Quantum mechanics is generally regarded as the physical theory that isour best candidate for a fundamental and universal description of thephysical world. The conceptual framework employed by this theorydiffers drastically from that of classical physics. Indeed, thetransition from classical to quantum physics marks a genuine revolutionin our understanding of the physical world.

One striking aspect of the difference between classical and quantumphysics is that whereas classical mechanics presupposes that exactsimultaneous values can be assigned to all physical quantities, quantummechanics denies this possibility, the prime example being the positionand momentum of a particle. According to quantum mechanics, the moreprecisely the position (momentum) of a particle is given, the lessprecisely can one say what its momentum (position) is. This is (asimplistic and preliminary formulation of) the quantum mechanicaluncertainty principle for position and momentum. The uncertaintyprinciple played an important role in many discussions on thephilosophical implications of quantum mechanics, in particular indiscussions on the consistency of the so-called Copenhageninterpretation, the interpretation endorsed by the founding fathersHeisenberg and Bohr.

This should not suggest that the uncertainty principle is the onlyaspect of the conceptual difference between classical and quantumphysics: the implications of quantum mechanics for notions as(non)-locality, entanglement and identity play no less havoc withclassical intuitions.

• 1. Introduction
• 2. Heisenberg
  • 2.1 Heisenberg's road to the uncertainty relations
  • 2.2 Heisenberg's argument
  • 2.3 The interpretation of Heisenberg's relation
  • 2.4 Uncertainty relations or uncertainty principle?
  • 2.5 Mathematical elaboration
• 3. Bohr
  • 3.1 From wave-particle duality to complementarity
  • 3.2 Bohr's view on the uncertainty relations
• 4. The Minimal Interpretation
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

The uncertainty principle is certainly one of the most famous andimportant aspects of quantum mechanics. It has often been regarded asthe most distinctive feature in which quantum mechanics differs fromclassical theories of the physical world. Roughly speaking, theuncertainty principle (for position and momentum) states that onecannot assign exact simultaneous values to the position and momentum ofa physical system. Rather, these quantities can only be determined withsome characteristic ‘uncertainties’ that cannot becomearbitrarily small simultaneously. But what is the exact meaning of thisprinciple, and indeed, is it really a principle of quantum mechanics?(In his original work, Heisenberg only speaks of uncertaintyrelations.) And, in particular, what does it mean to say that aquantity is determined only up to some uncertainty? These are the mainquestions we will explore in the following, focusing on the views ofHeisenberg and Bohr.

The notion of ‘uncertainty’ occurs in several differentmeanings in the physical literature. It may refer to a lack ofknowledge of a quantity by an observer, or to the experimentalinaccuracy with which a quantity is measured, or to some ambiguity inthe definition of a quantity, or to a statistical spread in an ensembleof similary prepared systems. Also, several different names are usedfor such uncertainties: inaccuracy, spread, imprecision,indefiniteness, indeterminateness, indeterminacy, latitude, etc. As weshall see, even Heisenberg and Bohr did not decide on a singleterminology for quantum mechanical uncertainties. Forestalling adiscussion about which name is the most appropriate one in quantummechanics, we use the name ‘uncertainty principle’ simplybecause it is the most common one in the literature.

2. Heisenberg

2.1 Heisenberg's road to the uncertainty relations

Heisenberg introduced his now famous relations in an article of1927, entitled "Ueber den anschaulichen Inhalt derquantentheoretischen Kinematik und Mechanik". A (partial)translation of this title is: "On theanschaulich content ofquantum theoretical kinematics and mechanics". Here, the termanschaulich is particularly notable. Apparently, it is one ofthose German words that defy an unambiguous translation into otherlanguages. Heisenberg's title is translated as "On the physicalcontent …" by Wheeler and Zurek (1983). His collected works(Heisenberg, 1984) translate it as "On the perceptible content…", while Cassidy's biography of Heisenberg (Cassidy, 1992),refers to the paper as "On the perceptual content …".Literally, the closest translation of the termanschaulich is‘visualizable’. But, as in most languages, words that makereference to vision are not always intended literally. Seeing is widelyused as a metaphor for understanding, especially for immediateunderstanding. Hence,anschaulich also means‘intelligible’ or ‘intuitive’.^[1]

Why was this issue of theAnschaulichkeit of quantummechanics such a prominent concern to Heisenberg? This question hasalready been considered by a number of commentators (Jammer, 1977;Miller 1982; de Regt, 1997; Beller, 1999). For the answer, it turnsout, we must go back a little in time. In 1925 Heisenberg had developedthe first coherent mathematical formalism for quantumtheory (Heisenberg, 1925). His leading idea was that only thosequantities that are in principle observable should play a role in thetheory, and that all attempts to form a picture of what goes on insidethe atom should be avoided. In atomic physics the observational datawere obtained from spectroscopy and associated with atomic transitions.Thus, Heisenberg was led to consider the ‘transitionquantities’ as the basic ingredients of the theory. Max Born,later that year, realized that the transition quantities obeyed therules of matrix calculus, a branch of mathematics that was not sowell-known then as it is now. In a famous series of papers Heisenberg,Born and Jordan developed this idea into the matrix mechanics versionof quantum theory.

Formally, matrix mechanics remains close to classical mechanics. Thecentral idea is that all physical quantities must be represented byinfinite self-adjoint matrices (later identified with operators on aHilbert space). It is postulated that the matricesq andprepresenting the canonical position and momentum variables of aparticle satisfy the so-called canonical commutation rule

qp −pq =iℏ

(1)

where ℏ =h/2π,h denotes Planck's constant, and boldface type is used torepresent matrices. The new theory scored spectacular empirical successby encompassing nearly all spectroscopic data known at the time,especially after the concept of the electron spin was included in thetheoretical framework.

It came as a big surprise, therefore, when one year later, ErwinSchrödinger presented an alternative theory, that became known aswave mechanics. Schrödinger assumed that an electron in an atomcould be represented as an oscillating charge cloud, evolvingcontinuously in space and time according to a wave equation. Thediscrete frequencies in the atomic spectra were not due todiscontinuous transitions (quantum jumps) as in matrix mechanics, butto a resonance phenomenon. Schrödinger also showed that the twotheories were equivalent.^[2]

Even so, the two approaches differed greatly in interpretation andspirit. Whereas Heisenberg eschewed the use of visualizable pictures,and accepted discontinuous transitions as a primitive notion,Schrödinger claimed as an advantage of his theory that it wasanschaulich. In Schrödinger's vocabulary, this meant thatthe theory represented the observational data by means of continuouslyevolving causal processes in space and time. He considered thiscondition ofAnschaulichkeit to be an essential requirement onany acceptable physical theory. Schrödinger was not alone inappreciating this aspect of his theory. Many other leading physicistswere attracted to wave mechanics for the same reason. For a while, in1926, before it emerged that wave mechanics had serious problems of itsown, Schrödinger's approach seemed to gather more support in thephysics community than matrix mechanics.

Understandably, Heisenberg was unhappy about this development. In aletter of 8 June 1926 to Pauli he confessed that "The more I thinkabout the physical part of Schrödinger's theory, the moredisgusting I find it", and: "What Schrödinger writes about theAnschaulichkeit of his theory, … I considerMist (Pauli, 1979, p. 328)". Again, this last German term istranslated differently by various commentators: as "junk" (Miller,1982) "rubbish" (Beller 1999) "crap" (Cassidy, 1992), and perhaps moreliterally, as "bullshit" (de Regt, 1997). Nevertheless, in publishedwritings, Heisenberg voiced a more balanced opinion. In a paper inDie Naturwissenschaften (1926) he summarized the peculiarsituation that the simultaneous development of two competing theorieshad brought about. Although he argued that Schrödinger'sinterpretation was untenable, he admitted that matrix mechanics did notprovide theAnschaulichkeit which made wave mechanics soattractive. He concluded: "to obtain a contradiction-freeanschaulich interpretation, we still lack some essentialfeature in our image of the structure of matter". The purpose of his1927 paper was to provide exactly this lacking feature.

2.2 Heisenberg's argument

Let us now look at the argument that led Heisenberg to hisuncertainty relations. He started by redefining the notion ofAnschaulichkeit. Whereas Schrödinger associated this termwith the provision of a causal space-time picture of the phenomena,Heisenberg, by contrast, declared:

We believe we have gainedanschaulichunderstanding of a physical theory, if in all simple cases, we cangrasp the experimental consequences qualitatively and see that thetheory does not lead to any contradictions. Heisenberg, 1927, p.172)

His goal was, of course, to show that, in this new sense of theword, matrix mechanics could lay the same claim toAnschaulichkeit as wave mechanics.

To do this, he adopted an operational assumption: terms like‘the position of a particle’ have meaning only if onespecifies a suitable experiment by which ‘the position of aparticle’ can be measured. We will call this assumption the‘measurement=meaning principle’. In general, there is nolack of such experiments, even in the domain of atomic physics.However, experiments are never completely accurate. We should beprepared to accept, therefore, that in general the meaning of thesequantities is also determined only up to some characteristicinaccuracy.

As an example, he considered the measurement of the position of anelectron by a microscope. The accuracy of such a measurement is limitedby the wave length of the light illuminating the electron. Thus, it ispossible, in principle, to make such a position measurement as accurateas one wishes, by using light of a very short wave length, e.g.,γ-rays. But for γ-rays, the Compton effect cannot beignored: the interaction of the electron and the illuminating lightshould then be considered as a collision of at least one photon withthe electron. In such a collision, the electron suffers a recoil whichdisturbs its momentum. Moreover, the shorter the wave length, thelarger is this change in momentum. Thus, at the moment when theposition of the particle is accurately known, Heisenberg argued, itsmomentum cannot be accurately known:

At the instant of time when the position is determined,that is, at the instant when the photon is scattered by the electron,the electron undergoes a discontinuous change in momentum. This changeis the greater the smaller the wavelength of the light employed, i.e.,the more exact the determination of the position. At the instant atwhich the position of the electron is known, its momentum therefore canbe known only up to magnitudes which correspond to that discontinuouschange; thus, the more precisely the position is determined, the lessprecisely the momentum is known, and conversely (Heisenberg, 1927, p.174-5).

This is the first formulation of the uncertainty principle. In itspresent form it is an epistemological principle, since it limits whatwe canknow about the electron. From "elementary formulae ofthe Compton effect" Heisenberg estimated the ‘imprecisions’to be of the order

δpδq ∼h

(2)

He continued: “In this circumstance we see the directanschaulich content of the relationqp −pq=iℏ.”

He went on to consider other experiments, designed to measure otherphysical quantities and obtained analogous relations for time andenergy:

δt δE ∼h

(3)

and actionJ and anglew

δw δJ ∼h

(4)

which he saw as corresponding to the "well-known" relations

tE −Et =iℏ    or  wJ −Jw =iℏ

(5)

However, these generalisations are not as straightforward asHeisenberg suggested. In particular, the status of the time variable inhis several illustrations of relation (3) is not at all clear(Hilgevoord 2005). See also onSection 2.5.

Heisenberg summarized his findings in a general conclusion: allconcepts used in classical mechanics are also well-defined in the realmof atomic processes. But, as a pure fact of experience ("reinerfahrungsgemäß"), experiments that serve to providesuch a definition for one quantity are subject to particularindeterminacies, obeying relations (2)-(4) which prohibit them fromproviding a simultaneous definition of two canonically conjugatequantities. Note that in this formulation the emphasis has slightlyshifted: he now speaks of a limit on the definition of concepts, i.e.not merely on what we canknow, but what we can meaningfullysay about a particle. Of course, this stronger formulationfollows by application of the above measurement=meaning principle: ifthere are, as Heisenberg claims, no experiments that allow asimultaneous precise measurement of two conjugate quantities, thenthese quantities are also not simultaneously well-defined.

Heisenberg's paper has an interesting "Addition in proof" mentioningcritical remarks by Bohr, who saw the paper only after it had been sentto the publisher. Among other things, Bohr pointed out that in themicroscope experiment it is not the change of the momentum of theelectron that is important, but rather the circumstance that thischange cannot be precisely determined in thesame experiment.An improved version of the argument, responding to this objection, isgiven in Heisenberg's Chicago lectures of 1930.

Here (Heisenberg, 1930, p. 16), it is assumed that the electron isilluminated by light of wavelength λ and that the scatteredlight enters a microscope with aperture angle ε. According tothe laws of classical optics, the accuracy of the microscope depends onboth the wave length and the aperture angle; Abbe's criterium for its‘resolving power’, i.e. the size of the smallestdiscernable details, gives

δq ∼ λ/sin ε

(6)

On the other hand, the direction of a scattered photon, when itenters the microscope, is unknown within the angle ε, renderingthe momentum change of the electron uncertain by an amount

δp ∼h sin ε/λ

(7)

leading again to the result (2).

Let us now analyse Heisenberg's argument in more detail. First notethat, even in this improved version, Heisenberg's argument isincomplete. According to Heisenberg's ‘measurement=meaningprinciple’, one must also specify, in the given context, whatthe meaning is of the phrase ‘momentum of the electron’,in order to make sense of the claim that this momentum is changed bythe position measurement. A solution to this problem can again befound in the Chicago lectures (Heisenberg, 1930, p. 15). Here, heassumes that initially the momentum of the electron is preciselyknown, e.g. it has been measured in a previous experiment with an inaccuracy δp, which may be arbitrarily small. Then, its position is measured withinaccuracy δq, and after this, its final momentum ismeasured with an inaccuracy δp. All three measurements can beperformed with arbitrary precision. Thus, the three quantitiesδp,δq, and δp can be made as small as one wishes.If we assume further that the initial momentum has not changed untilthe position measurement, we can speak of a definite momentum untilthe time of the position measurement. Moreover we can give operationalmeaning to the idea that the momentum is changed during the positionmeasurement: the outcome of the second momentum measurement (sayp) will generallydiffer from the initial valuep. In fact, one can also show thatthis change is discontinuous, by varying the time between the threemeasurements.

Let us now try to see, adopting this more elaborate set-up, if we cancomplete Heisenberg's argument. We have now been able to giveempirical meaning to the ‘change of momentum’ of theelectron,p −p. Heisenberg's argument claims that the order of magnitude of thischange is at least inversely proportional to the inaccuracy of theposition measurement:

p −p δq ∼h

(8)

However, can we now draw the conclusion that the momentum is onlyimprecisely defined? Certainly not. Before the position measurement,its value wasp,after the measurement it isp. One might, perhaps, claim that thevalue at the very instant of the position measurement is not yetdefined, but we could simply settle this by an assignment byconvention, e.g., we might assign the mean value (p +p)/2 to the momentum at this instant.But then, the momentum is precisely determined at all instants, andHeisenberg's formulation of the uncertainty principle no longerfollows. The above attempt of completing Heisenberg's argument thusovershoots its mark.

A solution to this problem can again be found in the ChicagoLectures. Heisenberg admits that position and momentum can be knownexactly. He writes:

If the velocity of the electron is at first known, and theposition then exactly measured, the position of the electron for timesprevious to the position measurement may be calculated. For these pasttimes, δpδq is smaller than the usualbound. (Heisenberg 1930, p. 15)

Indeed, Heisenberg says: "the uncertainty relation does not hold forthe past".

Apparently, when Heisenberg refers to the uncertainty or imprecisionof a quantity, he means that the value of this quantity cannot be givenbeforehand. In the sequence of measurements we have consideredabove, the uncertainty in the momentum after the measurement ofposition has occurred, refers to the idea that the value of themomentum is not fixed justbefore the final momentummeasurement takes place. Once this measurement is performed, andreveals a valuep,the uncertainty relation no longer holds; these values then belong tothe past. Clearly, then, Heisenberg is concerned withunpredictability: the point is not that the momentum of aparticle changes, due to a position measurement, but rather that itchanges by an unpredictable amount. It is, however always possible tomeasure, and hence define, the size of this change in a subsequentmeasurement of the final momentum with arbitrary precision.

Although Heisenberg admits that we can consistently attribute valuesof momentum and position to an electron in the past, he sees littlemerit in such talk. He points out that these values can never be usedas initial conditions in a prediction about the future behavior of theelectron, or subjected to experimental verification. Whether or not wegrant them physical reality is, as he puts it, a matter of personaltaste. Heisenberg's own taste is, of course, to deny their physicalreality. For example, he writes, "I believe that one can formulate theemergence of the classical ‘path’ of a particle pregnantlyas follows:the ‘path’ comes into being only because weobserve it" (Heisenberg, 1927, p. 185). Apparently, in his view, ameasurement does not only serve to give meaning to a quantity, itcreates a particular value for this quantity. This may becalled the ‘measurement=creation’ principle. It is anontological principle, for it states what is physically real.

This then leads to the following picture. First we measure themomentum of the electron very accurately. By ‘measurement=meaning’, this entails that the term "the momentum of theparticle" is now well-defined. Moreover, by the‘measurement=creation’ principle, we may say that thismomentum is physically real. Next, the position is measured withinaccuracy δq. At this instant, the position of theparticle becomes well-defined and, again, one can regard this as aphysically real attribute of the particle. However, the momentum hasnow changed by an amount that is unpredictable by an order of magnitude  p − p  ∼h/δq. The meaning and validity of this claimcan be verified by a subsequent momentum measurement.

The question is then what status we shall assign to the momentum ofthe electron just before its final measurement. Is it real? Accordingto Heisenberg it is not. Before the final measurement, the best we canattribute to the electron is some unsharp, or fuzzy momentum. Theseterms are meant here in an ontological sense, characterizing a realattribute of the electron.

2.3 The interpretation of Heisenberg's relation

The relations Heisenberg had proposed were soon considered to be acornerstone of the Copenhagen interpretation of quantum mechanics. Justa few months later, Kennard (1927) already called them the "essentialcore" of the new theory. Taken together with Heisenberg's contentionthat they provided the intuitive content of the theory and theirprominent role in later discussions on the Copenhagen interpretation, adominant view emerged in which the uncertainty relations were regardedas a fundamental principle of the theory.

The interpretation of these relations has often been debated. DoHeisenberg's relations express restrictions on the experiments we canperform on quantum systems, and, therefore, restrictions on theinformation we can gather about such systems; or do they expressrestrictions on the meaning of the concepts we use to describe quantumsystems? Or else, are they restrictions of an ontological nature, i.e.,do they assert that a quantum system simply does not possess a definitevalue for its position and momentum at the same time? The differencebetween these interpretations is partly reflected in the various namesby which the relations are known, e.g. as ‘inaccuracyrelations’, or: ‘uncertainty’,‘indeterminacy’ or ‘unsharpness relations’. Thedebate between these different views has been addressed by manyauthors, but it has never been settled completely. Let it suffice hereto make only two general observations.

First, it is clear that in Heisenberg's own view all the abovequestions stand or fall together. Indeed, we have seen that he adoptedan operational "measurement=meaning" principle according to which themeaningfulness of a physical quantity was equivalent to the existenceof an experiment purporting to measure that quantity. Similarly, his"measurement=creation" principle allowed him to attribute physicalreality to such quantities. Hence, Heisenberg's discussions movedrather freely and quickly from talk about experimental inaccuracies toepistemological or ontological issues and back again.

However, ontological questions seemed to be of somewhat lessinterest to him. For example, there is a passage (Heisenberg, 1927, p.197), where he discusses the idea that, behind our observational data,there might still exist a hidden reality in which quantum systems havedefinite values for position and momentum, unaffected by theuncertainty relations. He emphatically dismisses this conception as anunfruitful and meaningless speculation, because, as he says, the aim ofphysics is only to describe observable data. Similarly, in the ChicagoLectures (Heisenberg 1930, p. 11), he warns against the fact that thehuman language permits the utterance of statements which have noempirical content at all, but nevertheless produce a picture in ourimagination. He notes, "One should be especially careful in using thewords ‘reality’, ‘actually’, etc., since thesewords very often lead to statements of the type just mentioned." So,Heisenberg also endorsed an interpretation of his relations asrejecting a reality in which particles have simultaneous definitevalues for position and momentum.

The second observation is that although for Heisenberg experimental,informational, epistemological and ontological formulations of hisrelations were, so to say, just different sides of the same coin, thisis not so for those who do not share his operational principles or hisview on the task of physics. Alternative points of view, in which e.g.the ontological reading of the uncertainty relations is denied, aretherefore still viable. The statement, often found in the literature ofthe thirties, that Heisenberg hadproved the impossibility ofassociating a definite position and momentum to a particle is certainlywrong. But the precise meaning one can coherently attach toHeisenberg's relations depends rather heavily on the interpretation onefavors for quantum mechanics as a whole. And because no agreement hasbeen reached on this latter issue, one cannot expect agreement on themeaning of the uncertainty relations either.

2.4 Uncertainty relations or uncertainty principle?

Let us now move to another question about Heisenberg's relations: dothey express aprinciple of quantum theory? Probably the firstinfluential author to call these relations a ‘principle’was Eddington, who, in his Gifford Lectures of 1928 referred to them asthe ‘Principle of Indeterminacy’. In the English literaturethe name uncertainty principle became most common. It is used both byCondon and Robertson in 1929, and also in the English version ofHeisenberg's Chicago Lectures (Heisenberg, 1930), although, remarkably,nowhere in the original German version of the same book (see alsoCassidy, 1998). Indeed, Heisenberg never seems to have endorsed thename ‘principle’ for his relations. His favouriteterminology was ‘inaccuracy relations’(Ungenauigkeitsrelationen) or ‘indeterminacyrelations’ (Unbestimmtheitsrelationen). We know only onepassage, in Heisenberg's own Gifford lectures, delivered in 1955-56(Heisenberg, 1958, p. 43), where he mentioned that his relations "areusually called relations of uncertainty or principle of indeterminacy".But this can well be read as his yielding to common practice ratherthan his own preference.

But does the relation (2) qualify as a principle of quantummechanics? Several authors, foremost Karl Popper (1967), have contestedthis view. Popper argued that the uncertainty relations cannot begranted the status of a principle on the grounds that they arederivable from the theory, whereas one cannot obtain the theory fromthe uncertainty relations. (The argument being that one can neverderive any equation, say, the Schrödinger equation, or thecommutation relation (1), from an inequality.)

Popper's argument is, of course, correct but we think it misses thepoint. There are many statements in physical theories which are calledprinciples even though they are in fact derivable from other statementsin the theory in question. A more appropriate departing point for thisissue is not the question of logical priority but rather Einstein'sdistinction between ‘constructive theories’ and‘principle theories’.

Einstein proposed this famous classification in (Einstein, 1919).Constructive theories are theories which postulate the existence ofsimple entities behind the phenomena. They endeavour to reconstruct thephenomena by framing hypotheses about these entities. Principletheories, on the other hand, start from empirical principles, i.e.general statements of empirical regularities, employing no or only abare minimum of theoretical terms. The purpose is to build up thetheory from such principles. That is, one aims to show how theseempirical principles provide sufficient conditions for the introductionof further theoretical concepts and structure.

The prime example of a theory of principle is thermodynamics. Herethe role of the empirical principles is played by the statements of theimpossibility of various kinds of perpetual motion machines. These areregarded as expressions of brute empirical fact, providing theappropriate conditions for the introduction of the concepts of energyand entropy and their properties. (There is a lot to be said about thetenability of this view, but that is not the topic of this entry.)

Now obviously, once the formal thermodynamic theory is built, onecan alsoderive the impossibility of the various kinds ofperpetual motion. (They would violate the laws of energy conservationand entropy increase.) But this derivation should not misguide one intothinking that they were no principles of the theory after all. Thepoint is just that empirical principles are statements that do not relyon the theoretical concepts (in this case entropy and energy) for theirmeaning. They are interpretable independently of these concepts and,further, their validity on the empirical level still provides thephysical content of the theory.

A similar example is provided by special relativity, another theoryof principle, which Einstein deliberately designed after the ideal ofthermodynamics. Here, the empirical principles are the light postulateand the relativity principle. Again, once we have built up the moderntheoretical formalism of the theory (the Minkowski space-time) it isstraightforward to prove the validity of these principles. But againthis does not count as an argument for claiming that they were noprinciples after all. So the question whether the term‘principle’ is justified for Heisenberg's relations,should, in our view, be understood as the question whether they areconceived of as empirical principles.

One can easily show that this idea was never far from Heisenberg'sintentions. We have already seen that Heisenberg presented therelations as the result of a "pure fact of experience". A few monthsafter his 1927 paper, he wrote a popular paper with the title"Ueber die Grundprincipien der Quantenmechanik" ("On thefundamental principles of quantum mechanics") where he made the pointeven more clearly. Here Heisenberg described his recent break-throughin the interpretation of the theory as follows: "It seems to be ageneral law of nature that we cannot determine position and velocitysimultaneously with arbitrary accuracy". Now actually, and in spite ofits title, the paper does not identify or discuss any‘fundamental principle’ of quantum mechanics. So, it musthave seemed obvious to his readers that he intended to claim that theuncertainty relation was a fundamental principle, forced upon us as anempirical law of nature, rather than a result derived from theformalism of the theory.

This reading of Heisenberg's intentions is corroborated by the factthat, even in his 1927 paper, applications of his relation frequentlypresent the conclusion as a matter of principle. For example, he says"In a stationary state of an atom its phase isin principleindeterminate" (Heisenberg, 1927, p. 177, [emphasis added]). Similarly,in a paper of 1928, he described the content of his relations as: "Ithas turned out that it isin principle impossible to know, tomeasure the position and velocity of a piece of matter with arbitraryaccuracy. (Heisenberg, 1984, p. 26, [emphasis added])"

So, although Heisenberg did not originate the tradition of callinghis relations a principle, it is not implausible to attribute the viewto him that the uncertainty relations represent an empirical principlethat could serve as a foundation of quantum mechanics. In fact, his1927 paper expressed this desire explicitly: "Surely, one would like tobe able to deduce the quantitative laws of quantum mechanics directlyfrom theiranschaulich foundations, that is, essentially,relation [(2)]" (ibid, p. 196). This is not to say thatHeisenberg was successful in reaching this goal, or that he did notexpress other opinions on other occasions.

Let us conclude this section with three remarks. First, if theuncertainty relation is to serve as an empirical principle, one mightwell ask what its direct empirical support is. In Heisenberg'sanalysis, no such support is mentioned. His arguments concerned thoughtexperiments in which the validity of the theory, at least at arudimentary level, is implicitly taken for granted. Jammer (1974, p.82) conducted a literature search for high precision experiments thatcould seriously test the uncertainty relations and concluded they werestill scarce in 1974. Real experimental support for the uncertaintyrelations in experiments in which the inaccuracies are close to thequantum limit have come about only more recently. (See Kaiser, Wernerand George 1983, Uffink 1985, Nairz, Andt, and Zeilinger, 2001.)

A second point is the question whether the theoretical structure orthe quantitative laws of quantum theory can indeed be derived on thebasis of the uncertainty principle, as Heisenberg wished. Seriousattempts to build up quantum theory as a full-fledged Theory ofPrinciple on the basis of the uncertainty principle have never beencarried out. Indeed, the most Heisenberg could and did claim in thisrespect was that the uncertainty relations created "room" (Heisenberg1927, p. 180) or "freedom" (Heisenberg, 1931, p. 43) for theintroduction of some non-classical mode of description of experimentaldata, not that they uniquely lead to the formalism of quantummechanics. A serious proposal to construe quantum mechanics as a theoryof principle was provided only recently by Bub (2000). But, remarkably,this proposal does not use the uncertainty relation as one of itsfundamental principles.

Third, it is remarkable that in his later years Heisenberg put asomewhat different gloss on his relations. In his autobiographyDerTeil und das Ganze of 1969 he described how he had found hisrelations inspired by a remark by Einstein that "it is the theory whichdecides what one can observe" -- thus giving precedence to theory aboveexperience, rather than the other way around. Some years later he evenadmitted that his famous discussions of thought experiments wereactually trivial since "… if the process of observation itselfis subject to the laws of quantum theory, it must be possible torepresent its result in the mathematical scheme of this theory"(Heisenberg, 1975, p. 6).

2.5 Mathematical elaboration

When Heisenberg introduced his relation, his argument was based onlyon qualitative examples. He did not provide a general, exact derivationof his relations.^[3] Indeed, he did not even give a definition ofthe uncertainties δq, etc., occurring in theserelations. Of course, this was consistent with the announced goal ofthat paper, i.e. to provide some qualitative understanding of quantummechanics for simple experiments.

The first mathematically exact formulation of the uncertaintyrelations is due to Kennard. He proved in 1927 the theorem that for allnormalized state vectors |ψ> the following inequality holds:

ΔψpΔψq ≥ ℏ/2

(9)

Here, Δ_ψp andΔ_ψq are standarddeviations of position and momentum in the state vector |ψ>,i.e.,

(Δψp)² =<p²>ψ −(<p>ψ)²,     (Δψq)² =<q²>ψ −(<q>ψ)².

(10)

where <·>_ψ = <ψ|·|ψ>denotes the expectation value in state |ψ>. The inequality (9)was generalized in 1929 by Robertson who proved that for allobservables (self-adjoint operators)A andB

ΔψAΔψB   ≥  ½<[A,B]>ψ

(11)

where [A,B] :=AB −BA denotes thecommutator. This relation was in turn strengthened by Schrödinger(1930), who obtained:

(ΔψA)²(ΔψB)²   ≥    ¼<[A,B]>ψ² +¼<{A−<A>ψ,B−<B>ψ}>ψ²

(12)

where {A,B} :=(AB +BA) denotes theanti-commutator.

Since the above inequalities have the virtue of being exact andgeneral, in contrast to Heisenberg's original semi-quantitativeformulation, it is tempting to regard them as the exact counterpart ofHeisenberg's relations (2)-(4). Indeed, such was Heisenberg's own view.In his Chicago Lectures (Heisenberg 1930, pp. 15-19), he presentedKennard's derivation of relation (9) and claimed that "this proof doesnot differ at all in mathematical content" from the semi-quantitativeargument he had presented earlier, the only difference being that now"the proof is carried through exactly".

But it may be useful to point out that both in status and intendedrole there is a difference between Kennard's inequality andHeisenberg's previous formulation (2). The inequalities discussed inthe present section are not statements of empirical fact, but theoremsof the quantum mechanical formalism. As such, they presuppose thevalidity of this formalism, and in particular the commutation relation(1), rather than elucidating its intuitive content or to create‘room’ or ‘freedom’ for the validity of thisrelation. At best, one should see the above inequalities as showingthat the formalism is consistent with Heisenberg's empiricalprinciple.

This situation is similar to that arising in other theories ofprinciple where, as noted inSection 2.4,one often finds that, next to an empirical principle, the formalismalso provides a corresponding theorem. And similarly, this situationshould not, by itself, cast doubt on the question whether Heisenberg'srelation can be regarded as a principle of quantum mechanics.

There is a second notable difference between (2) and (9). Heisenbergdid not give a general definition for the ‘uncertainties’δp and δq. The most definite remark hemade about them was that they could be taken as "something like themean error". In the discussions of thought experiments, he and Bohrwould always quantify uncertainties on a case-to-case basis by choosingsome parameters which happened to be relevant to the experiment athand. By contrast, the inequalities (9)-(12) employ a single specificexpression as a measure for ‘uncertainty’: the standarddeviation. At the time, this choice was not unnatural, given that thisexpression is well-known and widely used in error theory and thedescription of statistical fluctuations. However, there was very littleor no discussion of whether this choice was appropriate for a generalformulation of the uncertainty relations. A standard deviation reflectsthe spread or expected fluctuations in a series of measurements of anobservable in a given state. It is not at all easy to connect this ideawith the concept of the ‘inaccuracy’ of a measurement, suchas the resolving power of a microscope. In fact, even though Heisenberghad taken Kennard's inequality as the precise formulation of theuncertainty relation, he and Bohr never relied on standard deviationsin their many discussions of thought experiments, and indeed, it hasbeen shown (Uffink and Hilgevoord, 1985; Hilgevoord and Uffink, 1988)that these discussions cannot be framed in terms of standarddeviation.

Another problem with the above elaboration is that the‘well-known’ relations (5) are actually false if energyE and actionJ areto be positive operators (Jordan 1927). In that case, self-adjointoperatorst andwdo not exist and inequalities analogous to (9) cannot be derived. Also,these inequalities do not hold for angle and angular momentum (Uffink1990). These obstacles have led to a quite extensive literature ontime-energy and angle-action uncertainty relations (Muga et al. 2002,Hilgevoord 2005).

3. Bohr

In spite of the fact that Heisenberg's and Bohr's views on quantummechanics are often lumped together as (part of) ‘the Copenhageninterpretation’, there is considerable difference between theirviews on the uncertainty relations.

3.1 From wave-particle duality to complementarity

Long before the development of modern quantum mechanics, Bohr hadbeen particularly concerned with the problem of particle-wave duality,i.e. the problem that experimental evidence on the behaviour of bothlight and matter seemed to demand a wave picture in some cases, and aparticle picture in others. Yet these pictures are mutually exclusive.Whereas a particle is always localized, the very definition of thenotions of wavelength and frequency requires an extension in space andin time. Moreover, the classical particle picture is incompatible withthe characteristic phenomenon of interference.

His long struggle with wave-particle duality had prepared him for aradical step when the dispute between matrix and wave mechanics brokeout in 1926-27. For the main contestants, Heisenberg andSchrödinger, the issue at stake was which view could claim toprovide a single coherent and universal framework for the descriptionof the observational data. The choice was, essentially between adescription in terms of continuously evolving waves, or else one ofparticles undergoing discontinuous quantum jumps. By contrast, Bohrinsisted that elements from both views were equally valid and equallyneeded for an exhaustive description of the data. His way out of thecontradiction was to renounce the idea that the pictures refer, in aliteral one-to-one correspondence, to physical reality. Instead, theapplicability of these pictures was to become dependent on theexperimental context. This is the gist of the viewpoint he called‘complementarity’.

Bohr first conceived the general outline of his complementarityargument in early 1927, during a skiing holiday in Norway, at the sametime when Heisenberg wrote his uncertainty paper. When he returned toCopenhagen and found Heisenberg's manuscript, they got into an intensediscussion. On the one hand, Bohr was quite enthusiastic aboutHeisenberg's ideas which seemed to fit wonderfully with his ownthinking. Indeed, in his subsequent work, Bohr always presented theuncertainty relations as the symbolic expression of his complementarityviewpoint. On the other hand, he criticized Heisenberg severely for hissuggestion that these relations were due to discontinuous changesoccurring during a measurement process. Rather, Bohr argued, theirproper derivation should start from the indispensability of bothparticle and wave concepts. He pointed out that the uncertainties inthe experiment did not exclusively arise from the discontinuities butalso from the fact that in the experiment we need to take into accountboth the particle theory and the wave theory. It is not so much theunknown disturbance which renders the momentum of the electronuncertain but rather the fact that the position and the momentum of theelectron cannot be simultaneously defined in this experiment. (See the"Addition in Proof" to Heisenberg's paper.)

We shall not go too deeply into the matter of Bohr's interpretationof quantum mechanics since we are mostly interested in Bohr's view onthe uncertainty principle. For a more detailed discussion of Bohr'sphilosophy of quantum physics we refer to Scheibe (1973), Folse (1985),Honner (1987) and Murdoch (1987). It may be useful, however, to sketchsome of the main points. Central in Bohr's considerations is thelanguage we use in physics. No matter how abstract and subtlethe concepts of modern physics may be, they are essentially anextension of our ordinary language and a means to communicate theresults of our experiments. These results, obtained under well-definedexperimental circumstances, are what Bohr calls the "phenomena". Aphenomenon is "the comprehension of the effects observed under givenexperimental conditions" (Bohr 1939, p. 24), it is the resultant of aphysical object, a measuring apparatus and the interaction between themin a concrete experimental situation. The essential difference betweenclassical and quantum physics is that in quantum physics theinteraction between the object and the apparatus cannot be madearbitrarily small; the interaction must at least comprise one quantum.This is expressed by Bohr's quantum postulate:

[… the] essence [of the formulation of the quantumtheory] may be expressed in the so-called quantum postulate, whichattributes to any atomic process an essential discontinuity or ratherindividuality, completely foreign to classical theories and symbolizedby Planck's quantum of action. (Bohr, 1928, p. 580)

A phenomenon, therefore, is an indivisible whole and the result of ameasurement cannot be considered as an autonomous manifestation of theobject itself independently of the measurement context. The quantumpostulate forces upon us a new way of describing physicalphenomena:

In this situation, we are faced with the necessity of aradical revision of the foundation for the description and explanationof physical phenomena. Here, it must above all be recognized that,however far quantum effects transcend the scope of classical physicalanalysis, the account of the experimental arrangement and the record ofthe observations must always be expressed in common languagesupplemented with the terminology of classical physics. (Bohr, 1948, p.313)

This is what Scheibe (1973) has called the "buffer postulate"because it prevents the quantum from penetrating into the classicaldescription: A phenomenon must always be described in classical terms;Planck's constant does not occur in this description.

Together, the two postulates induce the following reasoning. Inevery phenomenon the interaction between the object and the apparatuscomprises at least one quantum. But the description of the phenomenonmust use classical notions in which the quantum of action does notoccur. Hence, the interaction cannot be analysed in this description.On the other hand, the classical character of the description allows tospeak in terms of the object itself. Instead of saying: ‘theinteraction between a particle and a photographic plate has resulted ina black spot in a certain place on the plate’, we are allowed toforgo mentioning the apparatus and say: ‘the particle has beenfound in this place’. The experimental context, rather thanchanging or disturbing pre-existing properties of the object, defineswhat can meaningfully be said about the object.

Because the interaction between object and apparatus is left out inour description of the phenomenon, we do not get the whole picture.Yet, any attempt to extend our description by performing themeasurement of a different observable quantity of the object, orindeed, on the measurement apparatus, produces a new phenomenon and weare again confronted with the same situation. Because of theunanalyzable interaction in both measurements, the two descriptionscannot, generally, be united into a single picture. They are what Bohrcalls complementary descriptions:

[the quantum of action]...forces us to adopt a new mode ofdescription designated as complementary in the sense that any givenapplication of classical concepts precludes the simultaneous use ofother classical concepts which in a different connection are equallynecessary for the elucidation of the phenomena. (Bohr, 1929, p.10)

The most important example of complementary descriptions is providedby the measurements of the position and momentum of an object. If onewants to measure the position of the object relative to a given spatialframe of reference, the measuring instrument must be rigidly fixed tothe bodies which define the frame of reference. But this implies theimpossibility of investigating the exchange of momentum between theobject and the instrument and we are cut off from obtaining anyinformation about the momentum of the object. If, on the other hand,one wants to measure the momentum of an object the measuring instrumentmust be able to move relative to the spatial reference frame. Bohr hereassumes that a momentum measurement involves the registration of therecoil of some movable part of the instrument and the use of the law ofmomentum conservation. The looseness of the part of the instrument withwhich the object interacts entails that the instrument cannot serve toaccurately determine the position of the object. Since a measuringinstrument cannot be rigidly fixed to the spatial reference frame and,at the same time, be movable relative to it, the experiments whichserve to precisely determine the position and the momentum of an objectare mutually exclusive. Of course, in itself, this is not at alltypical for quantum mechanics. But, because the interaction betweenobject and instrument during the measurement can neither be neglectednor determined the two measurements cannot be combined. This means thatin the description of the object one must choose between the assignmentof a precise position or of a precise momentum.

Similar considerations hold with respect to the measurement of timeand energy. Just as the spatial coordinate system must be fixed bymeans of solid bodies so must the time coordinate be fixed by means ofunperturbable, synchronised clocks. But it is precisely thisrequirement which prevents one from taking into account of the exchangeof energy with the instrument if this is to serve its purpose.Conversely, any conclusion about the object based on the conservationof energy prevents following its development in time.

The conclusion is that in quantum mechanics we are confronted with acomplementarity between two descriptions which are united in theclassical mode of description: the space-time description (orcoordination) of a process and the description based on theapplicability of the dynamical conservation laws. The quantum forces usto give up the classical mode of description (also called the‘causal’ mode of description by Bohr^[4]): it is impossible toform a classical picture of what is going on when radiation interactswith matter as, e.g., in the Compton effect.

Any arrangement suited to study the exchange of energy andmomentum between the electron and the photon must involve a latitude inthe space-time description sufficient for the definition of wave-numberand frequency which enter in the relation [E =hνandp =hσ]. Conversely, any attempt oflocating the collision between the photon and the electron moreaccurately would, on account of the unavoidable interaction with thefixed scales and clocks defining the space-time reference frame,exclude all closer account as regards the balance of momentum andenergy. (Bohr, 1949, p. 210)

A causal description of the process cannot be attained; we have tocontent ourselves with complementary descriptions. "The viewpoint ofcomplementarity may be regarded", according to Bohr, "as a rationalgeneralization of the very ideal of causality".

In addition to complementary descriptions Bohr also talks aboutcomplementary phenomena and complementary quantities. Position andmomentum, as well as time and energy, are complementary quantities.^[5]

We have seen that Bohr's approach to quantum theory puts heavyemphasis on the language used to communicate experimental observations,which, in his opinion, must always remain classical. By comparison, heseemed to put little value on arguments starting from the mathematicalformalism of quantum theory. This informal approach is typical of allof Bohr's discussions on the meaning of quantum mechanics. One mightsay that for Bohr the conceptual clarification of the situation hasprimary importance while the formalism is only a symbolicrepresentation of this situation.

This is remarkable since, finally, it is the formalism which needsto be interpreted. This neglect of the formalism is one of the reasonswhy it is so difficult to get a clear understanding of Bohr'sinterpretation of quantum mechanics and why it has aroused so muchcontroversy. We close this section by citing from an article of 1948 toshow how Bohr conceived the role of the formalism of quantummechanics:

The entire formalism is to be considered as a tool forderiving predictions, of definite or statistical character, as regardsinformation obtainable under experimental conditions described inclassical terms and specified by means of parameters entering into thealgebraic or differential equations of which the matrices or thewave-functions, respectively, are solutions. These symbols themselves,as is indicated already by the use of imaginary numbers, are notsusceptible to pictorial interpretation; and even derived realfunctions like densities and currents are only to be regarded asexpressing the probabilities for the occurrence of individual eventsobservable under well-defined experimental conditions. (Bohr, 1948, p.314)

3.2 Bohr's view on the uncertainty relations

In his Como lecture, published in 1928, Bohr gave his own version ofa derivation of the uncertainty relations between position and momentumand between time and energy. He started from the relations

E =hν andp =h/λ

(13)

which connect the notions of energyE and momentumpfrom the particle picture with those of frequency ν and wavelengthλ from the wave picture. He noticed that a wave packet oflimited extension in space and time can only be built up by thesuperposition of a number of elementary waves with a large range ofwave numbers and frequencies. Denoting the spatial and temporalextensions of the wave packet by Δx andΔt, and the extensions in the wave number σ :=1/λ and frequency by Δσ and Δν, it followsfrom Fourier analysis that in the most favorable case ΔxΔσ ≈ Δt Δν ≈ 1, and,using (13), one obtains the relations

Δt ΔE ≈ ΔxΔp ≈h

(14)

Note that Δx, Δσ, etc., are not standarddeviations but unspecified measures of the size of a wave packet. (Theoriginal text has equality signs instead of approximate equality signs,but, since Bohr does not define the spreads exactly the use ofapproximate equality signs seems more in line with his intentions.Moreover, Bohr himself used approximate equality signs in laterpresentations.) These equations determine, according to Bohr: "thehighest possible accuracy in the definition of the energy and momentumof the individuals associated with the wave field" (Bohr 1928, p. 571).He noted, "This circumstance may be regarded as a simple symbolicexpression of the complementary nature of the space-time descriptionand the claims of causality" (ibid).^[6] We note a few pointsabout Bohr's view on the uncertainty relations. First of all, Bohr doesnot refer todiscontinuous changes in the relevant quantitiesduring the measurement process. Rather, he emphasizes the possibilityofdefining these quantities. This view is markedly differentfrom Heisenberg's. A draft version of the Como lecture is even moreexplicit on the difference between Bohr and Heisenberg:

These reciprocal uncertainty relations were given in arecent paper of Heisenberg as the expression of the statistical elementwhich, due to the feature of discontinuity implied in the quantumpostulate, characterizes any interpretation of observations by means ofclassical concepts. It must be remembered, however, that theuncertainty in question is not simply a consequence of a discontinuouschange of energy and momentum say during an interaction betweenradiation and material particles employed in measuring the space-timecoordinates of the individuals. According to the above considerationsthe question is rather that of the impossibility of definingrigourously such a change when the space-time coordination of theindividuals is also considered. (Bohr, 1985 p. 93)

Indeed, Bohr not only rejected Heisenberg's argument that theserelations are due to discontinuous disturbances implied by the act ofmeasuring, but also his view that the measurement processcreates a definite result:

The unaccustomed features of the situation with which weare confronted in quantum theory necessitate the greatest caution asregard all questions of terminology. Speaking, as it is often done ofdisturbing a phenomenon by observation, or even of creating physicalattributes to objects by measuring processes is liable to be confusing,since all such sentences imply a departure from conventions of basiclanguage which even though it can be practical for the sake of brevity,can never be unambiguous. (Bohr, 1939, p. 24)

Nor did he approve of an epistemological formulation or one in terms ofexperimental inaccuracies:

[…] a sentence like "we cannot know both themomentum and the position of an atomic object" raises at once questionsas to the physical reality of two such attributes of the object, whichcan be answered only by referring to the mutual exclusive conditionsfor an unambiguous use of space-time concepts, on the one hand, anddynamical conservation laws on the other hand. (Bohr, 1948, p. 315;also Bohr 1949, p. 211)

It would in particular not be out of place in this connection towarn against a misunderstanding likely to arise when one tries toexpress the content of Heisenberg's well-known indeterminacy relationby such a statement as ‘the position and momentum of a particlecannot simultaneously be measured with arbitrary accuracy’.According to such a formulation it would appear as though we had to dowith some arbitrary renunciation of the measurement of either the oneor the other of two well-defined attributes of the object, which wouldnot preclude the possibility of a future theory taking both attributesinto account on the lines of the classical physics. (Bohr 1937, p.292)

Instead, Bohr always stressed that the uncertainty relations arefirst and foremost an expression of complementarity. This may seem oddsince complementarity is a dichotomic relation between two types ofdescription whereas the uncertainty relations allow for intermediatesituations between two extremes. They "express" the dichotomy in thesense that if we take the energy and momentum to be perfectlywell-defined, symbolically ΔE = Δp = 0,the postion and time variables are completely undefined,Δx = Δt = ∞, and vice versa. Butthey also allow intermediate situations in which the mentioneduncertainties are all non-zero and finite. This more positive aspect ofthe uncertainty relation is mentioned in the Como lecture:

At the same time, however, the general character of thisrelation makes it possible to a certain extent to reconcile theconservation laws with the space-time coordination of observations, theidea of a coincidence of well-defined events in space-time points beingreplaced by that of unsharply defined individuals within space-timeregions. (Bohr 1928, p. 571)

However, Bohr never followed up on this suggestion that we might beable to strike a compromise between the two mutually exclusive modes ofdescription in terms of unsharply defined quantities. Indeed, anattempt to do so, would take the formalism of quantum theory moreseriously than the concepts of classical language, and this step Bohrrefused to take. Instead, in his later writings he would be contentwith stating that the uncertainty relations simply defy an unambiguousinterpretation in classical terms:

These so-called indeterminacy relations explicitly bear outthe limitation of causal analysis, but it is important to recognizethat no unambiguous interpretation of such a relation can be given inwords suited to describe a situation in which physical attributes areobjectified in a classical way. (Bohr, 1948, p.315)

It must here be remembered that even in the indeterminacy relation[Δq Δp ≈h] we are dealingwith an implication of the formalism which defies unambiguousexpression in words suited to describe classical pictures. Thus asentence like "we cannot know both the momentum and the position of anatomic object" raises at once questions as to the physical reality oftwo such attributes of the object, which can be answered only byreferring to the conditions for an unambiguous use of space-timeconcepts, on the one hand, and dynamical conservation laws on the otherhand. (Bohr, 1949, p. 211)

Finally, on a more formal level, we note that Bohr's derivation doesnot rely on the commutation relations (1) and (5), but on Fourieranalysis. These two approaches are equivalent as far as therelationship between position and momentum is concerned, but this isnot so for time and energy since most physical systems do not have atime operator. Indeed, in his discussion with Einstein (Bohr, 1949),Bohr considered time as a simple classical variable. This even holdsfor his famous discussion of the ‘clock-in-the-box’thought-experiment where the time, as defined by the clock in the box,is treated from the point of view of classical generalrelativity. Thus, in an approach based on commutation relations, theposition-momentum and time-energy uncertainty relations are not onequal footing, which is contrary to Bohr's approach in terms ofFourier analysis (Hilgevoord 1996 and 1998).

4. The Minimal Interpretation

In the previous two sections we have seen how both Heisenberg andBohr attributed a far-reaching status to the uncertainty relations.They both argued that these relations place fundamental limits on theapplicability of the usual classical concepts. Moreover, they bothbelieved that these limitations were inevitable and forced upon us.However, we have also seen that they reached such conclusions bystarting from radical and controversial assumptions. This entails, ofcourse, that their radical conclusions remain unconvincing for thosewho reject these assumptions. Indeed, the operationalist-positivistviewpoint adopted by these authors has long since lost its appeal amongphilosophers of physics.

So the question may be asked what alternative views of theuncertainty relations are still viable. Of course, this problem isintimately connected with that of the interpretation of the wavefunction, and hence of quantum mechanics as a whole. Since there is noconsensus about the latter, one cannot expect consensus about theinterpretation of the uncertainty relations either. Here we onlydescribe a point of view, which we call the ‘minimalinterpretation’, that seems to be shared by both the adherents ofthe Copenhagen interpretation and of other views.

In quantum mechanics a system is supposed to be described by itsquantum state, also called its state vector. Given the state vector,one can derive probability distributions for all the physicalquantities pertaining to the system such as its position, momentum,angular momentum, energy, etc. The operational meaning of theseprobability distributions is that they correspond to the distributionof the values obtained for these quantities in a long series ofrepetitions of the measurement. More precisely, one imagines a greatnumber of copies of the system under consideration, all prepared in thesame way. On each copy the momentum, say, is measured. Generally, theoutcomes of these measurements differ and a distribution of outcomes isobtained. The theoretical momentum distribution derived from thequantum state is supposed to coincide with the hypotheticaldistribution of outcomes obtained in an infinite series of repetitionsof the momentum measurement. The same holds,mutatis mutandis,for all the other physical quantities pertaining to the system. Notethat no simultaneous measurements of two or more quantities arerequired in defining the operational meaning of the probabilitydistributions.

Uncertainty relations can be considered as statements about thespreads of the probability distributions of the several physicalquantities arising from the same state. For example, the uncertaintyrelation between the position and momentum of a system may beunderstood as the statement that the position and momentumdistributions cannot both be arbitrarily narrow -- in some sense of theword "narrow" -- in any quantum state. Inequality (9) is an example ofsuch a relation in which the standard deviation is employed as ameasure of spread. From this characterization of uncertainty relationsfollows that a more detailed interpretation of the quantum state thanthe one given in the previous paragraph is not required to studyuncertainty relations as such. In particular, a further ontological orlinguistic interpretation of the notion of uncertainty, as limits onthe applicability of our concepts given by Heisenberg or Bohr, need notbe supposed.

Indeed, this minimal interpretation leaves open whether it makessense to attribute precise values of position and momentum to anindividual system. Some interpretations of quantum mechanics, e.g.those of Heisenberg and Bohr, deny this; while others, e.g. theinterpretation of de Broglie and Bohm insist that each individualsystem has a definite position and momentum (see the entry onBohmian mechanics). The only requirement is that, as an empirical fact, it is notpossible to prepare pure ensembles in which all systems have the samevalues for these quantities, or ensembles in which the spreads aresmaller than allowed by quantum theory. Although interpretations ofquantum mechanics, in which each system has a definite value for itsposition and momentum are still viable, this is not to say that theyare without strange features of their own; they do not imply a returnto classical physics.

We end with a few remarks on this minimal interpretation. First, itmay be noted that the minimal interpretation of the uncertaintyrelations is little more than filling in the empirical meaning ofinequality (9), or an inequality in terms of other measures of width,as obtained from the standard formalism of quantum mechanics. As such,this view shares many of the limitations we have noted above about thisinequality. Indeed, it is not straightforward to relate the spread in astatistical distribution of measurement results with theinaccuracy of this measurement, such as, e.g. the resolvingpower of a microscope. Moreover, the minimal interpretation does notaddress the question whether one can makesimultaneousaccurate measurements of position and momentum. As a matter of fact,one can show that the standard formalism of quantum mechanics does notallow such simultaneous measurements. But this is not a consequence ofrelation (9).

If one feels that statements about inaccuracy of measurement, or thepossibility of simultaneous measurements, belong to any satisfactoryformulation of the uncertainty principle, the minimal interpretationmay thus be too minimal.

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Other Internet Resources

• Exhibit on Heisenberg and the uncertainty principle, from the American Institute of Physics
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quantum mechanics |quantum mechanics: Bohmian mechanics |quantum mechanics: Copenhagen interpretation of