ambiguity: aspects of the wave–particle duality

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Ambiguity: aspects of the wave–particle duality

BARBARA K. STEPANSKY

The British Journal for the History of Science / Volume 30 / Issue 03 / September 1997, pp 375 385DOI: 10.1017/S0007087497003142, Published online: 30 October 2008

Link to this article: http://journals.cambridge.org/abstract_S0007087497003142

How to cite this article:BARBARA K. STEPANSKY (1997). Ambiguity: aspects of the wave–particle duality. The British Journal for the History of Science, 30, pp 375385 doi:10.1017/S0007087497003142

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BJHS, 1997, 30, 375–85

Ambiguity : aspects of the wave–particle duality

BARBARA K. STEPANSKY*

As has become evident from historical studies, science does not proceed in the coherent and

predictable way that basic science texts would have us believe. I will argue that an excellent

counter-example is an episode from the historical development of quantum mechanics in

which the incompatibility of the particle and the wave representations of the electron and

light were destined to be encompassed by two mathematically equivalent, but conceptually

quite different theories.

I shall argue that the appearance of two such different, yet equivalent, quantum theories

was not surprising at all and I claim even predictable. As Einstein himself wrote in 1909:

‘ It is my opinion that the next phase in the development of theoretical physics will bring

us a theory of light that can be interpreted as a kind of fusion of the wave and the [particle]

theory. ’" Certainly, the interpretative content of Werner Heisenberg’s and Erwin

Schro$ dinger’s theories could not have been more different. By mid-1926, the theoretical

foundations had been laid for a scientific and emotional battleground between the particle

and the wave. I suggest that an important element in the debate was not the incompatibility

itself but actually coming to terms with ambiguity in science. For, in the end, ambiguous

and vague interpretations of the same phenomena became part of science, where science

was supposed to give a clear and unambiguous description of nature.

EARLY THEORIES OF QUANTA AND MATTER WAVES

At first sight, Max Planck’s early results on the quantization of black body radiation may

not seem to stand in direct connection to the subsequent paradox of light and matter.

Planck’s original derivation of his radiation law was grounded in classical thermodynamics.

However, his approach of using an atomic foundation for his theory with the help of

Boltzmann’s statistical entropy conception introduced the universal constant h, which

would in time change the course of theoretical physics. What was regarded as just another

constant to describe finite amounts of energy turned out to be invaluable for research on

energy and radiation and consequently on light itself. The day Planck’s paper, ‘On the law

of the energy distribution in the normal spectrum’,# was read to the German Physical

* Department of Science and Technology Studies, University College London, Gower Street, London WC1E

6BT.

1 A. Einstein, ‘Entwicklung unserer Anschauungen u$ ber das Wesen und die Konstitution der Strahlung’,

Zeitschrift fuX r Physik (1909), 10, 817.

2 M. Planck, ‘U> ber das Gesetz der Energieverteilung im Normalspektrum’, Annalen der Physik (1901), 4,

553–63.

376 Barbara K. Stepansky

Society, 14 December 1900, can legitimately be regarded as the ‘birthday of quantum

theory’.$

Though Albert Einstein did not contribute directly to the formulation of quantum

mechanics and wave mechanics in the mid-1920s, he nevertheless set the essential

foundation for the properties of light based on Planck’s radiation theory. In the first of

three epoch-making papers in 1905 Einstein posed the dazzling ‘heuristic hypothesis ’ that

light can be represented as particles, which he called light quanta.% His basic premise was

essentially of an ‘aesthetic ’ nature,& which he then backed up with arguments from the

entropy of radiation. Among the consequences of light quanta was an explanation for such

phenomena as the photoelectric effect. Einstein considered light quanta to be ‘very

revolutionary’,' as he wrote to his friend Konrad Habicht, something he never said about

special relativity theory. Contemporary scientists rejected Einstein’s theory of light quanta

mainly for the reason that it provided no usual model for interference,( whereas such a

model existed in Maxwell’s theory. For example, after Robert Millikan verified Einstein’s

law for the photoelectric effect eleven years later, he concluded: ‘Einstein’s photoelectric

equation…appears in every case to predict exactly the observed results. Yet the

semicorpuscular theory by which Einstein arrived at this equation seems at present wholly

untenable.’) When Einstein was awarded the Nobel prize in 1921, it was for the law of the

photoelectric effect, not for his theory of light quanta.

Although the scientific community became more aware of light quanta in these ways,

most of them remained sceptical. Niels Bohr claimed that in order to arrive at a

‘contradiction-free description [of atomic processes, one could not make] use of

conceptions borrowed from classical dynamics ’.* He resolved these fundamental problems

with his correspondence principle, which presented atoms as being analogous to harmonic

oscillators. This coupling mechanism permitted his renouncing light quanta. The

correspondence principle was a formalistic solution, creating a linkage between the

microscopic and the macroscopic realms. But it was from light quanta that the apparently

paradoxical behaviour of light emerged, and Bohr’s linkage could not explain it.

Maurice and Louis de Broglie’s research during 1913 and 1921 in the inner orbits of the

3 M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd edn, The History of Modern

Physics, 12, Los Angeles, 1989, 55, example given by Max von Laue in his Memorial Address, delivered at Planck’s

funeral in the Albani Church, Go$ ttingen, on 7 October 1947.

4 A. Einstein, ‘U> ber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen

Gesichtspunkt ’, Annalen der Physik (1905), 17, 132–48.

5 A. I. Miller, Albert Einstein’s Special Theory of Relativity : Emergence (1905) and Early Interpretation

(1905–1911), Reading, MA, 1981, 136.

6 A. Einstein in a letter to K. Habicht, early spring 1905, in C. Seelig, Albert Einstein : Eine dokumentarische

Biographie, Zu$ rich, 1954, 185.

7 In lectures ‘On the Series Spectra of the Elements ’ delivered to the Physical Society in Berlin on 27 April 1920,

Niels Bohr discussed the conceptual problem of light quanta, the ‘ insurmountable difficulties from the point of

view of optical interference’.

8 R. A. Millikan, ‘Einstein’s photoelectric equation and contact electromotive force ’, Physical Review (1916)

7, 18.

9 N. Bohr, ‘On the application of the quantum theory to atomic structure : Part I. Postulates of the theory ’,

Proceedings of the Cambridge Philosophical Society (Supplement) (1924), published in Zeitschrift fuX r Physik

(1923), 13, 117.

Ambiguity 377

atom using X-rays and gamma rays introduced, as Bruce Wheaton has found, ‘corpuscular

spectra as a means to analyse the atom’."! Their studies on X-ray absorption in atoms and

the photoelectric effect led de Broglie to explore light quanta further. He became intrigued

by the paradoxical nature of light because he did not believe that a quantum theory could

consist of a corpuscular interpretation only. He arrived at the educated guess of using

relativity to relate periodicity and locality of an electron in his famous equation p¯h}λ,

where p is the electron’s momentum and λ its wavelength. Momentum denotes locality or

discreteness, whereas wavelength stands for periodicity or continuity. De Broglie developed

this theory during 1922 and 1924. Erwin Schro$ dinger ‘derived most benefit from his study

of de Broglie’s thesis ’,"" in particular for the discovery of the electron wave equation.

The so-called de Broglie waves can be used to provide ‘confirmation of the existence of

matter waves ’,"# where a wave at an incoming angle is diffracted at a plane of atoms and

scattered at a whole number of wavelengths. As Martin J. Klein has put it, ‘where Einstein

assigned particle properties to radiation, de Broglie assigned wave properties to matter ’."$

By 1923, however, Arthur H. Compton’s scattering experiments with crystals seemed to

indicate a corpuscular property of light. In Compton’s work scattering X-rays collide with

an electron. The electron is scattered with a certain momentum at a certain angle, similarly

the photon is scattered with a changed momentum and a changed beam direction.

Compton’s experiments could be regarded as evidence for light quanta. It was Compton’s

decision to redirect his X-ray data into investigating the light quantum hypothesis, which

at the time was regarded with mixed emotions. In subsequent experiments Compton

showed that light quanta were indivisible.

A paper by Bohr, Hendrik Kramers and John C. Slater in 1924 set out to reconcile the

contradictory mechanism that governed the relationship between matter and radiation in

general and the Compton effect in particular."% Slater’s suggestion of a radiation theory

without the notion of ‘ little lumps carried along on the waves ’"& was enormously appealing

to Bohr and Kramers. So the Bohr, Kramers and Slater (BKS) paper was completed and sent

for publication exactly one month after Slater’s arrival in Copenhagen on 21 December

1923. The BKS theory succeeded in establishing a framework from which quanta were

eliminated. The price that was paid was to relegate conservation of energy and momentum

in individual events to a statistical law.

The lack of a visual model for interference led, as Einstein wrote in 1909,"' to a clash

between a wave and particle mode of light. Owen W. Richardson expressed the paradox

of the situation in the following way: ‘ It is difficult, in fact it is not too much to say that

at present it appears to be impossible, to reconcile the divergent claims of the photoelectric

10 B. R. Wheaton, The Tiger and the Shark – Empirical Roots of Wave–Particle Dualism, Cambridge, 1983,

269.

11 M. Klein, ‘Einstein and the wave–particle duality ’, Natural Philosophy (1964), 3, 40.

12 A. I. M. Rae, Quantum Mechanics, 3rd edn, Bristol, 1993, 7.

13 Klein, op. cit. (11), 32.

14 N. Bohr, H. Kramers and J. C. Slater, ‘U> ber die Quantentheorie der Strahlung’, Zeitschrift fuX r Physik

(1924), 24, 69–87.

15 Letter from Slater to his parents, 18 January 1924.

16 Einstein, op. cit. (1), 820.


and interference groups of phenomena. The same energy of the radiation behaves as

though it possessed at the same time the opposite properties of extension and localisation.’"(

Let us pause for a moment to consider one of the experimental aspects of the wave–

particle duality. Although today we have clear experimental evidence for the existence of

light quanta, no one claims to understand completely the paradoxical behaviour of light.

Indeed, the coexistence of the particle and the wave caused much discomfort even for the

most confident of physicists. An often quoted example is the double-slit experiment that

Richard Feynman makes use of in his book The Character of Physical Law.") Feynman

draws an analogy between the particles as bullets and the waves as water. He interprets

the pattern made by bullets on the other side of the double slit as a distribution curve,

whereas the water waves emerge as secondary wavelets that create a distinctive interference

pattern."*

Yet something entirely counter-intuitive happens when we repeat the same experiments

with electrons or light quanta, because an interference pattern appears of the same sort as

for waves. How could this possibly be? Is it not the case that electrons and light quanta

are indivisible? Must we nevertheless assume that the light quantum or electron has

somehow to go through both slits in order to make an interference pattern possible? To

be absolutely certain which slit the light quantum goes through means doing an experiment

to detect the entity’s path. This, however, results in the loss of the interference pattern.

What can we conclude from this experiment? Shall we suppose that owing to the results

of the double-slit experiments light has to be regarded as a continuous wave phenomenon?

Would it be possible to regard its particle properties as mere side-effects of the

experimental situation? The answer must be ‘no’. No intrinsic property of any entity could

be reduced to a side-effect.

MATRIX MECHANICS AND WAVE MECHANICS

As I have shown above, two strands of theoretical and experimental physics had developed

step by step, leaving no doubt that the light quantum would not disappear if you ignored

it, nor would waves be completely replaced by particles. As the plot thickens around the

particle and the wave, let us now consider the two atomic theories that arose in 1925 and

1926.

In 1925, while obtaining a postdoctoral research post in Go$ ttingen, Werner Heisenberg

published his paper on a theory of quantum mechanics with the title ‘On a quantum-

theoretical reinterpretation of kinematic and mechanical relations ’.#! He wanted to

establish a theory where all quantities and the relations between them were, as he wrote,

in principle ‘observable ’#" Heisenberg’s notion of observability was based on what he

17 O. W. Richardson, The Electron Theory of Matter, Cambridge, 1916, 62.

18 R. P. Feynman, The Character of Physical Law, Harmondsworth, 1992, 127–48.

19 For numerous examples see D. Bohm, Quantum Theory, New York, 1979, especially 99–115.

20 W. Heisenberg, ‘U> ber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen’,

Zeitschrift fuX r Physik (1925), translated by B. L. van der Waerden in Sources of Quantum Mechanics (ed. B. L.

van der Waerden), New York, 1967.

21 Heisenberg, op. cit. (20), 261.

Ambiguity 379

thought had been Einstein’s philosophy for developing special relativity theory. However,

Einstein changed his mind about this philosophy by 1926. I will come back to this point

when I discuss ambiguity in the quantum domain.

In his paper, Heisenberg compared classical physics with quantum-theoretical physics

and arrived at a new multiplication law that could determine transition probabilities and

the kinematic measurement of quantum-theoretical quantities. For the description of a

state of a mechanical system, he used only observable amplitudes and fluctuation

frequencies, which he represented by a new mathematics that obeyed matrix algebra.

In collaboration with Max Born and Pascuale Jordan, Heisenberg further developed the

mathematical foundations of matrix mechanics. Go$ ttingen was the mathematical centre of

German science at the time, and it was here that GoX ttinger quantum mechanics originated.

In October 1925 Born and Jordan were working in collaboration with Heisenberg on the

so-called three-man paper in Go$ ttingen. The paper was received by Zeitschrift fuX r Physik

on 16 November of the same year but was not published until February 1926.## Apparently,

Heisenberg did not think much of the name matrix mechanics, as it sounded more

mathematical than physical. Indeed, its physical interpretation was far from obvious

because kinematical quantities like position and momentum became operators satisfying

the mathematics of matrices.

Although matrix mechanics looked extremely promising, its difficulties in calculation

were formidable. In addition to this, Erwin Schro$ dinger disliked its intrinsic discontinuities

and felt ‘ repelled…by the lack of visualisability ’ in matrix mechanics.#$ He aspired to

bring back classical theory into the quantum domain, suitably redefined. In his opinion,

continuity could be obtained only in a wave picture. In March 1926, Schro$ dinger published

the first of four communications for his wave mechanics under the title ‘Quantisation as

an eigenvalue problem’.#% His approach was to express quantum mechanics by a wave

function. In a sequel comparing the Heisenberg–Born–Jordan quantum mechanics to his

own semi-classical version Schro$ dinger writes that he simply replaced the ‘system of an

infinite number of algebraic equations ’#& with a single differential equation. In this way, he

could account for phenomena much more expediently and within a more agreeable

conceptual framework.

In §5 of his paper Schro$ dinger draws a comparison between the two conflicting theories

and considers the advantages of concentrating only on observable quantities. Though he

admits that a lack of visualizability keeps science from proposing misleading space-time

pictures, he says : ‘ It appears to me to be extraordinarily difficult to attack problems of the

kind sketched above, as long as one feels due to our theory of knowledge, obliged in atom

dynamics to suppress intuition and to operate only with abstract concepts such as

transition probabilities, energy levels, and the like.’#' Schro$ dinger did not hide his joy

22 W. Heisenberg, M. Born and P. Jordan, ‘Zur Quantenmechanik II ’, Zeitschrift fuX r Physik (1926), 35,

557–615.

23 E. Schro$ dinger, ‘On the relationship of the Heisenberg–Born–Jordan quantum mechanics to mine’,

translated in Wave Mechanics (ed. G. Ludwig), 1968, 128.

24 E. Schro$ dinger, ‘Quantisierung als Eigenwertproblem’, Annalen der Physik (1926), 79, 361–75.

25 Schro$ dinger, op. cit. (23), 128.



about wave mechanics, and about the ‘human advance of all these things ’.#( This implies

his low regard for Heisenberg’s quantum mechanics.

The wave–particle duality forced the scientific community to face two versions of

quantum mechanics, which were, as Schro$ dinger himself wrote, mathematically but not

physically equivalent : ‘This thesis that mathematical equivalence is the same thing as

physical equivalence, can be assigned only limited validity.’#) He said ‘ that the complete

parallelism of the two new quantum theories will remain valid…can hardly be doubted’.#*

The essential difference, though, that led to a preference for his wave mechanics among

physicists was that its mathematical apparatus was ‘ familiar…and restore[d] customary

intuitions ’.$! Matrix mechanics on the other hand lacked this familiarity. When Heisenberg

had the opportunity to go to Schro$ dinger’s lectures in Munich in 1926, he opposed

Schro$ dinger’s interpretations so vehemently that the eminent Wilhelm Wien nearly kicked

Heisenberg out of the lecture theatre.$"

Heisenberg considered his mathematical approach was free of the limitations of mental

pictures that can describe only daily life. He held that ‘ fortunately…it has been possible

to invent a mathematical scheme – the quantum theory – which seems entirely adequate

for the treatment of atomic processes ’.$# In a series of letters to his friend and colleague

Wolfgang Pauli in Hamburg, Heisenberg continued to complain about Schro$ dinger’s

theory. He writes on the 8 June 1926: ‘The more I think about the physical part of

Schro$ dinger’s theory, the more disgusting I find it…in other words, I think it’s rubbish.’$$

Over a month later Heisenberg again writes to Pauli on the 28 July 1926: ‘As nice as

Schro$ dinger is in person, as strange I find his physics : one feels 26 years younger listening

to him. Schro$ dinger throws everything ‘‘quantum theoretical ’’, such as photoelectric

effect, Frank’s collisions, Stern–Gerlach effect etc. overboard, then it’s not difficult to make

up a theory.’$%

Schro$ dinger’s success enraged Heisenberg. He was disturbed about ‘ the confusion which

[he] believed would burden atomic theory’.$& In an article entitled ‘Fluctuation phenomena

and quantum mechanics ’.$' Heisenberg presented arguments against ‘a continuous

interpretation of the quantum mechanical formalism’.$( This continuous interpretation

could obviously not account for the nature of the discontinuous relationships in atomic




30 A. I. Miller, ‘ Imagery, probability and the roots of Werner Heisenberg’s uncertainty principle paper ’, in

Sixty-Two Years of Uncertainty (ed. A. I. Miller), New York, 1990, 7.

31 Interview with Werner Heisenberg for the ‘History of Quantum Physics ’. Wolfgang Pauli, Wissen-

shaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg a.o. (ed. A. Hermann, K. V. Meyenn and V. F. Weisskopf),

2 vols., New York, 1979, i, 336.

32 W. Heisenberg, The Physical Principles of the Quantum Theory, New York, 1930, 11.

33 Heisenberg in a letter to Pauli on 8 June 1926, from Pauli, op. cit. (31), 328.

34 Heisenberg in a letter to Pauli on 28 July 1926, from Pauli, op. cit. (31), 337–8.

35 ‘Commentary of Heisenberg (1967) ’ in Quantum Theory and Measurement (ed. J. A. Wheeler and W. H.

Zurek), Princeton, 1983, 56.

36 W. Heisenberg, ‘Schwankungserscheinungen und Quantenmechanik’, Zeitschrift fuX r Physik (1926), 40,

501–6.


Ambiguity 381

phenomena. He closes with an expression that seems counter-intuitive on first sight : ‘after

these calculations the fact of harmonious discontinuities is encompassed in the

mathematical scheme of quantum mechanics ’.$) The counter-intuitive nature emerges

when one considers that discontinuities could be regarded as harmonious in the first place

in any theoretical framework. One has to understand Heisenberg’s philosophy in order to

exchange the notion of counter-intuition with normality. For Heisenberg and Bohr,

discontinuities were intrinsic to quantum phenomena. Without discontinuities, there

would not be a quantum domain. Discontinuities were in this sense part of a scenario,

without which there would not have been a scenario in the first place. This is why

Heisenberg thought of his quantum mechanics as complete. There could be nothing added

to what he could observe.

Nevertheless, physicists did not share Heisenberg’s impulses, so it seems, for longer than

was absolutely necessary. Since the two theories were mathematically equivalent, what

kept anyone from using wave mechanics exclusively? This was very upsetting to

Heisenberg, who proved with the help of Bohr that Schro$ dinger’s wave mechanics could

not even explain Planck’s radiation law.$* Schro$ dinger, having accepted an invitation by

Bohr to Copenhagen in the same year as the publication of his wave mechanics, had to

endure long and tiring discussions with Bohr about the insufficiency of the wave picture.%!

Schro$ dinger nevertheless remained highly opposed to discontinuity and imprecision in

physical science. In particular, he stressed the absurdity of quantum jumps and in fact

expressed regrets about ever having been involved in quantum theory, whereupon Bohr

praised Schro$ dinger’s great promotion of quantum theory and thanked him.

Physicists who supported Heisenberg’s discontinuous quantum theory referred to

Schro$ dinger’s continuous wave mechanics as ‘ZuX richer Aberglaube ’ (‘Zurich’s super-

stition’), a term introduced by Pauli in September 1926. He was thinking of the time when

Schro$ dinger was in Copenhagen and so firmly clung to his side of quantum-theoretical

interpretation. Pauli wrote to Schro$ dinger on 22 November 1926 to apologize for this

banter because ‘ it was not meant to be personal hostility, but an expression of the factual

conviction that quantum phenomena show features in nature which cannot alone be

encompassed with terms of continuum physics (field physics) ’.%" Pauli admitted that he did

not like these features any more, but had been forced to put up with them: ‘Do not think

that this conviction makes life easier, I have been very troubled by it, and will probably

have to be in future time! ’%#

In fact in 1929 John von Neumann developed a quantum mechanics ‘as a calculus of

Hermitian operators in Hilbert space’.%$ He was able to show that Heisenberg’s and

Schro$ dinger’s theories are the only representative parts of this calculus. This refutes the

anti-historical present-day notion that the particle and the wave were two different

paradigms. I disagree with the assumption that the wave–particle duality implies


39 W. Heisenberg, ‘Quantum theory and its interpretation’, in Niels Bohr: His Life and Work as Seen by his

Friends and Colleagues (ed. S. Rozental), New York, 1967, 101.


41 Pauli in a letter to Schro$ dinger on 22 November 1926, from Pauli, op. cit. (31), 357.

42 Pauli, op. cit. (31), 357, italics in original.

43 M. Jammer, Philosophy of Quantum Mechanics, New York, 1974, 22.


incommensurability of two theories. This would indicate that the scientists of the 1920s

who were arguing over and improving their understanding of quantum physics were

actually not able to communicate this reciprocal understanding. Such an assumption is

highly implausible considering their achievements that stem from communication and

exchange.

UNCERTAINTY

It is my contention that up to this point in 1926 the development of the two different

quantum theories was pretty much as could be expected. Did Einstein, for one, as I pointed

out at the beginning, not anticipate such a development in 1909? Since the seventeenth

century the scientific community had based much of its reasoning on there being two

distinct phenomena, the particle and the wave. Heisenberg’s matrix mechanics on the one

hand, and Schro$ dinger’s wave mechanics on the other, set a mark on formally separating

the particle properties from the wave properties in atomic entities. In my view it is only

after 1926, when attempts were made to free quantum physics from such a separation, that

things become much more conceptually disordered. By the end of 1926 Heisenberg

developed his well-known uncertainty principle based on the corpuscular properties of

quantum phenomena only, and again his theory collided with other physicists’ imagination.

In the first part of his uncertainty paper, Heisenberg argues that position, path, velocity

and energy have been defined by classical measurements and are therefore classical

concepts. However, the same concepts in the world of atoms are defined with the notion

of discontinuity and uncertainty. As Heisenberg writes : ‘We therefore have good reason

to be suspicious about an uncritical application of those words ‘‘position’’ and

‘‘velocity ’’.’%% As a consequence, ordinary language has to be redefined for the quantum

domain.

In the paper the formal basis for transforming between wave and particle is the

Dirac–Jordan transformation theory. In order to prepare the reader for the formal

derivation of the uncertainty principle using Dirac–Jordan transformations, Heisenberg

offers the following thought experiment. An electron is observed under a gamma-ray

microscope in order to determine its exact position. But when a light quantum hits the

electron, the Compton effect is automatically involved, as the light quantum inevitably

‘changes [the electron’s] momentum discontinuously ’.%& To determine the electron’s

position as exactly as possible one needs light of very small wavelength. The smaller the

wavelength becomes, the energy of the light quantum increases and causes a significant

change in the electron’s momentum. Heisenberg’s conclusion about these reciprocal

relations was to be the core of his uncertainty principle : the more accurate the position

determination, the less accurate the momentum determination and vice versa.

Bohr found a fundamental error in this gamma-ray microscope experiment. He pointed

out to Heisenberg that with the help of the Compton effect, one could indeed know the

44 W. Heisenberg, ‘The physical content of quantum kinematics and mechanics ’, in Wheeler and Zurek, op.

cit. (35), 70. (Originally published as ‘U> ber den aunschaulichen Inhalt der quantentheoretischen Kinematik und

Mechanik’, Zeitschrift fuX r Physik (1927), 43, 173.)

45 Jammer, op. cit. (43), 63.

Ambiguity 383

position and momentum very exactly, that is, more exactly than ∆p ∆qCh, where ∆p is

the error in determining momentum and ∆q the error in determining position, if one knows

the direction of the incoming and the reflected light quanta. In reality this is not possible,

because of the limits of the resolving power of the microscope, which cannot give the exact

result for the wavelength. As Heisenberg wrote to Pauli on 16 May 1927: ‘So the relation

∆p∆qCh certainly does result from this, but not exactly as I have thought.’%'

Bohr agreed with Heisenberg’s uncertainty relations, but strongly disagreed on

Heisenberg’s sole use of particle properties. The inabilities and limitations that

Heisenberg’s paper implied were, for Bohr, an indication of the impossibility of using both

modes of expression simultaneously. Bohr did not blame intrinsic discontinuities for the

uncertainty of experimental results but the wave–particle duality. When in early spring of

1927 Heisenberg presented Bohr with the draft of the uncertainty paper that was to go into

print very shortly, Bohr refused to give his blessing. Bohr agreed with Heisenberg’s

conclusions, but could not agree ‘with the general trend of its reasoning’.%(

As Heisenberg’s reasoning was based exclusively on particle properties of the quantum

system, he and Bohr were soon locked in a dispute over the conceptual basis of uncertainty.

Bohr interpreted the wave–particle duality as two mutually exclusive and complementary

modes of the same phenomenon. Bohr’s notion of complementary threatened everything

to which Heisenberg was committed. Bohr claimed as inevitable what most were afraid to

accept : the atomic entity is neither a wave nor a particle, it is both together. If the

experimental apparatus for atomic entities is one for waves, they will act like a wave. If

the apparatus is set up for particles, the atomic entities will act like particles. It is not

possible for these entities to be observed as both a wave and a particle in a single

experiment.

Where Bohr on the one hand ascribed the inability of a precise measurement not to a

simple intrinsic limitation, but to the fact that one could not observe both wave and

particle modes simultaneously, a limitation rooted in the wave–particle duality, Heisenberg

interpreted his results as a limitation on our ability to know the two variables of a quantum

system with classical accuracy. It was this fundamental consequence of the uncertainty

paper that caused these heated debates – during one of which Heisenberg recalls he broke

down in tears.%) What Pauli considered merely a dispute about the Rangordnung der

Begriffe (precedence-order of concepts),%* Heisenberg found quite impossible to accept. If

Bohr stated that it was possible to use classical theories as analogies for atomic

phenomena, then he implied that there was a procedure to measure and to define these

concepts. Heisenberg, however, had shown that such a simultaneous measurement was

impossible and therefore former concepts had to be redefined for atomic phenomena. He

was convinced that quantum theory was completed with the establishment of the

uncertainty principle. Using the Dirac–Jordan transformation theory, it was possible to

assert the inevitability and irrefutability of all uncertainty relations.

46 Heisenberg in a letter to Pauli on 16 May 1927, from Pauli, op. cit. (31), 394.

47 Jammer, op. cit. (43), 65.

48 Heisenberg to his parents, 16 May 1927. D. C. Cassidy, Uncertainty – The Life and Science of Werner

Heisenberg, New York, 1992, 242.

49 Jammer, op. cit. (43), 68.


When a compromise was finally reached between Heisenberg’s limitations and Bohr’s

wave–particle interpretations, Heisenberg published the uncertainty paper in 1927 with a

‘Note added in proof ’, in which he admitted the mistake of ascribing the origin of

uncertainty solely to the occurrence of discontinuities.&! Heisenberg thanked Bohr for the

discussion on the conceptual structure of quantum theory and announced that Bohr

himself would soon publish a paper on his investigations about the nature of uncertainty.

The wave–particle duality and Bohr’s complementarity notion that he developed into a

general principle are the essence of the Copenhagen spirit.&"

AMBIGUITY

Ambiguity is defined in any dictionary as an expression for the fact that a word can have

two or more different meanings. The meaning becomes clear from the context in which the

word is used. This connects to the line of argument I have given here. From the

phenomenon of light and electrons, two different concepts have originated from experiment

and theory, the particle and the wave. Bohr’s complementarity implies that the

wave–particle duality can be explained from the experimental context. Experimental

results depend on either the particle or the wave theory context, and cannot be explained

by both contexts at the same time. However, the dictionary also says that in formal

languages such as in mathematics, logic and informatics, ambiguities have to be avoided.

What do you do if such an ambiguity, as in the case of the wave–particle duality, becomes

unavoidable?

One reply to this question might run as follows. Ambiguity of phenomena resulted in

ambiguous interpretations, which proved to be the basis for any further unification. Thus

ambiguity proved to be more fruitful for science than forthcoming unification could ever

be. There was a fundamental shift between the years of quantum and wave mechanics in

1925}26 and the year of the uncertainty principle and complementarity in 1927. It seems

that ambiguity functioned as a creative catalyst that converted the ambiguous interpretative

contexts of quantum and wave mechanics into quantum physics as we know it today.

Another example of the way ambiguity directed theoretical research can be found in

Einstein’s philosophy of scientific theories. When in Berlin in early 1926 Heisenberg had the

opportunity to make Einstein’s personal acquaintance, Heisenberg found to his surprise

that Einstein had revised his commitment to basing theories only on observable quantities.

Einstein was now clearly opposed to theories that employ only observable quantities.

Einstein argued, as Heisenberg recalls, that ‘every observation…presupposes that there is

an unambiguous connection known to us, between the phenomenon to be observed and the

sensation which eventually penetrates our consciousness ’.&# As is the case in modern

atomic physics, the laws by which this connection is determined are called into question;


51 As Copenhagen was the centre of theoretical physics with Bohr as the founder of the Copenhagen Institute,

Bohr’s complementarity principle was the interpretative context of quantum mechanics. It is regarded as part of

the Copenhagen Interpretation, which soon became an orthodox way of not questioning the strangeness behind

phenomena such as the wave–particle duality, but accepting it.

52 W. Heisenberg, Encounters with Einstein, Princeton, 1989, 114.

Ambiguity 385

Einstein concludes that observation loses its meaning. Since this means that it is the theory

that determines what can be observed, we can see clearly that these ambiguous connections

between phenomenon and observation that Einstein implies were indeed encompassed by

two different theories. Heisenberg’s matrix mechanics determined the particle point of

view of the atomic quantities, whereas Schro$ dinger’s wave mechanics established the wave

point of view.

I feel, and I am sure most physicists will agree, that the description of the world around

us seems to become more and more intuitively abstract. The wave–particle duality caused

much tension during the period 1925–27. This tension originated in the ambiguity of

interpreting atomic phenomena. The development of two different atomic theories is

logical in so far as it stems from the nature of atomic phenomena. In 1925 Heisenberg

took up the strand of corpuscular properties of light and radiation, whereas Schro$ dinger

in 1926 concentrated on the wave properties. How shall this be regarded as surprising?

Today, there is little interest among physicists in reviving the tension that ambiguous

phenomena caused in the mid-1920s. The wave–particle duality has been accepted as a part

of quantum mechanics that is set within the context of complementarity and referred to as

the Copenhagen Interpretation. Equations such as Einstein’s E¯hν for quanta (where E

stands for energy, and ν for frequency) and de Broglie’s p¯h}λ are nowadays taught in

the very first hour of a quantum physics class. Very rarely does someone wonder why and

how a corpuscular property, here energy and momentum, can be equated with a wave

property, here frequency and wavelength. At last the wave and the particle seem to be

harmoniously reconciled in abstract mathematical equations, and by the universal constant

h. We know today that wherever h is employed in a quantum-mechanical equation, one

can be sure of dealing with an atomic world where not everything is quite as it seems.

ambiguity: aspects of the wave–particle duality

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