patentcad

When you show up on the Saturday AM ride, and it's 25º, it is hardly necessary to dress like this:



Judicious use of layers combined with adequate hand, foot and head warming gear will suffice.

You wouldn't believe the Fredilicious winter gear some of these guys show up in on rides in late November and December. Hey, at least they're out there, I'll grant them that.

mzeffex 

There goes more bandwidth.

patentcad

I perform a valuable BF Community Service and this is the thanks I get.

Thankless Freds.

VA_Esquire

Do we need to warm up the tyres or shall this not be needed?

mzeffex 

Not that I don't appreciate it of course.

caloso

Come on out to NorCal. Guys show up dressed like that when it's 55F.

Sixty Fiver

When did you go climbing ?

jdon

Yes, but it requires a little bit of imagination. Better these posts for comic relief than another " I just started riding and my underparts hurt" thread.

BillyD

Is she pregnant?

Hot Potato

Uh-oh.

I wore the goggles riding once. But hey, I was riding in 25mph wind driven falling snow near zero degrees. And it wasn't a Saturday am club ride. I think that absolves me?

patentcad

Goggles?

adriano 

the pc ad strikes again.

spry

No more winter pictures in your own driveway wearing cheerleader socks.
Let's see some distant locations during these so called big winter blows in Chester.

KiddSisko

If I were a moderator concerned with BF bandwidth issues, I'd combine all these Pcad's Cycling 101 For Freds parts 1 - 5 into one thread. The funny behind that would be a moderator actually taking any one of them serious enough to keep open and unmoved.

patentcad

Where would you move them? They have serious Road Cycling content.

Ungrateful Fred Scum that you are. I blow my snot rockets at you.

rufvelo

Quantum mechanics is generally regarded as blah physical blahory that is our best candidate for a fundamental and universal description of blah physical world. blah conceptual framework employed by this blahory differs drastically from that of classical physics. Indeed, blah transition from classical to quantum physics marks a genuine revolution in our understanding of blah physical world.

One striking aspect of blah difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, blah prime example being blah position and momentum of a particle. According to quantum mechanics, blah more precisely blah position (momentum) of a particle is given, blah less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) blah quantum mechanical uncertainty principle for position and momentum. blah uncertainty principle played an important role in many discussions on blah philosophical implications of quantum mechanics, in particular in discussions on blah consistency of blah so-called Copenhagen interpretation, blah interpretation endorsed by blah founding fablahrs Heisenberg and Bohr.

This should not suggest that blah uncertainty principle is blah only aspect of blah conceptual difference between classical and quantum physics: blah implications of quantum mechanics for notions as (non)-locality, entanglement and identity play no less havoc with classical intuitions.

* 1. Introduction
* 2. Heisenberg
o 2.1 Heisenberg's road to blah uncertainty relations
o 2.2 Heisenberg's argument
o 2.3 blah interpretation of Heisenberg's relation
o 2.4 Uncertainty relations or uncertainty principle?
o 2.5 Mablahmatical elaboration
* 3. Bohr
o 3.1 From wave-particle duality to complementarity
o 3.2 Bohr's view on blah uncertainty relations
* 4. blah Minimal Interpretation
* Bibliography
* Oblahr Internet Resources
* Related Entries

1. Introduction

blah uncertainty principle is certainly one of blah most famous and important aspects of quantum mechanics. It has often been regarded as blah most distinctive feature in which quantum mechanics differs from classical blahories of blah physical world. Roughly speaking, blah uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to blah position and momentum of a physical system. Rablahr, blahse quantities can only be determined with some characteristic ‘uncertainties’ that cannot become arbitrarily small simultaneously. But what is blah exact meaning of this principle, and indeed, is it really a principle of quantum mechanics? (In his original work, Heisenberg only speaks of uncertainty relations.) And, in particular, what does it mean to say that a quantity is determined only up to some uncertainty? blahse are blah main questions we will explore in blah following, focusssing on blah views of Heisenberg and Bohr.

blah notion of ‘uncertainty’ occurs in several different meanings in blah physical literature. It may refer to a lack of knowledge of a quantity by an observer, or to blah experimental inaccuracy with which a quantity is measured, or to some ambiguity in blah definition of a quantity, or to a statistical spread in an ensemble of similary prepared systems. Also, several different names are used for such uncertainties: inaccuracy, spread, imprecision, indefiniteness, indeterminateness, indeterminacy, latitude, etc. As we shall see, even Heisenberg and Bohr did not decide on a single terminology for quantum mechanical uncertainties. Forestalling a discussion about which name is blah most appropriate one in quantum mechanics, we use blah name ‘uncertainty principle’ imply because it is blah most common one in blah literature.
2. Heisenberg
2.1 Heisenberg's road to blah uncertainty relations

Heisenberg introduced his now famous relations in an article of 1927, entitled "Ueber den anschaulichen Inhalt der quantenblahoretischen Kinematik und Mechanik". A (partial) translation of this title is: "On blah anschaulich content of quantum blahoretical kinematics and mechanics". Here, blah term anschaulich is particularly notable. Apparently, it is one of those German words that defy an unambiguous translation into oblahr languages. Heisenberg's title is translated as "On blah physical content …" by Wheeler and Zurek (1983). His collected works (Heisenberg, 1984) translate it as "On blah perceptible content …", while Cassidy's biography of Heisenberg (Cassidy, 1992), refers to blah paper as "On blah perceptual content …". Literally, blah closest translation of blah term anschaulich is ‘visualizable’. But, as in most languages, words that make reference to vision are not always intended literally. Seeing is widely used as a metaphor for understanding, especially for immediate understanding. Hence, anschaulich also means ‘intelligible’ or ‘intuitive’.[1]

Why was this issue of blah Anschaulichkeit of quantum mechanics such a prominent concern to Heisenberg? This question has already been considered by a number of commentators (Jammer, 1977; Miller 1982; de Regt, 1997; Beller, 1999). For blah answer, it turns out, we must go back a little in time. In 1925 Heisenberg had developed blah first coherent mablahmatical formalism for quantum blahory (Heisenberg, 1925). His leading idea was that only those quantities that are in principle observable should play a role in blah blahory, and that all attempts to form a picture of what goes on inside blah atom should be avoided. In atomic physics blah observational data were obtained from spectroscopy and associated with atomic transitions. Thus, Heisenberg was led to consider blah ‘transition quantities’ as blah basic ingredients of blah blahory. Max Born, later that year, realized that blah transition quantities obeyed blah rules of matrix calculus, a branch of mablahmatics that was not so well-known blahn as it is now. In a famous series of papers Heisenberg, Born and Jordan developed this idea into blah matrix mechanics version of quantum blahory.

Formally, matrix mechanics remains close to classical mechanics. blah central idea is that all physical quantities must be represented by infinite self-adjoint matrices (later identified with operators on a Hilbert space). It is postulated that blah matrices q and p representing blah canonical position and momentum variables of a particle satisfy blah so-called canonical commutation rule

qp − pq = iℏ (1)

where ℏ = h/2π, h denotes Planck's constant, and boldface type is used to represent matrices. blah new blahory scored spectacular empirical success by encompassing nearly all spectroscopic data known at blah time, especially after blah concept of blah electron spin was included in blah blahoretical framework.

It came as a big surprise, blahrefore, when one year later, Erwin Schrödinger presented an alternative blahory, that became known as wave mechanics. Schrödinger assumed that an electron in an atom could be represented as an oscillating charge cloud, evolving continuously in space and time according to a wave equation. blah discrete frequencies in blah atomic spectra were not due to discontinuous transitions (quantum jumps) as in matrix mechanics, but to a resonance phenomenon. Schrödinger also showed that blah two blahories were equivalent.[2]

Even so, blah two approaches differed greatly in interpretation and spirit. Whereas Heisenberg eschewed blah use of visualizable pictures, and accepted discontinuous transitions as a primitive notion, Schrödinger claimed as an advantage of his blahory that it was anschaulich. In Schrödinger's vocabulary, this meant that blah blahory represented blah observational data by means of continuously evolving causal processes in space and time. He considered this condition of Anschaulichkeit to be an essential requirement on any acceptable physical blahory. Schrödinger was not alone in appreciating this aspect of his blahory. Many oblahr leading physicists were attracted to wave mechanics for blah same reason. For a while, in 1926, before it emerged that wave mechanics had serious problems of its own, Schrödinger's approach seemed to gablahr more support in blah physics community than matrix mechanics.

Understandably, Heisenberg was unhappy about this development. In a letter of 8 June 1926 to Pauli he confessed that "blah more I think about blah physical part of Schrödinger's blahory, blah more disgusting I find it", and: "What Schrödinger writes about blah Anschaulichkeit of his blahory, … I consider Mist (Pauli, 1979, p. 328)". Again, this last German term is translated differently by various commentators: as "junk" (Miller, 1982) "rubbish" (Beller 1999) "crap" (Cassidy, 1992), and perhaps more literally, as "bull****" (de Regt, 1997). Neverblahless, in published writings, Heisenberg voiced a more balanced opinion. In a paper in Die Naturwissenschaften (1926) he summarized blah peculiar situation that blah simultaneous development of two competing blahories had brought about. Although he argued that Schrödinger's interpretation was untenable, he admitted that matrix mechanics did not provide blah Anschaulichkeit which made wave mechanics so attractive. He concluded: "to obtain a contradiction-free anschaulich interpretation, we still lack some essential feature in our image of blah structure of matter". blah purpose of his 1927 paper was to provide exactly this lacking feature.
2.2 Heisenberg's argument

Let us now look at blah argument that led Heisenberg to his uncertainty relations. He started by redefining blah notion of Anschaulichkeit. Whereas Schrödinger associated this term with blah provision of a causal space-time picture of blah phenomena, Heisenberg, by contrast, declared:

We believe we have gained anschaulich understanding of a physical blahory, if in all simple cases, we can grasp blah experimental consequences qualitatively and see that blah blahory does not lead to any contradictions. Heisenberg, 1927, p. 172)

His goal was, of course, to show that, in this new sense of blah word, matrix mechanics could lay blah same claim to Anschaulichkeit as wave mechanics.

To do this, he adopted an operational assumption: terms like ‘blah position of a particle’ have meaning only if one specifies a suitable experiment by which ‘blah position of a particle’ can be measured. We will call this assumption blah ‘measurement=meaning principle’. In general, blahre is no lack of such experiments, even in blah domain of atomic physics. However, experiments are never completely accurate. We should be prepared to accept, blahrefore, that in general blah meaning of blahse quantities is also determined only up to some characteristic inaccuracy.

As an example, he considered blah measurement of blah position of an electron by a microscope. blah accuracy of such a measurement is limited by blah wave length of blah light illuminating blah electron. Thus, it is possible, in principle, to make such a position measurement as accurate as one wishes, by using light of a very short wave length, e.g., γ-rays. But for γ-rays, blah Compton effect cannot be ignored: blah interaction of blah electron and blah illuminating light should blahn be considered as a collision of at least one photon with blah electron. In such a collision, blah electron suffers a recoil which disturbs its momentum. Moreover, blah shorter blah wave length, blah larger is this change in momentum. Thus, at blah moment when blah position of blah particle is accurately known, Heisenberg argued, its momentum cannot be accurately known:

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

This is blah first formulation of blah uncertainty principle. In its present form it is an epistemological principle, since it limits what we can know about blah electron. From "elementary formulae of blah Compton effect" Heisenberg estimated blah ‘imprecisions’ to be of blah order

δpδq ∼ h (2)

He continued: “In this circumstance we see blah direct anschaulich content of blah relation qp − pq = iℏ.”

He went on to consider oblahr experiments, designed to measure oblahr physical quantities and obtained analogous relations for time and energy:

δt δE ∼ h (3)

and action J and angle w

δw δJ ∼ h (4)

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

tE − Et = iℏ or wJ − Jw = iℏ (5)

However, blahse generalisations are not as straightforward as Heisenberg suggested. In particular, blah status of blah time variable in his several illustrations of relation (3) is not at all clear (Hilgevoord 2005). See also on Section 2.5.

Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in blah realm of atomic processes. But, as a pure fact of experience ("rein erfahrungsgemäß"), experiments that serve to provide such a definition for one quantity are subject to particular indeterminacies, obeying relations (2)-(4) which prohibit blahm from providing a simultaneous definition of two canonically conjugate quantities. Note that in this formulation blah emphasis has slightly shifted: he now speaks of a limit on blah definition of concepts, i.e. not merely on what we can know, but what we can meaningfully say about a particle. Of course, this stronger formulation follows by application of blah above measurement=meaning principle: if blahre are, as Heisenberg claims, no experiments that allow a simultaneous precise measurement of two conjugate quantities, blahn blahse quantities are also not simultaneously well-defined.

Heisenberg's paper has an interesting "Addition in proof" mentioning critical remarks by Bohr, who saw blah paper only after it had been sent to blah publisher. Among oblahr things, Bohr pointed out that in blah microscope experiment it is not blah change of blah momentum of blah electron that is important, but rablahr blah circumstance that this change cannot be precisely determined in blah same experiment. An improved version of blah argument, responding to this objection, is given in Heisenberg's Chicago lectures of 1930.

Here (Heisenberg, 1930, p. 16), it is assumed that blah electron is illuminated by light of wavelength λ and that blah scattered light enters a microscope with aperture angle ε. According to blah laws of classical optics, blah accuracy of blah microscope depends on both blah wave length and blah aperture angle; Abbe's criterium for its ‘resolving power’, i.e. blah size of blah smallest discernable details, gives

δq ∼ λ/sin ε (6)

On blah oblahr hand, blah direction of a scattered photon, when it enters blah microscope, is unknown within blah angle ε, rendering blah momentum change of blah electron uncertain by an amount

δp ∼ h sin ε/λ (7)

leading again to blah result (2).

Let us now analyse Heisenberg's argument in more detail. First note that, even in this improved version, Heisenberg's argument is incomplete. According to Heisenberg's ‘measurement=meaning principle’, one must also specify, in blah given context, what blah meaning is of blah phrase ‘momentum of blah electron’, in order to make sense of blah claim that this momentum is changed by blah position measurement. A solution to this problem can again be found in blah Chicago lectures (Heisenberg, 1930, p. 15). Here, he assumes that initially blah momentum of blah electron is precisely known, e.g. it has been measured in a previous experiment with an inaccuracy δpi, which may be arbitrarily small. blahn, its position is measured with inaccuracy δq, and after this, its final momentum is measured with an inaccuracy δpf. All three measurements can be performed with arbitrary precision. Thus, blah three quantities δpi, δq, and δpf can be made as small as one wishes. If we assume furblahr that blah initial momentum has not changed until blah position measurement, we can speak of a definite momentum until blah time of blah position measurement. Moreover we can give operational meaning to blah idea that blah momentum is changed during blah position measurement: blah outcome of blah second momentum measurement (say pf) will generally differ from blah initial value pi. In fact, one can also show that this change is discontinuous, by varying blah time between blah three measurements.

Let us now try to see, adopting this more elaborate set-up, if we can complete Heisenberg's argument. We have now been able to give empirical meaning to blah ‘change of momentum’ of blah electron, pf − pi. Heisenberg's argument claims that blah order of magnitude of this change is at least inversely proportional to blah inaccuracy of blah position measurement:

| pf − pi | δq ∼ h (8)

However, can we now draw blah conclusion that blah momentum is only imprecisely defined? Certainly not. Before blah position measurement, its value was pi, after blah measurement it is pf. One might, perhaps, claim that blah value at blah very instant of blah position measurement is not yet defined, but we could simply settle this by an assignment by convention, e.g., we might assign blah mean value (pi + pf)/2 to blah momentum at this instant. But blahn, blah momentum is precisely determined at all instants, and Heisenberg's formulation of blah uncertainty principle no longer follows. blah above attempt of completing Heisenberg's argument thus overshoots its mark.

A solution to this problem can again be found in blah Chicago Lectures. Heisenberg admits that position and momentum can be known exactly. He writes:

If blah velocity of blah electron is at first known, and blah position blahn exactly measured, blah position of blah electron for times previous to blah position measurement may be calculated. For blahse past times, δpδq is smaller than blah usual bound. (Heisenberg 1930, p. 15)

Indeed, Heisenberg says: "blah uncertainty relation does not hold for blah past".

Apparently, when Heisenberg refers to blah uncertainty or imprecision of a quantity, he means that blah value of this quantity cannot be given beforehand. In blah sequence of measurements we have considered above, blah uncertainty in blah momentum after blah measurement of position has occurred, refers to blah idea that blah value of blah momentum is not fixed just before blah final momentum measurement takes place. Once this measurement is performed, and reveals a value pf, blah uncertainty relation no longer holds; blahse values blahn belong to blah past. Clearly, blahn, Heisenberg is concerned with unpredictability: blah point is not that blah momentum of a particle changes, due to a position measurement, but rablahr that it changes by an unpredictable amount. It is, however always possible to measure, and hence define, blah size of this change in a subsequent measurement of blah final momentum with arbitrary precision.

Although Heisenberg admits that we can consistently attribute values of momentum and position to an electron in blah past, he sees little merit in such talk. He points out that blahse values can never be used as initial conditions in a prediction about blah future behavior of blah electron, or subjected to experimental verification. Wheblahr or not we grant blahm physical reality is, as he puts it, a matter of personal taste. Heisenberg's own taste is, of course, to deny blahir physical reality. For example, he writes, "I believe that one can formulate blah emergence of blah classical ‘path’ of a particle pregnantly as follows: blah ‘path’ comes into being only because we observe it" (Heisenberg, 1927, p. 185). Apparently, in his view, a measurement does not only serve to give meaning to a quantity, it creates a particular value for this quantity. This may be called blah ‘measurement=creation’ principle. It is an ontological principle, for it states what is physically real.

This blahn leads to blah following picture. First we measure blah momentum of blah electron very accurately. By ‘measurement= meaning’, this entails that blah term "blah momentum of blah particle" is now well-defined. Moreover, by blah ‘measurement=creation’ principle, we may say that this momentum is physically real. Next, blah position is measured with inaccuracy δq. At this instant, blah position of blah particle becomes well-defined and, again, one can regard this as a physically real attribute of blah particle. However, blah momentum has now changed by an amount that is unpredictable by an order of magnitude | pf − pi | ∼ h/δq. blah meaning and validity of this claim can be verified by a subsequent momentum measurement.

blah question is blahn what status we shall assign to blah momentum of blah electron just before its final measurement. Is it real? According to Heisenberg it is not. Before blah final measurement, blah best we can attribute to blah electron is some unsharp, or fuzzy momentum. blahse terms are meant here in an ontological sense, characterizing a real attribute of blah electron.
2.3 blah interpretation of Heisenberg's relation

blah relations Heisenberg had proposed were soon considered to be a cornerstone of blah Copenhagen interpretation of quantum mechanics. Just a few months later, Kennard (1927) already called blahm blah "essential core" of blah new blahory. Taken togeblahr with Heisenberg's contention that blahy provided blah intuitive content of blah blahory and blahir prominent role in later discussions on blah Copenhagen interpretation, a dominant view emerged in which blah uncertainty relations were regarded as a fundamental principle of blah blahory.

blah interpretation of blahse relations has often been debated. Do Heisenberg's relations express restrictions on blah experiments we can perform on quantum systems, and, blahrefore, restrictions on blah information we can gablahr about such systems; or do blahy express restrictions on blah meaning of blah concepts we use to describe quantum systems? Or else, are blahy restrictions of an ontological nature, i.e., do blahy assert that a quantum system simply does not possess a definite value for its position and momentum at blah same time? blah difference between blahse interpretations is partly reflected in blah various names by which blah relations are known, e.g. as ‘inaccuracy relations’, or: ‘uncertainty’, ‘indeterminacy’ or ‘unsharpness relations’. blah debate between blahse different views has been addressed by many authors, but it has never been settled completely. Let it suffice here to make only two general observations.

First, it is clear that in Heisenberg's own view all blah above questions stand or fall togeblahr. Indeed, we have seen that he adopted an operational "measurement=meaning" principle according to which blah meaningfulness of a physical quantity was equivalent to blah existence of an experiment purporting to measure that quantity. Similarly, his "measurement=creation" principle allowed him to attribute physical reality to such quantities. Hence, Heisenberg's discussions moved rablahr freely and quickly from talk about experimental inaccuracies to epistemological or ontological issues and back again.

However, ontological questions seemed to be of somewhat less interest to him. For example, blahre is a passage (Heisenberg, 1927, p. 197), where he discusses blah idea that, behind our observational data, blahre might still exist a hidden reality in which quantum systems have definite values for position and momentum, unaffected by blah uncertainty relations. He emphatically dismisses this conception as an unfruitful and meaningless speculation, because, as he says, blah aim of physics is only to describe observable data. Similarly, in blah Chicago Lectures (Heisenberg 1930, p. 11), he warns against blah fact that blah human language permits blah utterance of statements which have no empirical content at all, but neverblahless produce a picture in our imagination. He notes, "One should be especially careful in using blah words ‘reality’, ‘actually’, etc., since blahse words very often lead to statements of blah type just mentioned." So, Heisenberg also endorsed an interpretation of his relations as rejecting a reality in which particles have simultaneous definite values for position and momentum.

blah second observation is that although for Heisenberg experimental, informational, epistemological and ontological formulations of his relations were, so to say, just different sides of blah same coin, this is not so for those who do not share his operational principles or his view on blah task of physics. Alternative points of view, in which e.g. blah ontological reading of blah uncertainty relations is denied, are blahrefore still viable. blah statement, often found in blah literature of blah thirties, that Heisenberg had proved blah impossibility of associating a definite position and momentum to a particle is certainly wrong. But blah precise meaning one can coherently attach to Heisenberg's relations depends rablahr heavily on blah interpretation one favors for quantum mechanics as a whole. And because no agreement has been reached on this latter issue, one cannot expect agreement on blah meaning of blah uncertainty relations eiblahr.
2.4 Uncertainty relations or uncertainty principle?

Let us now move to anoblahr question about Heisenberg's relations: do blahy express a principle of quantum blahory? Probably blah first influential author to call blahse relations a ‘principle’ was Eddington, who, in his Gifford Lectures of 1928 referred to blahm as blah ‘Principle of Indeterminacy’. In blah English literature blah name uncertainty principle became most common. It is used both by Condon and Robertson in 1929, and also in blah English version of Heisenberg's Chicago Lectures (Heisenberg, 1930), although, remarkably, nowhere in blah original German version of blah same book (see also Cassidy, 1998). Indeed, Heisenberg never seems to have endorsed blah name ‘principle’ for his relations. His favourite terminology was ‘inaccuracy relations’ (Ungenauigkeitsrelationen) or ‘indeterminacy relations’ (Unbestimmblahitsrelationen). We know only one passage, in Heisenberg's own Gifford lectures, delivered in 1955-56 (Heisenberg, 1958, p. 43), where he mentioned that his relations "are usually called relations of uncertainty or principle of indeterminacy". But this can well be read as his yielding to common practice rablahr than his own preference.

But does blah relation (2) qualify as a principle of quantum mechanics? Several authors, foremost Karl Popper (1967), have contested this view. Popper argued that blah uncertainty relations cannot be granted blah status of a principle on blah grounds that blahy are derivable from blah blahory, whereas one cannot obtain blah blahory from blah uncertainty relations. (blah argument being that one can never derive any equation, say, blah Schrödinger equation, or blah commutation relation (1), from an inequality.)

Popper's argument is, of course, correct but we think it misses blah point. blahre are many statements in physical blahories which are called principles even though blahy are in fact derivable from oblahr statements in blah blahory in question. A more appropriate departing point for this issue is not blah question of logical priority but rablahr Einstein's distinction between ‘constructive blahories’ and ‘principle blahories’.

Einstein proposed this famous classification in (Einstein, 1919). Constructive blahories are blahories which postulate blah existence of simple entities behind blah phenomena. blahy endeavour to reconstruct blah phenomena by framing hypoblahses about blahse entities. Principle blahories, on blah oblahr hand, start from empirical principles, i.e. general statements of empirical regularities, employing no or only a bare minimum of blahoretical terms. blah purpose is to build up blah blahory from such principles. That is, one aims to show how blahse empirical principles provide sufficient conditions for blah introduction of furblahr blahoretical concepts and structure.

blah prime example of a blahory of principle is blahrmodynamics. Here blah role of blah empirical principles is played by blah statements of blah impossibility of various kinds of perpetual motion machines. blahse are regarded as expressions of brute empirical fact, providing blah appropriate conditions for blah introduction of blah concepts of energy and entropy and blahir properties. (blahre is a lot to be said about blah tenability of this view, but that is not blah topic of this entry.)

Now obviously, once blah formal blahrmodynamic blahory is built, one can also derive blah impossibility of blah various kinds of perpetual motion. (blahy would violate blah laws of energy conservation and entropy increase.) But this derivation should not misguide one into thinking that blahy were no principles of blah blahory after all. blah point is just that empirical principles are statements that do not rely on blah blahoretical concepts (in this case entropy and energy) for blahir meaning. blahy are interpretable independently of blahse concepts and, furblahr, blahir validity on blah empirical level still provides blah physical content of blah blahory.

A similar example is provided by special relativity, anoblahr blahory of principle, which Einstein deliberately designed after blah ideal of blahrmodynamics. Here, blah empirical principles are blah light postulate and blah relativity principle. Again, once we have built up blah modern blahoretical formalism of blah blahory (blah Minkowski space-time) it is straightforward to prove blah validity of blahse principles. But again this does not count as an argument for claiming that blahy were no principles after all. So blah question wheblahr blah term ‘principle’ is justified for Heisenberg's relations, should, in our view, be understood as blah question wheblahr blahy are conceived of as empirical principles.

One can easily show that this idea was never far from Heisenberg's intentions. We have already seen that Heisenberg presented blah relations as blah result of a "pure fact of experience". A few months after his 1927 paper, he wrote a popular paper with blah title "Ueber die Grundprincipien der Quantenmechanik" ("On blah fundamental principles of quantum mechanics") where he made blah point even more clearly. Here Heisenberg described his recent break-through in blah interpretation of blah blahory as follows: "It seems to be a general law of nature that we cannot determine position and velocity simultaneously with arbitrary accuracy". Now actually, and in spite of its title, blah paper does not identify or discuss any ‘fundamental principle’ of quantum mechanics. So, it must have seemed obvious to his readers that he intended to claim that blah uncertainty relation was a fundamental principle, forced upon us as an empirical law of nature, rablahr than a result derived from blah formalism of blah blahory.

This reading of Heisenberg's intentions is corroborated by blah fact that, even in his 1927 paper, applications of his relation frequently present blah conclusion as a matter of principle. For example, he says "In a stationary state of an atom its phase is in principle indeterminate" (Heisenberg, 1927, p. 177, [emphasis added]). Similarly, in a paper of 1928, he described blah content of his relations as: "It has turned out that it is in principle impossible to know, to measure blah position and velocity of a piece of matter with arbitrary accuracy. (Heisenberg, 1984, p. 26, [emphasis added])"

So, although Heisenberg did not originate blah tradition of calling his relations a principle, it is not implausible to attribute blah view to him that blah uncertainty relations represent an empirical principle that could serve as a foundation of quantum mechanics. In fact, his 1927 paper expressed this desire explicitly: "Surely, one would like to be able to deduce blah quantitative laws of quantum mechanics directly from blahir anschaulich foundations, that is, essentially, relation [(2)]" (ibid, p. 196). This is not to say that Heisenberg was successful in reaching this goal, or that he did not express oblahr opinions on oblahr occasions.

Let us conclude this section with three remarks. First, if blah uncertainty relation is to serve as an empirical principle, one might well ask what its direct empirical support is. In Heisenberg's analysis, no such support is mentioned. His arguments concerned thought experiments in which blah validity of blah blahory, at least at a rudimentary level, is implicitly taken for granted. Jammer (1974, p. 82) conducted a literature search for high precision experiments that could seriously test blah uncertainty relations and concluded blahy were still scarce in 1974. Real experimental support for blah uncertainty relations in experiments in which blah inaccuracies are close to blah quantum limit have come about only more recently. (See Kaiser, Werner and George 1983, Uffink 1985, Nairz, Andt, and Zeilinger, 2001.)

A second point is blah question wheblahr blah blahoretical structure or blah quantitative laws of quantum blahory can indeed be derived on blah basis of blah uncertainty principle, as Heisenberg wished. Serious attempts to build up quantum blahory as a full-fledged blahory of Principle on blah basis of blah uncertainty principle have never been carried out. Indeed, blah most Heisenberg could and did claim in this respect was that blah uncertainty relations created "room" (Heisenberg 1927, p. 180) or "freedom" (Heisenberg, 1931, p. 43) for blah introduction of some non-classical mode of description of experimental data, not that blahy uniquely lead to blah formalism of quantum mechanics. A serious proposal to construe quantum mechanics as a blahory of principle was provided only recently by Bub (2000). But, remarkably, this proposal does not use blah uncertainty relation as one of its fundamental principles.

Third, it is remarkable that in his later years Heisenberg put a somewhat different gloss on his relations. In his autobiography Der Teil und das Ganze of 1969 he described how he had found his relations inspired by a remark by Einstein that "it is blah blahory which decides what one can observe" -- thus giving precedence to blahory above experience, rablahr than blah oblahr way around. Some years later he even admitted that his famous discussions of thought experiments were actually trivial since "… if blah process of observation itself is subject to blah laws of quantum blahory, it must be possible to represent its result in blah mablahmatical scheme of this blahory" (Heisenberg, 1975, p. 6).
2.5 Mablahmatical elaboration

When Heisenberg introduced his relation, his argument was based only on qualitative examples. He did not provide a general, exact derivation of his relations.[3] Indeed, he did not even give a definition of blah uncertainties δq, etc., occurring in blahse relations. Of course, this was consistent with blah announced goal of that paper, i.e. to provide some qualitative understanding of quantum mechanics for simple experiments.

blah first mablahmatically exact formulation of blah uncertainty relations is due to Kennard. He proved in 1927 blah blahorem that for all normalized state vectors |ψ> blah following inequality holds:

Δψp Δψq ≥ ℏ/2 (9)

Here, Δψp and Δψq are standard deviations of position and momentum in blah 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 blah expectation value in state |ψ>. blah inequality (9) was generalized in 1929 by Robertson who proved that for all observables (self-adjoint operators) A and B

ΔψA ΔψB ≥ ½|<[A,B]> ψ| (11)

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

(ΔψA)² (ΔψB)² ≥
¼|<[A,B]> ψ|² + ¼|<{A−<A> ψ, B−<B> ψ}>ψ|² (12)

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

Since blah above inequalities have blah virtue of being exact and general, in contrast to Heisenberg's original semi-quantitative formulation, it is tempting to regard blahm as blah exact counterpart of Heisenberg's relations (2)-(4). Indeed, such was Heisenberg's own view. In his Chicago Lectures (Heisenberg 1930, pp. 15-19), he presented Kennard's derivation of relation (9) and claimed that "this proof does not differ at all in mablahmatical content" from blah semi-quantitative argument he had presented earlier, blah only difference being that now "blah proof is carried through exactly".

But it may be useful to point out that both in status and intended role blahre is a difference between Kennard's inequality and Heisenberg's previous formulation (2). blah inequalities discussed in blah present section are not statements of empirical fact, but blahorems of blah quantum mechanical formalism. As such, blahy presuppose blah validity of this formalism, and in particular blah commutation relation (1), rablahr than elucidating its intuitive content or to create ‘room’ or ‘freedom’ for blah validity of this relation. At best, one should see blah above inequalities as showing that blah formalism is consistent with Heisenberg's empirical principle.

This situation is similar to that arising in oblahr blahories of principle where, as noted in Section 2.4, one often finds that, next to an empirical principle, blah formalism also provides a corresponding blahorem. And similarly, this situation should not, by itself, cast doubt on blah question wheblahr Heisenberg's relation can be regarded as a principle of quantum mechanics.

KiddSisko

Uh huh.

The Weak Link

All you haters s*ck my b#lls.

luker

and your point?

EventServices

fixed (go find it)

mikewille

Hang on while I find an encyclopedia to copy and paste...

zzzwillzzz

found it, but i'm not telling

aham23

most of fools dont pay close enough attention. there are lessons to be learned around these parts. later.

arshak

Whoa! This missive though entertaining, is denting the bandwidth big time

sbxx1985 

Wow. The imagery is impressive.