Revista Brasileira de Ensino de Fısica, vol. 43, e20200508 (2021) Physics Education Researchwww.scielo.br/rbef cb

DOI: https://doi.org/10.1590/1806-9126-RBEF-2020-0508 Licenca Creative Commons

Indistinguishability and Counting: Teaching the role ofdifferent interpretations in Quantum Mechanics

O.L.S. Filho1 , Marcello Ferreira*1

1Universidade de Brasılia, Instituto de Fısica, Brasilia, DF, Brasil.

Received on December 9, 2020. Revised on February 24, 2021. Accepted on March 23, 2021.

The theme of indistinguishability in the context of Quantum Mechanics as opposed to a distinguishabilityin the context of Classical Mechanics has been treated for almost a century in terms of an essential differencebetween the way of counting in Quantum and Classical Mechanics. Such differences are based on the adoption ofthe principles of Heisenberg, Duality and Complementarity. In this work, we show that it is possible to avoid suchconsiderations and derive the same results considering only the Correspondence Principle. This is done consideringthat, in the classic context, the quantum discrete energy levels become continuous or almost continuous. Fromthese results, we indicate how to develop a discussion within the scope of teaching Quantum Mechanics thatinvolves such principles and concepts and possible comparisons, working on ideas related to the interpretation ofQuantum Mechanics with regard to this phenomenon, rarely developed in traditional courses.Keywords: Indistinguishability, Counting principles, Quantum Mechanics, Physics teaching.

1. Introduction

In usual courses of Quantum Mechanics, one is gen-erally taught the three different probability weightfunctions, known as Boltzmann’s, Bose-Einstein’s andFermi-Dirac’s. It is then argued that they come fromthree different counting strategies, each one with its ownproperties and characteristics.

From Boltzmann’s and Gibbs’ seminal works [1],the investigations of Ehrenfest [2] and the presentapproaches on the subject [3], it is generally assumedthat the major difference between classical and quantumcounting was due to the ability to distinguish amongclassical particles, even if they are identical.

There are two arguments in favor of this classicaldistinguishability: (a) one always has trajectories in theclassical domain and (b) they suffice to “follow” particlesand, thus, keep track of their identities.

This framework is contrasted to the quantum onewhich assumes, in the context of most interpretations,some slightly different perspectives regarding the issueof trajectories within Quantum Mechanics:

1. one may have trajectories in Quantum Mechanics,but Heisenberg’s Principle impedes one to followthese trajectories without error to keep track ofparticles – the errors grow in such a way thatthe particles trajectories become mixed up in anirreducible way;

2. there are no trajectories in Quantum Mechan-ics because, without being measured, quantum

* Correspondence email address: marcellof@unb.br

objects present a dual aspect, as sustained bythe Duality Principle and the ComplementaryPrinciple;

Both these perspectives come from the way interpre-tations of Quantum Mechanics (mainly those comingfrom the Copenhagen Interpretation) interrelate thesethree fundamental principles. They are slightly differentbecause the first can be called “epistemological”, since itresides in the ability of knowing (measuring) somethingthat do exist; while the second should be called “ontolog-ical”, since it affirms that it is the physical object itself(the trajectories) that has no being [4].

At the end, these interpretations relate distinguisha-bility with tracking capability. The fact is that it is gener-ally assumed that classical entities, otherwise identical,can always be distinguished from each other becausethey can be tracked, while quantum ones should beassumed indistinguishable because we lack the habilityto do so.

As is amply known, Boltzmann’s weight is assumedto represent the inadequacy of Classical counting, sinceits use in actual calculations gives rise to Gibbs Para-dox in Classical Statistical Mechanics, when it fur-nishes a non-extensive entropy [5]. Quantum counting,on the contrary, gives the correct extensive entropyresults.

Since these results lay on quite specific perspectivesrelated to the interpretation of the quantum formalismfor this context, it would be interesting if one couldpresent a different approach to counting that make norecourse to trajectories and the like. Such an approachwould be a source for teaching some aspects regarding

Copyright by Sociedade Brasileira de Fısica. Printed in Brazil.

e20200508-2 Indistinguishability and Counting: Teaching the role of different interpretation in Quantum Mechanics

the role of interpretation within Quantum Mechanics,something not usual in courses within this theme.

In this paper, we present such an approach: we showthat one can obtain the correct results for Boltzmann’sweight (giving extensive entropy) assuming only thatthere exists a classical indistinguishability and a con-tinuous spectrum of energy. This would serve as anexample, within the subject of this paper, of the impactof assuming different interpretations for the quantumformalism.

We will show that this alternative approach not onlyunifies the classical and quantum ways of counting, but italso gives the correct Boltzmann’s weight, removing theissue of Gibb’s paradox. This latter fact can then be usedto compare both approaches, revealing other possibleepistemological discussions related to the interpretationof physical theory, but considered in approaches such asthat of Greca & Freire [6], which propose, in additionto the relativity of the importance of mathematicalformalism, teaching in the area based on a philosophicalsensitivity mediated by the following didactic binomial:balanced introduction of fundamental concepts – andstrictly quantum – and consideration of interpretiveissues. This perspective is timely linked to the ideathat we bring about the use of Johnson-Laird’s mentalmodels as a potentially qualifying cognitive strategy ofinterpretation in the teaching of Quantum Mechanics.

2. Some Trivial Results on Counting

Let’s take, as example, the following trivial situationwhere everyone (presumably) will agree on the way oneshould be counting [7].

One has nine geometric objects as in Figure 1. Threeof them are red (R), two are yellow (Y) and four areblue (B), while the shapes are as indicated in the figure.We then ask:

• [Q1]: in how many different ways can we arrangethe geometrical figures in Figure 1 with respectto color?

The terms in bold are extremely important for ourinterests: “arrange” tells us that we are assuming that

Figure 1: Nine geometrical figures with colors. Three red (R),two yellow (Y) and four blue (B).

the order of the presentation of the colored geometricalfigures matters; (b) “with respect to color” tells us thecounting principle – that is, what is to be counted.

This is a trivial problem that can be found in anytextbook on probability and statistics. The answer is,obviously, the weight related to a permutation withrepetition:

WQ1 = 9!3!2!4! , (1)

since, with respect to color, the first three geometricalobjects are considered indistinguishable, as with thefourth and fifth yellow objects and also the last fourblue objects. This is why one has a 3!, a 2! and a 4! inthe denominator of (1).

Of course, in the case of n objects that can be consid-ered indistinguishable with respect to some property, wewould get simply:

Wn = n!n1!n2! · · ·nK ! , (2)

where K is the number of different values for theproperty, n is the number of objects and ni is the numberof indistinguishable objects within some class definedby the property used to count (e.g. color, color plusconstitution, etc).

To make our point as clearly as possible, let us nowchange our question to:

• [Q2]: in how many different ways can we arrangethe geometrical figures of Figure 1 with respectto color and shape?

The answer is again trivial and is given by:

WQ2 = 9!1!2!1!1!1!2!1! , (3)

since, now, the categories of indistinguishable objectschanged when we changed our property used to count.Note that the denominators also changed.

We then learn that each ni! in the denominatorof (2) reflects exactly one class of indistinguishableobjects, given the property used to count (the property,of course, can be complex and referring to more thanone characteristic of the objects, like color and shape).

The obvious conclusion is that the concept of distin-guishability is always used in an operational perspective,not an ontological one. It means nothing if, in Q1,one can “see” (or track) that some colored geometricalobjects have different shapes; the property used to countbeing color, their distinguishability can be phrased onlywith respect to their color – irrespective of what “theyreally are”.

The previous argument remains unchanged if we applyour counting skills to classical or quantum worlds. Giventhe properties of the objects, counting proceeds withoutreference to such worlds.

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In the usual argument, classical particles, when havingexactly the same physical properties (as mass, charge,etc), must be considered identical. However, they alsoshould be considered distinguishable because we can puton them some mark or simply track them from theirtrajectories.

However, in the usual physical situation we count howenergy cells can be filled by these particles. The propertyused to count is energy. It is not energy & trajectory.It is, thus, irrelevant if we can or cannot track theparticles. If these particles are considered identical, theonly thing that can differentiate them is the energy theyassume, and this is precisely what is inscribed into theni! in the denominator of (2) – there are ni particlesassuming energy in the cell [εi, εi + ∆εi], nj particlesassuming energy in cell [εj , εj + ∆εj ], etc.

In fact, if tracking capabilities were to be assumedwithin the counting principle, then there should be noni! in the denominator, since we would have to considerall particles “distinguishable”, since all particles havedifferent trajectories (or are associated to different num-bers, or colors, or can be seen by us as being different),and the [sole complex] property used to count shouldbe “particles with different trajectories and energies” –this is completely analogous to the “shape and colorgeometrical objects” already mentioned. Thus, therewould be no nj in the denominator. The number of waysto arrange them would then be equal to the number ofways to permute them, and counting would fail to giveBoltzmann’s probability density function.

3. A Different Approach

In agreement with the above arguments, which disre-gards some possible “tracking capability”, we may nowask in how many different ways we can fill our energyboxes (or colored boxes) [8]. This means that, in allsituations in which we have only one set of quantumobjects (e.g. only photons or only electrons) presentingenergies filling different energy cells, we would have tocombine (not arrange) these objects into the energycells with possible repetitions. Lets consider, again, theproblem in the language of geometrical figures:

Consider three colored boxes with sub boxes, one Red(sub boxes light red and dark red, that is, degenerateby gR = 2 with respect to red color), one Yellow (subboxes light yellow and dark yellow, gY = 2), and oneBlue (sub boxes light blue, medium blue and dark blue,gB = 3) and let us ask:

• [Q3]: in how many ways we can fill these coloredboxes with N geometrical figures such that we putnR into the Red box, nY into the Yellow box andnB into the Blue box?

Note that, by uttering the counting principle as relatedto “geometrical figures”, all geometrical figures becomeidentical (no matter if they have different colors orshapes).

Figure 2: Ways to combine nine geometrical objects into threedegenerate stages with degenerate degree numbers equal to 2,2 and 3.

It is easy to see that we get simply (see Figure 2):

WQ3 = (nR + gR − 1)!nR! (gR − 1)! ·

(nY + gY − 1)!nY ! (gY − 1)! ·

(nB + gB − 1)!nB ! (gB − 1)! ,

(4)such that N = nR + nY + nB , since, fixing thislast expression as a constraint, the three factors aboverepresent independent probabilities. The result we get ofthe previous counting furnishes the ways we can get nRRed, nY Yellow and nB Blue geometric figures (wherewe are counting each shade of color as “degenerate” withrespect to the underlying color).

The differences are obvious if we take a look atFigure 10-2 and compare it with Figure 10-1. In Figure10-2 the geometrical figures lost their identificationlabels (colors), which were now passed to the boxes,meaning that the geometrical figures are all identical (asgeometrical objects [as quantum objects]) no matter howdifferent they may be in shape [in trajectories, labels,etc.] That is, all figures [objects] are entitled to occupyany one of the degenerate shade-states [degenerateenergy states] of the color-boxes [energy boxes].

This colorful example is completely analogous to theusual Quantum Mechanical counting of bosons. Again,we must stress that the fact that the geometrical figuresmay be “visually distinguishably” meant nothing to theoutcome of our counting—they are being counted asgeometrical figures [identical objects], not as geometricalfigures of some shape or another [identical objects withsome trajectory or another].

4. The Usual Way of Counting

Boltzmann’s weight function is generally presented com-pared to the quantum weight functions. Table 1 showsthe usual approach of textbooks in giving an example ofthe three ways of counting for two particles occupyingtwo degenerate states.

The first counting assumes that the particles aredistinguishable; thus, there are two possibilities forstate (1, 1); the other two counting strategies assumeindistinguishability, the last one assuming also Pauli’sExclusion Principle.

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e20200508-4 Indistinguishability and Counting: Teaching the role of different interpretation in Quantum Mechanics

Table 1: Usual counting process for Boltzmann’s, Bose-Einstein’s and Fermi-Dirac’s distributions.

DistributionsBoltzmann Bose-Einstein Fermi-Dirac

(2,0) ab ∅ aa ∅ ∅ ∅(0,2) ∅ ab ∅ aa ∅ ∅

(1,1) a b a a a ab a

We note that the way of counting for Bose-Einsteinand Fermi-Dirac are equivalent, but Fermi-Dirac’scounting principle includes half-integral spin particles,which are governed by Pauli’s Principle. Thus the firsttwo lines of Fermi-Dirac’s distribution cannot be filled,since they will assume two quantum objects in the sameenergy cell. These ways of counting (combinations) arequalitatively different from the way of counting thatgives rise to Boltzmann’s weight (arrange).

If one generalizes the results of Table 1 for gi stateswith ni particles, one gets the following weight functions:

W ∗B ({ni}) =N !

K∏j=1

gnj

j

nj !

WBE ({ni}) =K∏j=1

(gj − 1 + nj)!(gj − 1)!nj !

WFD ({ni}) =K∏j=1

gj !(gj − nj)!nj !

, (5)

for Boltzmann’s, Bose-Einstein’s and Dirac’s weightfunctions. In this case, W ∗

B gives an incorrect answerbecause of the N ! factor, which makes, for instance, theentropy non-extensive in calculations for ideal gases [9].However, we may take another path to Boltzmann’sdistribution, which clarifies what we have previouslysaid.

5. Deducing the Correct Boltzmann’sWeight

Our first assumption was to assume the same countingprinciple to all three situations (Fermi-Dirac’s assumingalso Pauli’s principle), we can find WBE or WFD asour weights (Cf. with [10]). How then can we getBoltzmann’s weight from one of these (or even both)?

We thus present our second assumption: our energystates are continuous (or semi-continuous). Thus, wemay take the limit gj →∞, ∀j in both WBE and WFD.In this case, Stirling’s approximation allows us to writen! = nne−n and thus:

WBE({nj}) =K∏j=1

(gj + nj)gj+nj

nnj

j ggj

j

=K∏j=1

gnj

j (1 + nj/gj)gj

nnj

j

; (6)

using the fact that (1 + x/s)s → ex if s → ∞, we endwith:

WBE({nj})→K∏j=1

gnj

j

nnj

j e−nj

→K∏j=1

gnj

j

nj != WB({nj})

, (7)

where WB already lacks the N ! term, which W ∗B shows,

and is devoid of problems regarding the extensive char-acter of the entropy.

The same approach can be used for WFD. We thushave:

WFD({nj}) =K∏j=1

gnj

j

(1− nj/gj)gjenjnj !

→K∏j=1

gnj

j

n! = WB({nj})

(8)

Figure 3 shows the asymptotic behavior of WB/WFD

and WB/WBE as g grows with a fixed n = 200.Now we must ask for the interpretation of the limit

gj/nj → ∞. For the example regarding colors, thiswould mean that each coloured box has an infinity ofpossible shades, as shown in Figure 4.

Clearly, this means, when it comes to quantum objectsand energy boxes, that for any number of objects nj ,the amount of quantum states to be filled by them

Figure 3: Asymptotic behavior of Bose-Einstein’s an Fermi-Dirac’s weights as the density of states goes to infinity.

Figure 4: The same example of combination of colors, but witha continuous range of shades for each color.

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goes to a continuous or semi-continuous. This is inevery sense equivalent to our combination of colorswhen our colored-boxes are chosen to be ranges of theelectromagnetic spectrum, as in Figure 4.

The conclusion is obvious enough: the only differencebetween Boltzmann’s and Bose-Einstein’s distributions(and also Fermi-Dirac’s) comes from the assumption,in the derivation of the former, that energy forms acontinuum. This has nothing to do with trajectories,Heisenberg’s uncertainty principle, Duality principle orthe Complementary principle.

In fact, making gj → ∞ is related to the Corre-spondence Principle. This shows that this principle isindependent of the other three mentioned above, andcan be used in alternative interpretations of QuantumMechanics that do not take recourse to the them.

6. Comparison Between the TwoApproaches

Now, if we count as in the first column of Table 1 forBoltzmann’s weight (that is, assuming distinguishabilityof particles with respect to the energy), we get thephysically wrong expression (because of the term N !),while if we count as in the second or third columnsof Table 1 for Bose-Einstein’s or Fermi-Dirac’s weight(that is, assuming indistinguishability with respect to theenergy) and take the limit of continuous energy stateswe get the physically correct expression for Boltzmann’sweight function.

It seems obvious that one should take the secondapproach, not the first. In such a case, all three weightfunctions come not from the assumption of some ontolog-ical indistinguishability of particles, but the assumptionof a continuity of accessible states. In this framework, ina way analogous to situation Q3 already mentioned, welearn that our ability to track particles is irrelevant tothe statistical result, given the counting property, whichis the filling of quantum mechanical energy states.

This approach can give comprehensive grounds tothe usual “large temperature limit” usually mentionedin the context of these statistical probability density.Thus, the fact that the two quantum mechanical dis-tributions (µ is the chemical potential):

ni,BE = gie(εi−µ)/kbT − 1

ni,FD = gie(εi−µ)/kbT + 1

, (9)

tend to Boltzmann’s statistical distribution

ni,B = e−εi/kbT , (10)

can be interpreted as showing that the increase intemperature and low density will increase the numberof accessible states, making them to tend to a quasicon-tinuous spectrum.

It remains for us to show how the above presentedfeatures can be approached in an actual class. We presentone possibility in the next section.

7. The Role of Mental Models inLearning: Johnson-Laird

Johnson-Laird’s mental models integrate the set of cog-nitive theories in which the human mind corresponds toa complex and hierarchical system: a cognitive structurecapable of receiving, understanding, storing and usinginformation. To this end, it admits that the mentalmodels of individuals are structural analogs of theworld [11], that is, internal representations of a cognitivestructure which have a direct relationship with elementsof the outside world.

The central role of mental models is to mediate theindividual’s comprehension, explanation and inferencecapacities about their non-cognitive analog [12]. Theypresent themselves at different levels of complexity,ranging from those which represent simple objects, ofthe broad domain and acquaintance, to those that serveas scientific theories, and are characterized by nineprinciples [11, 12]:

1. Computability: they must be described in the formof non-dubious procedures, perfectly executable bya process or machine.

2. Finitude: deriving from the human cognition, theyare associated with a vast, but limited set ofpossibilities for realization.

3. Constructivism: they are derivatives of essentialelements that are or represent a state (or a setof states) of the outside world.

4. Economy: each configuration of the outside worldis represented by only one mental model, howeverincipient or incomplete it may be.

5. Definiteness: they can only represent indetermina-tion if they are circumvented by mental processingand without the exponential growth of their com-plexity.

6. Predictability: they must be constructed by char-acteristics that intersect at a point; a predicatewhich can apply to any element to which anotherpredicate also applies, as long as not commuta-tively, to provide the distinction between naturaland artificial concepts.

7. Innatism: all the conceptual primitives necessaryfor the construction of mental models are naturalto individuals and make up their perceptual expe-riences.

8. A finite number of conceptual primitives: theconceptual primitives of a given semantic field arenot infinite.

9. Structural identity: the structures of mental mod-els must be identical to those of the states of theworld they refer to – each element has a symbolicfunction and meaning.

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Being individual representations of the world state ofaffairs, the mental models are not necessarily accurateand consistent. Also, for this reason, they are notstationary and can be revised, complemented or evenabandoned at any time [12]. Thus, whenever a givenmental model encounters a situation that it is unableto explain, the individual will be driven to update or, ifthis is not possible, to discard that model. On the otherhand, competing models may be evaluated with respectto the elements just mentioned, economy being one ofthe most prominent and one that particularly connectsto the present approach.

In the teaching field, mental models can be used tofacilitate the understanding of conceptual structures,which are human constructions developed to facilitatethe understanding of physical systems. Unlike mentalmodels, conceptual models are accurate, consistent andcomplete representations of physical system [13].

From the teaching perspective, the teacher’s role isto recover, curate, create, present and evaluate theconceptual models for their students. This set of actionsshould help the students to build, in their cognitivestructures, mental models that are consistent with thephysical systems they represent. In order to identify howthese students construct such models, the teacher mustseek to understand them, bearing in mind that, by theirown individual nature, mental models are confusing,incomplete and unstable and that these characteristicsare not indicators of an inadequate mental model [12].

The previous sections of this article, when describ-ing the counting principles applied to the Boltzmann,Bose-Einstein and Fermi-Dirac distributions, providethe reader with competing conceptual models of thedescribed physical systems. At the same time, whenmaking an analogy of such systems with objects occu-pying spaces in colored boxes, they also try to describea mental model under which it is possible to understandthe distributions mentioned before.

In this mental model, the objects represent quantumobjects, and each colored box and sub-box an energyquantum state and its degenerate states, respectively.The different combinations in which the objects canoccupy the colored sub-boxes represent the differentenergetic degenerate states that the quantum objectscould occupy.

It is essential to highlight that the mental modeldescribed, unlike the conceptual model related to thecounting principles described in the previous sections,is incomplete, inconsistent and inaccurate. This idea isemphasized if it is considered that each reader formsher own mental model related to the one described toher. Again, this does not mean that they are inadequate,as they have a mutable nature and potential to evolveinto models closer to physical reality. Anywise, it ispossible to perceive and demonstrate that its structureis analogous to that of the described phenomena.

It is also worth noting that the article contrasts twomental models within the same theory and the same

subject, thus showing how physical phenomena, notrarely, can be seen from different mental structures. Thistype of conflict is particularly interesting in the presentcase since the presented model (involving only the Prin-ciple of Correspondence, of an epistemological character)confronts an almost universally accepted model (involv-ing the ontological character of the indistinguishabilityof quantum objects).

This is shown in the results obtained, which showthat, whenever the number of boxes tends to infinity,the energy states can be represented as being continuousor almost continuous, which would eliminate, in theBoltzmann distribution, the error related to the entropy.The proposed mental model also allows a continuity (ofan ontological nature) between the characteristics of theobjects in the quantum and classical worlds (identifyingthem in terms of an statistical indistinguishability –not ontological – hence being organized in the epis-temological dimension). We believe it is unnecessaryto say how much such contrast enhances the criticaldimension of apprehension of the theme, in particular,and of Quantum Theory itself, in general.

Finally, it is worth noting that, in his theory, Johnson-Laird had no epistemological commitment and did notpropose methods capable of guiding an instructionalprocess through mental models. What his theory did,it should be noted, was to describe how the processof assimilation of knowledge by the cognitive structureusing mental models would occur. Thereby, as we havealready highlighted in another moment [14], a well-founded and structured didactic on the theme dealtin this study should combine a descriptive theoreticalperspective (cognitivist, in essence, like Laird’s) withother theories or educational methodologies – therefore,of an eminently normative nature – having in mind thedual nature of such intent.

8. Conclusion

The obvious interpretation of the previous argumentsshould be that there is no inherent problem with Classi-cal Physics (in what respects to counting). The problemresides (as it always did) in the (wrong) way many of usdecided to count. In fact, the inadequacy of W ∗

B shouldhave been considered from the start as an indication ofproblems with the chosen process of counting, not as anindication of problems with some particular domain ofphysics (classical or quantum). It seems that we becameso eager to blame Classical Physics for each failure ofsome approach to a particular problem in Physics thatwe lost our ability to scrutinize (and, thus, understand)our own mistakes.

These arguments, together with those of previoussections, should suffice to show how classical and quan-tum counting can be understood as equal with respectto the notion of distinguishability, although differentwith respect to the number of possible degenerate

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Filho and Ferreira e20200508-7

states (where quantization reveals its importance), asexpressed in the Correspondence Principle.

We believe that this would be an important con-tribution to explanations of the topic in our classeson Quantum Mechanics, and can give a very nice(and simple) example of the role of interpretations inQuantum Mechanics.

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