quantization rules.Classical Boltzmann statistics are re-placed by Bose statistics, named afterSatyandra Nath Bose. Einstein rec-ognized at once the fundamental im-portance of this change. The fluctua-tions of energy in a Bose gas are onceagain represented by two terms; oneof them corresponds to the behavior ofparticles according to classical me-chanics, the other to the behavior ofclassical waves. We find the sameduality as with light.

Later, Bose statistics were discov-ered to rule only a certain class ofparticles (for example, photons, me-sons and atoms of the normal heliumisotope He4). The other class of ele-mentary particles is governed by dif-ferent statistics, discovered by EnricoFermi and Paul Dirac. Electrons, neu-trons, protons and the rare heliumisotope He3 belong to this class. Forsystems consisting of particles of this

kind we have for the mean squarefluctuation

(Arc2) = - < K ) 2 + <«) (3)

The negative sign of the quadraticterm prevents us from interpreting theequation the same way as we didequation 2. We shall return to Fermistatistics later on.

Bose statistics were discovered al-most at the same time as Louis deBroglie's wave mechanics, which isbased on the idea that, in the theoryof relativity, energy E is not a scalarbut the fourth component of a four-vector, whose other three componentsrepresent the momentum p. There-fore, as well as using Planck's quantumrule for the change in energy of a sys-tem periodic in time with frequencyv, AE = hv, de Broglie postulates thechange of momentum Ap = hk, wherek is the wave number of the frequencyin space (k = I/A; A = wave length).

Duane's formulation

Lande lays much stress on the factthat William Duane formulated themomentum rule before de Broglie,without relativistic foundation, in orderto explain x-ray diffracton in crystalsby means of the corpuscular theory oflight (see figure 1). Lande uses

Duane's formulation only for particles,not for x rays, which are waves classi-cally and must be treated as waves ifone follows Lande.

He tries to explain the interferenceof electrons when passing a crystal lat-tice by a corpuscular theory; that is,he insists that the crystal interferencesof light and electrons are caused byquite different processes. We thinkthis is hardly plausible. Anyone whohas seen Laue and Debye-Scherrerphotographs taken with electrons andwith x rays for the same material (fig-ure 2) knows that they cannot be dis-tinguished from one another withoutdifficulty. We can not see how mech-anisms that are physically totally dif-ferent from each other should produceidentical phenomena. Above all thereis no reason why a unified interpreta-tion that already exists, the quantumtheory as it is generally understood,should be wrong and why it should bereplaced by a new interpretation thatexplains apparently related phenomenain two different ways, and quite dog-matically lays stress on these differ-ences.

Duane's "quantum rule" has hardlybeen noticed, and this is understand-able for the rule is obscure without deBroglie's idea of the correspondence

PHYSICS TODAY • AUGUST 1968 53

Time 0

Time t

MOTION OF A WAVE PACKET in quantum mechanics. The same packet is shown at two instantsof time; originally concentrated, the wave packet is later smeared out in space. —FIG. 3

of particle and wave and without hisproof that the group velocity of a wavepacket coincides with the velocity ofthe corresponding particle. In thesame sense, Planck's quantum ruleAE = hv was obscure at that time.The problem for physicists was to givea meaning to all these rules. It wasnecessary to see that such a meaningwas impossible in the domain of tradi-tional concepts. The first steps toreach the desired aim were matrixmechanics, inspired by Heisenberg andexecuted by him together with Bornand Pascual Jordan, and Dirac's theoryof noncommuting quantities, whichwas also inspired by Heisenberg's ideasbut was independently developed.Both theories were quite abstract, andthe words "particle," "motion," "mo-mentum," etc, were only used symboli-cally. Then Erwin Schrodinger pub-lished his wave mechanics, which wasbased on de Broglie's ideas.

Was Schrodinger an opponent?

Lande quotes Schrodinger, as well asEinstein, as an opponent to the dual-istic theory. It appears to us thatLande also misunderstood Schrod-inger. Schrodinger believed that mat-ter is a wave phenomenon. He de-nied the existence of particles and"quantum jumps" that occur in Bohr'stheory of the electron clouds of atoms,and maintained that the facts ex-plained by this theory could only bedescribed by means of waves and wavepackets. He adhered to this opinionduring his lifetime. Lande, however,writes about "Einstein's and Schrod-

54 • AUGUST 1968 • PHYSICS TODAY

inger's realistic point of view" as ifboth of them had defended the sameposition, opposite to dualism.

In fact dualism is a discovery, notan invention, of Einstein, as we ex-plained above; it was proved by theequations on fluctuations in gases andin radiation and there are no substan-tial arguments against it. Einsteinhimself never tried to deny dualism.It was quite clear for him that onecould not avoid dualism with the usualconcept of particles. He tried to mas-ter the whole question in quite anotherway, namely with his unified fieldtheory, which combined the fields ofgravitation and electromagnetism intoa formal unity. The substrata of thistheory (or, strictly, theories, for Ein-stein laid down several theories dur-ing his lifetime) were fields of a highlyabstract kind, nonsymmetrical tensors,ruled by complicated differential equa-tions. In 1920 he wrote about theseideas:5 "During my spare time Ibrood over the quantum problem fromthe point of view of relativity .But I do not succeed in giving a con-crete shape to my favorite idea—thecomprehension of the quantum struc-ture from an overdetermination by dif-ferential equations."

Singularities from a continuum

Einstein rejected quantum mechanicsfor reasons totally different fromLande's. He never denied the dual-ism of particle and wave, which hehimself had discovered, but pursuedthe idea that particles as singulari-ties could be derived from a contin-

uum theory in which there is an over-determination of the variables. Hedid not succeed in carrying throughthis project.

Lande's remarks give the impressionthat Schrodinger defended Lande'sthesis that matter consists only of par-ticles, light only of waves; the verycontrary, however, was true. Schrod-inger wished to regard both matterand light as waves, and maintainedthat he had found the way back to thegood old classical theory.

Schrodinger's opinion is not valid, ashas often been shown. First, a con-centrated wave packet that, accordingto his concept, is to represent a par-ticle does not hold together but dis-perses as shown in figure 3. Second,his wave function ^ is distributed inthree-dimensional space only for oneparticle. For two particles one needsa function in six dimensional space \?(x-j, t/j, zx; x2, y2, Zo), and for n par-ticles a function in a 3n-dimensionalspace. These functions are not reduc-ible to functions in three-dimensionalspace. Thus the desired approach tothe classical wave theory has becomean illusion.

Space and momentum pictures

The discussion on dualism or nondual-ism appears to be superfluous. SinceEinstein's discovery of the fluctuationequation it has become more and moreobvious that nature can be describednot by particles or waves alone, butby a more sophisticated mathematicaltheory. This is the quantum theory,which supersedes both models and

only in certain limits represents one orthe other. Quantum theory has be-come known to us as a complete wholesince the end of the 1920's. We neednot turn from a particle picture to awave picture arbitrarily, and we neednot be without real comprehensionwhen using it. It is, on the contrary,possible to represent the states of asystem in different ways, and theserepresentations are connected byunique transformations.

Among these representations thereis a space representation from whichone can easily derive the probabilityof finding the particles at a certainpoint in space. There is a momentumrepresentation as well from which onecan easily read off the probability ofthe particles having certain momenta(that is, velocities). As the momen-tum, according to de Broglie, definesthe wavelength one has found thewave properties of the system in thisrepresentation. These two representa-tions correspond respectively to the"particle picture" and the "wave pic-ture;" they date from the period whenquantum theory as a whole had not yetbeen developed. But there is an un-limited number of other representa-tions of the states of a system, for ex-ample the energy representation, fromwhich one can easily calculate the en-ergies of the system. Moreover, allproperties of the system can be calcu-lated in any of these representations(for example, the distribution in spaceof the particles can also be calculatedin the momentum distribution). Thuswith quantum theory one treats allsystems in the same way, whether theyconsist classically of particles or are de-scribed classically by fields (waves).

Bosons and fermions

Much more fundamental than the dis-tinction between particles and wavesis the classification of particles betweenthose that follow Bose statistics (bos-ons) and those that are ruled by Fermistatistics (fermions). Whereas bosonscan be compared to classical particlesand waves, as we have shown above bymeans of their fluctuations, this com-parison is not altogether true for fer-mions, as we have already seen whenlooking at the fluctuation equation 3.

As to Lande's claim to have given anew derivation of quantum mechanicsbased on classical ideas though not assuch taken from classical physics, itmust of course be examined thor-

oughly; Shimony wrote of such an ex-amination in his review.3 We will addonly the following remarks. Landestarts from statistical postulates whentrying to give a new foundation toquantum theory. These postulates areunknown in classical physics, fromwhich Landc takes all his other con-cepts. It appears to lie not very sur-prising that one can derive theoriessimilar to quantum mechanics fromstatistical postulates. Such a deriva-tion can be interesting in itself anddocs not need to be accompanied byattacks on supposed enemies. Thestrangest aspect of Lande's treatmentis his dogmatic use of classical andmacroscopic concepts of particles andwaxes in atomic dimensions and hisrejection of obvious explanations ofsimple experimental results on accountof this dogma or prejudice.

References

1. A. Lande, Foundations of QuantumTheory, Yale Univ. Press, New Haven,L955; From Dualism lo Unity in Quan-tum Physics, Cambridge Univ. Press,London, 1960; New Foundations ofQuantum Mechanics, CambridgeUniv. Press, London, 1965.

2. A. Lancle, Dualismus, Wissenschaftund Hypothese, Heisenbergfestschrift,1961; Philosophia naturalis, Vienna, 8,232 (1964); Brit. I. Phil. Sci., 15, 307(1965); Am. J. Phys., 33, 123 (1965)and 34, 1160 (1966); PHYSICS TODAY,20, in). 2, 55 (1967).

3. A. Shimony, PHYSICS TODAY, 19, no. 9,85, 1966.

4. A. Einstein, Ann. Physik, 17, 132(1905) and 20, 199 (1906); Phys. Z.,

10, 185 (1909).

5. A. Einstein, in a letter to Max Born, 3March 1920. The Einstein-Born cor-respondence is to be published.

DIALOG ON DUALISMReplies by ALFRED LANDE to pointsraised in the preceding article andfurther comments by MAX BORN andWALTER BIEM.

LANDE: Concerning "the historicalorigin of the dualistic interpretation"which I "have not realized:" I knowof course of Einstein's light quanta inopposition to light waves. But I alsoknow that there is a unitary quantumtheory of radiation that has relegatedthe "photon" to the role of a quantumnumber attached to the periodic com-ponents of the continuous Maxwellfield; thereby it has become unneces-sary to attribute various ad hoc in-vented quantities—spin, interdepen-dence of electric and magnetic proper-ties of the photon—in order to save aparticle picture dual to the wave pic-ture of light. Light waves are real,matter waves are imaginary, in morethan one sense.

Duality began to be taken seriouslyonly after the experiment of electrondiffraction seemed to allow no otherexplanation than the assumption thatparticles of matter pass through awave interlude near a crystal or screenwith slits; "An electron spreads outfrom its original size millions of timesto cover both slits; thereafter it inter-

feres with itself." This oddity, to-gether with an associated "new con-ceptual situation" accounting for theunphysical transmutation magic, couldhave been avoided if quantum theo-rists around 1927 had been aware ofthe quantum rule for linear momentum(Duane, 1923). This rule explainsthe electronic diffraction patterns in anatural way as due to the quantizedmomentum activity of the diffractor,including coherence effects as shownin my books and articles. It has beenquite a revelation to many youngerphysicists trained in the dualistic doc-trine. To belittle the quantum rulefor the momentum /) beside those forE and pv as is done by Born, is as un-physicat as if one would belittle themechanical conservation law for p be-side those for E and p# in classicaltheory. One here really must ask;"Why do quantum theorists ignore thequantum theory?" I would be de-lighted to be shown a single place inthe literature on interpretation whereDuane's unitary explanation of diffrac-tion, applied to matter particles, isquoted. Born was one of the few whoknew of Duane's March 1923 paper.

BORN AND BIEM: Every physicistmust accept Duane's rule,8 which de-

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