Human language is a substance consisting of combinations of concepts giving rise to meaning. We will show that a good model for this substance is the one of a gas of entangled bosonic quantum particles such as they appear in physics in the situation close to a Bose-Einstein condensate. In this respect we also introduce the new notion of ‘cogniton’ as the entity (particle) playing the same role within human language of the ‘bosonic quantum particle’ for the ‘quantum gas’. There is a gas of bosonic quantum particles that we all know very well, and that is ‘light’, which is a substance of photons. Often we will use ‘light’ as a model and inspiration of how we will talk and reason about human language where ‘concepts’ (words) as ‘states of the cogniton’, are then like ‘photons of different energies (frequencies, wave lengths)’. With the new findings we present here we also make an essential and new step forward in the elaboration of our ‘conceptuality interpretation of quantum theory’, where quantum particles are the concepts of a proto-language, in a similar way that concepts (words), are the cognitons of human language (Aerts, 2009a, 2010a, 2010b, 2013, 2014; Aerts et al., 2018c, 2019c).
There are several new results and insights that we put forward in the coming sections. We shortly mention some of them here in the introduction referring also to earlier work on which they are built, guaranteeing however that the article is self-contained, such that it is not necessary to have studied this earlier work. The reason we can present here now a self-contained theory of human language is because most of the earlier work’s results take a simple and transparent form in the model of a boson gas that we elaborate here for human language. Since we also introduce the basics of the physics of a boson gas our presentation remains self-contained also from a physics’ perspective.
We will see that the state of the gas of bosonic quantum particles which we identify explicitly to also be the state of a piece of text such as that of a story is one of very low temperature, i.e. a temperature in the neighborhood of where also the fifth state of matter appears, namely the Bose-Einstein condensate. This means that the interactions between ‘words’, which are the boson particles of language in our description, is mainly one of ‘quantum superposition’ and ‘quantum entanglement’, or more precisely one of ‘overlapping de Broglie wave functions’. This corresponds well with some of our earlier findings when studying the combinations of concepts (words) in human language, namely that superposition and entanglement are abundant, and the type of entanglement is deep, namely it also violates additionally to Bell’s inequality the marginal laws (Aerts, 2009b; Aerts, Broekaert Gabora, 2011; Aerts & Sozzo, 2011; Aerts et al., 2012; Aerts & Sozzo, 2014; Aerts, Sozzo & Veloz, 2015, 2016; Aerts et al., 2018a, b, 2019a, 2019b; Aerts Arguëlles, 2018; Beltran & Geriente, 2019).
When we present our model in the next sections, we will see that it contains several new explanations of aspects of human language which we brought up in earlier work. For example, we elaborated an axiomatic quantum model for human concepts, which we called SCoP (state context property system), and in which different exemplars of concepts are considered as different states of this concept (Gabora & Aerts, 2002; Aerts & Gabora, 2005a, b; Aerts, 2009b; Aerts et al., 2013a, b). In the theory of the boson gas for human language that we develop here now we will not only introduce these states explicitly, but also introduce them as eigenstates for specific values of the energy. A detailed energy scale for all the words appearing in a considered piece of text will be introduced, which means that the states we consider for the exemplars presenting these words are eigenstates of energy. If we compare with light, it means that the cognitons of our piece of text of human language will radiate their meaning with different frequencies to the human mind engaging in the meaning of this piece of text.
Let us consider an example of a text namely the Winnie the Pooh story entitled ‘In Which Piglet Meets a Haffalump’ (Milne, 1926) to make this introduction of ‘energy’ in our theory of language more concrete. We define the ‘energy level’ of a concept (word, cogniton) in the story by means of the frequency of appearance of that word in that story. The most frequently appearing word, namely 133 times, is the concept And (we will denote concepts in italics starting with a capital, like in our earlier works) and hence we attribute to it the lowest energy level . The second most frequently appearing word with a frequency of 111 times is the concept He, hence we attribute to it the second lowest energy level , and so on …. till we reach words, such as the concept Able, which only appear once. If we think of a story as a ‘gas of bosonic particles’ in ‘thermal equilibrium with its environment’, then these frequencies would indeed indicate different energy levels of the particles of the gas, following the ‘energy distribution law governing in the gas’, hence this is our inspiration in the introduction of ‘energy’ in human language. Remember indeed that each of these words (concepts) is a ‘state of the cogniton particle’, exactly like different energy levels of photons (different wave lengths of light) are each ‘states of the photon’. Proceeding in this way we arrive at 452 energy levels
for the story ‘In Which Piglet Meets a Haffalump’. We denote the frequency of appearance of the concept (word, cogniton) with energy level , and if we denote the total number of energy levels, we have
where is the total number of concepts (words, cognitons) of the considered piece of text, which is 2655 for the story ‘In Which Piglet Meets a Haffalump’.
For each of the energy levels we calculate the total amount of energy ‘radiated’ by the story ‘In Which Piglet Meets a Haffalump’ with the ‘frequency or wave length’ connected to this energy level. For example, the energy level is populated by the concept Thought and the word Thought appeared times in the story ‘In Which Piglet Meets a Haffalump’. Each of the 10 appearances of Thought radiates energy 54, which means that the total radiation with the wave length connected to Thought of the story ‘In Which Piglet Meets a Haffalump’ equals 10 times 54, hence 540.
We introduce the total energy radiated by the considered piece of text
For the story ‘In Which Piglet Meets a Haffalump’ we have . Let us represent now some of the other findings that we will expose in the following sections.
When we applied the Bose-Einstein distribution
to model the data we had collected on the story ‘In Which Piglet Meets a Haffalump’, determining the parameters and by the two requirements
we found an almost complete fit with the data (see section 2, Table 1, Figure 1, Figure 2 and Figure 3). We tested numerous other texts, short stories (see section 3, Table 4, Figure 4, Figure 5 and long stories of the size of novels (see section 4, (b) of Figure 8), and each time it showed that a modeling by means of a boson gas with a Bose-Einstein statistical energy distribution, like explained above, gives rise to an almost complete fit with the data.
We started this investigation with the idea that ‘concepts within human language behave like bosonic entities’, an idea we expressed earlier as one of the basic pieces of evidence for the ‘conceptuality interpretation’ (Aerts, 2009a), and the origin of the idea is the simple direct understanding that if one considers, for example, the concept combination Eleven Animals, then, on the level of the ‘conceptual realm’ each one of the eleven animals is completely ‘identical with’ and ‘indistinguishable from’ each other of the eleven animals. It is also a simple direct understanding that in the case of ‘eleven physical animals’, there will always be differences between each one of the eleven animals, because as ‘objects’ present in the physical world, they have an individuality, and as individuals, with physical bodies, none of them will be really identical with the other one, which means that each one of them will also always be able to be distinguished from the others. Even if all the animals are horses, simply because they are ‘objects’ and not ‘concepts’, they will not be completely identical and hence they will be distinguishable. The idea is that it is ‘this not being completely identical and hence being distinguishable’ which makes the Maxwell-Boltzmann statistics being applicable to them. However, when we consider ‘eleven animals’ as concepts, such that their ontological nature is conceptual, they are all ‘completely identical and hence intrinsically indistinguishable’. Within the conceptuality interpretation of quantum theory, where we put forward the hypothesis that quantum entities are ‘conceptual’ and hence are not ‘objects’, their ‘being completely identical and hence intrinsically indistinguishable’, would also be due to their being conceptual instead of objectual entities.
In earlier work we already investigated this idea by looking at simple numeral quantities of concepts, such as indeed Eleven Animals and then considering two states of Animal, namely Cat and Dog. We then checked whether the twelve different exemplars of them that form in these two states, namely Eleven Dogs, One Cat and Ten Dogs, Two Cats and Nine Dogs, …, Ten cats and One Dog, Eleven Cats, in their appearance in texts follow a Maxwell-Boltzmann or rather a Bose-Einstein statistical pattern. In a less convincing way because of a collection of limited data (Aerts, 2009a; Aerts, Sozzo & Veloz, 2015), but with an abundance of data and very convincingly (Beltran, 2019), it was shown that indeed the Bose-Einstein statistics delivers a better model for the data as compared to the Maxwell-Boltzmann statistics.
The result that we put forward in the present article, namely that the Bose-Einstein statistics as explained above models entire texts of any size is of course a much stronger result still, although it expresses the same idea. Consider any text, and then consider two instances of the word Cat appearing in the text, if then one of the concepts Cat is exchanged with the other concept Cat, nothing and completely absolutely nothing, changes to the text. Hence, a text contains a perfect symmetry for the exchange of cognitons (concepts, words) in the same state. This is not true for physical reality and its physical objects. Suppose one considers a physical landscape where two cats are within the landscape, exchanging the two cats will always change the landscape, because the cats are not identical and are distinguishable as physical objects. If we introduce a quantum description of the text, the wave function must be invariant for the exchange of the two cats, which would again be not the case if the wave function would describe the physical landscape containing two cats as objects. This is the result we will show in section 2.
Section 3 we devote to a self-contained presentation of the phenomenon of Bose-Einstein condensation in physics. We illustrate the different aspects of the Bose-Einstein condensation valuable for our use of it by means of the two examples of the Bose gases, the rubidium 87 atom gas and the sodium atom gas, that also originally where the first one to be used to realize a Bose-Einstein condensate of them (Anderson et al., 1995; Davis et al., 1995).
Another finding that we will put forward in section 4 was completely unexpected. The method of attributing an energy level to a concept depending on the frequency of appearance of the concept in the considered text, introduces the typical ranking considered in the well-known Zipf’s law analysis of texts (Zipf, 1935, 1949). When we look at the graph of ranking in function of the frequency of appearance we indeed see appearing the linear function, or a slight deviation of it, which represents the most common version of Zipf’s law. Zipf’s law is an experimental law, there has not yet been given any theoretical foundation of it, hence perhaps our finding of its unexpected connection with Bose-Einstein statistics might give rise to such a theoretical foundation? We show in section 4 also how the connection with Zipf’s law allows us to develop in more depth the Bose-Einstein model of the texts of different sizes, short stories and long stories of the size of novels.
In section 5 we reflect about the issue of ‘identify and indistinguishability’ from the perspective we developed in the foregoing sections taking into account the conundrum this issue actually still is in quantum theory with respect to quantum particles (Dieks & Lubberdink, 2019). Confronting the theoretical view where bosons and fermions are considered to be identical and indistinguishable even if they are in different states we note that experimentalists take another stance in this respect considering, for example, photons of different frequencies as distinguishable. A recent experiment shows that if this experimentally accepted possibility to distinguish them is erased by means of a quantum eraser, these different frequency photons behave as indistinguishable (Zhao et al., 2014). This make us put forward the proposal that ‘the way in which we clearly see and understand the identify and indistinguishability’ of concepts (words, cognitons) in human language’ and ‘the way in which this also depends on the interface between the language and the human mind, e.g. if the means to recognize a concept by a human mind is erased, the concept will loose its possibility to distinguish it from another concept for which also the means to recognize it has been erased’, could well be the way in which the conundrum that now still exists of how to consider ‘identity and indistinguishability’ for quantum particles needs to be resolved. If this view finds still more supporting evidence that the one we put forward in the present article it is a strong confirmation of our conceptuality interpretation of quantum theory.
2 Human language as a Bose gas
Let us consider again the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’ as published in Milne (1926). In Table 1 we have presented the list of all concepts (words, cognitons) that appear in the story, with their ‘frequencies of appearance’, and hence ordered from lowest energy level to highest energy level, where the energy levels are attributed according to this frequency of appearance.
The word And is the most frequent word and appears 133 times in the text of the story, and hence the cognitions in this state populate the ground state energy level , which we put equal to zero. The word He is the second most frequent word and appears 111 times and hence the cognitons in this state populate the first energy level , which we put equal to 1. The ‘words’, their ‘energy levels’ and ‘frequencies of appearance’ are in the first three columns, under ‘Words’, ‘Energy Levels’ and ‘Frequencies’ of Table 1.
The question can be asked ‘what is the unity of energy in this model that we put forward?’, is the number ‘1’ that we choose for energy level a quantity expressed in Joule, or in Electron Volt, or still in another unity? This question gives us the opportunity to reveal already one of the very new aspects of our approach. Energy will not be expressed in ‘kg m/sec’ like it is the case in physics. Why not? Well, a human language is not situated somewhere in space, like we believe it to be the case with a physics boson gas of atoms, or a photon gas of light. Hence, ‘energy’ is here in our approach a basic quantity, and if we manage to introduce, one of our aims in further work, what the ‘human language equivalent’ of ‘physical space’ is, then it will be oppositely, namely this ‘equivalent of space’ will be expressed in unities where ‘energy appears’. Hence, the ‘1’ indicating that ‘He radiates with energy 1’, or ‘the cogniton in state He carries energy 1’, stands with a basic measure of energy, just like ‘distance (length)’ is a basic measure in ‘the physics of space and objects inside space’, not to be expressed as a combination of other physical quantities. We used the expressions ‘He radiates with energy 1’, and ‘the cognition in state He carries energy 1’, and we will use this way of speaking about ‘human language within the view of a boson gas of entangled cognitons that we develop here’, in similarity with how we speak in physics about light and photons.
The words The, It, A and To are the four next most frequent words of the Winnie the Pooh story, and hence the energy levels , , and are populated by cognitons respectively in the states The, It, A and To carrying respectively energies 2, 3, 4 and 5 of basic energy units. Hence, the first three columns in Table 1 describe the experimental data that we extract from the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’. The story contains in total 2655 words, which give rise to 542 energy levels, where energy levels are connected with words, hence different words radiate with different energies, and the size of the energies are determined by ‘the frequencies of appearances of the words in question’, the most frequently appearing words being states of lowest energy of the cognition and the least frequently appearing words being states of highest energy of the cognition. In Table 1 we have not presented all 542 energy levels, because that would lead to a very long table, but we have presented the most important part of the energy levels, with respect to the further aspects we will point out.
More concretely, we have represented the range from energy level , the ground state of the cognition, which is the cognition in state And, to energy level , which is the cognition in state Put. Then we skipped the energy levels to , and took up back at the high frequency range of the energy spectrum, namely from energy level , which is the cogniton in state Whishing, to the highest energy level of the Winnie the Pooh story, which is the cogniton in state You’ve.
|Words||Energy Levels||Frequencies||Bose-Einstein||Maxwell-Boltzmann||Energies||Energies BE||Energies MB|
These last five highest energy levels, from to , corresponding respectively to the cogniton in states Whishing, Word, Worse, Year and You’ve, all appear with a frequency of ‘one time’ in the story. They do however radiate with different energies, but the story is not given us enough information to determine whether Whishing is the lower energy radiating as compared to Year
or vice versa. Since this does not play a role in our actual analysis, we have ordered them alphabetically. So, different words which radiate with different energies that appear with equal frequencies in this specific Winnie the Pooh story will be classified from lower to higher energy level alphabetically.
In column six we represent the ‘total amount of energy radiated by the Winnie the Pooh story by the cognitons of a specific word, hence a specific energy level ’. The formula for this total amount is given by
the product of the number of cognitons in state of the word with energy level multiplied by the amount of energy radiated by such a cogniton in state of the word with energy level . In the last row of Table 1 we give the Totalities, namely in the third column of this last row we give the total number of words
which is 2655, and in the sixth column of the last row we give the total amount of energy
radiated by the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’, which is 242891 energy units. Hence columns one, two, three and six contain all the experimental data of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’.
In columns four and five of Table 1 we give the values of respectively a Bose-Einstein and a Maxwell-Boltzmann model of the data of the considered story. Let us explain what these two models are. The Bose-Einstein distribution function is given by
where is the number of bosons obeying the Bose-Einstein statistics in energy level and and are two constants that are determined by expressing that the total number of bosons equals the total number of words, and the total energy radiated equals the total energy of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’, hence
We remark that the Bose-Einstein distribution function is derived in quantum statistical mechanics by expressing explicitly the ‘identity and indistinguishability’ of the bosons (Huang, 1987). We come back to this in section 5 when we analyse our findings and our aim is, given our conceptuality interpretation of quantum theory, to understand better how ‘identity and indistinguishability’ can be understood for the physical Bose gas using our understanding of it in human language.
Since we want to show that validity of Bose-Einstein statistics for concepts in human language we compared our Bose-Einstein distribution model with a Maxwell-Boltzmann distribution model, hence let us introduce also the Maxwell-Boltzmann distribution explicitly. It is the distributing described by the following function
where is the number of classical identical particles obeying the Maxwell-Boltzmann statistics in energy level and and are two constants that will be determined, like in the case of the Bose-Einstein statistics, by expressing that the total number of Boltzmann particles equals the total number of words, and the total energy radiated equals the total energy of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’, hence
The Maxwell-Boltzmann distribution function is derived for ‘classical identical and distinguishable’ particles, and can also be shown in quantum statistical mechanics to be a good approximation if the quantum particles are such that their ‘the Broglie waves’ do not overlap (Huang, 1987). In the last two columns of Table 1 we show the ‘energies’ related to the Bose-Einstein modeling and to the Maxwell-Boltzmann respectively.
We have now introduced all what is necessary to announce the principle result of our investigation.
When we determine the two constants A and B, respectively C and D, in both the Bose-Einstein distribution function (9) and the Maxwell-Boltzmann distribution function (12) by putting the total number of particles of the model equal to the total number of words of the considered piece of text (10) and (13) and putting the total energy of the model to the total energy of the considered piece of text (11) and (14), we find a remarkable good fit of the Bose-Einstein modeling function with the data of the piece of text and a big deviation of the Maxwell-Boltzmann modeling function with the data of the piece of text.
The result is expressed in the graphs of Figure 1, where the blue graph represents the data, hence the numbers in column three of Table 1, the red graph represents the quantities obtained by the Bose-Einstein model, hence the quantities in column four of Table 1, and the green graph represents the quantities obtained by the Maxwell-Boltzmann model, hence the quantities of column five of Table 1. We can easily see in Figure 1 how the blue and red graphs almost coincide, while the green graph deviates abundantly from the two other graphs which shows how Bose-Einstein statistics is a very good model for the data we collected from the Winnie the Pooh story while Maxwell-Boltzmann statistics completely fails to model these data.
To construct the two models we also considered the energies, and expressed as second constraint (11, 14) that the total energy of the Bose-Einstein model and the total energy of the Maxwell-Boltzmann model both equal the total energy of the data of the Winnie the Pooh story. The result of both constraints (10, 13) and (11, 14) on the energy functions that express the amount of energy radiation per energy level – or, to use the language customarily used for light, the frequency spectrum of light – can be seen in Figure 2. We see again that the red graph, which represent the Bose-Einstein radiation spectrum, is a much better model for the blue graph, which represent the experimental radiation spectrum, as compared to the green graph, which represent the Maxwell-Boltzmann radiation spectrum.
Both solutions, the Bose-Einstein shown in the red graph, and the Maxwell-Boltzman shown in the green graph, have been found by making use of a computer calculating the values of and such that (10) and (11) are satisfied, which gives
In the graphs of Figure 3 we can see that a maximum is reached for the energy level corresponding to the word First, which appears with a frequency 7 in the Winnie the Pooh story. If we use the analogy with light, we can say that the radiation spectrum of the story ‘In Which Piglet Meets a Haffalump’ has a maximum at First, which would hence be, again in analogy with light, the dominant colour of the story, and we are happy, although it is of course a coincidence, that it is also the ‘first’ story we analyse and use in this article. We have indicated this radiation peak in Table 1 where we can see that the amount of energy radiated in this peak following the Bose-Einstein model is 522.79.
Due to their form the graphs in Figure 1 are not easily comparable, and although quite obviously the blue and red graphs are almost overlapping while the blue and green graphs are very different, which shows that the data are well modeled by Bose-Einstein statistics and not well modeled by Maxwell-Boltzmann statistics, it is interesting to consider a transformation where we apply the function to both -values, i.e. the domain values, and -values, i.e. the image values, of the functions underlying the graphs. This is a well-known technique to render functions giving rise to this type of graphs more easily comparable. An amazing surprise awaited us when we did so for the first time.
In Figure 3 the graphs can be seen where we have taken the of the -coordinates and also the of the -coordinates of the graph representing the data, this is again the blue graph in Figure 3, of the Bose-Einstein distribution model of these data, which is the red graph in Figure 3, and of the Maxwell-Boltzmann distribution model of the data, which is the green graph in Figure 3. For those of our readers who are acquainted with Zipf’s law as it appears in human language, they will recognize Zipf’s graph in the blue graph of Figure 3. It is indeed the graph of ‘ranking’ versus ’frequency’ of the text of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’, which is the ‘definition’ of Zipf’s graph. As to be expected, we see Zipf’s law being satisfied, the blue graph is well approximated by a straight line with negative gradient close to -1. We see that the Bose-Einstein graph still models very well this Zipf’s graph, and what is more, it also models the (small) deviation of Zipf’s graph of the straight line. Zipf’s law and the corresponding straight line when a graph is drawn is an empirical law. Intrigued by the well modeling of the Bose-Einstein graph of the Zipf graph, we will analyse this correspondence in detail in section 4.
In the next section however we want to describe what a Bose gas is in physics when it is brought nearby its state of Bose-Einstein condensate with the aim of identifying the physical equivalent to the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’ and other pieces of texts which we will consider.
3 The corresponding Bose-Einstein condensate
We will explain in this section different aspects related to the experimental realization of a Bose gas near to it being a Bose-Einstein condensate where most of the bosons are in the lowest energy state. The awareness of the existence of this special state of a Bose gas came about as a consequence of a peculiar exchange between the Indian physicist Satyendra Nath Bose and Albert Einstein (Bose, 1924; Einstein, 1924, 1925). Bose actually devised a new way to derive Planck’s radiation law for light – which has the form of a Bose-Einstein statistics, hence, like we now know, being a consequence of the indistinguishability of the photon as a boson, but that was not known in these pre-quantum theory times – and send the draft of his calculation to Einstein. Although what Bose did was far from fully understood in that time, the new method of calculation must have caught right away the full attention of Einstein, because he translated the article from English to German and supported its publication in one of the most important scientific journals of that time (Bose, 1924). Einstein himself then, inspired by Bose’s method, worked our a new model and calculation for an atom gas consisting of bosons, and predicted the existence of what we now call a Bose-Einstein condensate, an amazing accomplishment, taken into account that the difference between bosons and fermions and the Pauli exclusion principle were not yet known (Einstein, 1924, 1925). Because of the intense study of Bose-Einstein condensates that took of after their first experimental realizations (Anderson et al., 1995; Bradley et al., 1995; Davis et al., 1995), a lot of new knowledge, experimental, but also theoretical, has been obtained, material which we built upon for some of the details of the present article (Ketterle & van Druten, 1996; Parkins & Walls, 1998; Dalfovo et al., 1999; Ketterle, Durfee & Stamper-Kum, 1999; Görlitz et al., 2001; Henn at al., 2008).
The principle idea is still the one foreseen by Einstein, namely to realize a dilute gas of boson particles and then stepwise lower the temperature and as a consequence the total energy of the gas such that at a certain moment there is so little energy in the gas that all boson particles are forced to change to the lowest energy state. At that moment all boson particles are in the same state, namely this lowest energy state, and the gas behaves then in a way for which there is no classical equivalent – we will see that given our conceptuality interpretation and the boson gas model we built here for human language we will be able to put forward a new way to view the indistinguishability that lies at the heart of the Bose-Einstein condensate (see section5).
The Bose-Einstein condensates that have been realized so far all consist mainly of massive boson particles, hence generally atoms with entire spins, which makes them bosons. Indeed, the situation of the bosons of light, i.e. the photons, is more complicated, because photons interact so abundantly with matter that their number is never constant which makes it difficult to realize a thermal equilibrium for a Bose-Einstein condensate consisting of photons, albeit not impossible (Klaers, Verwinger & Weitz, 2010a, b; Klaers et al., 2011; Klaers & Weitz, 2013)
. We do want to keep using our analogy of language with light, of course the pieces of texts that we will study contain a fixed number of words, but a more dynamic use of human language will of course also give rise to a continuous coming into existence of new words, which means that for this dynamic use of human language the example of light is probably even more representative that the atom gases with a fixed number of atoms. In this stage of our analysis, also because they are the more easy to realize Bose-Einstein condensates, we focus on massive bosons, hence atoms with entire spins.
The underlying idea is that the gas consists of such atoms with entire spins that do not interact with each other, and hence only carry kinetic energy generated by random movements due to the temperature . It can be shown that in this situation the kinetic energy of a free particle equals where is Boltzmann’s constant, and hence we have
where is the mass of the atoms and their momentum. From (17) and Planck’s formula we can calculate the ‘thermal de Broglie wave length’ of the atoms of the gas
Let us make things more concrete and calculate this thermal de Broglie wave lengths for the atoms of the Bose gas that were used in the Bose-Einstein condensate realized by Eric Cornell and Carl Wieman at the University of Colorado at Boulder in their NIST-JILA lab (Anderson et al., 1995), and by the group led by Wolgang Ketterle at MIT, for which they jointly were attributed the Nobel Prize in physics in 1999. At Cornell they used a vapor of rubidium 87 atoms in a number density of
atoms per cubic centimeter cooled down to a temperature of 170 nanokelvin to see the condensate fraction appear containing an estimated 2000 atoms and be preserved for more than 15 seconds. At MIT they used a dilute gas of sodium atoms in a number density of higher thanatoms per cubic centimeter and a condensate containing up to atoms was forming at a temperature of 2 microkelvin with a lifetime of 2 seconds.
Let us calculate the ‘thermal de Broglie length ’ for both realizations, the rubidium 87 Bose-Einstein condensate at Cornell and the sodium condensate at MIT. Next to the values of Planck’s and Boltzmann’s constants, and the value of , we only need the value of the mass of a rubidium 87 atom and of a sodium atom to calculate the thermal de Broglie wave length of the atoms in the considered gas. The atomic mass of a rubidium 87 atom and of the sodium atom are respectively and unified atomic mass units and given that one such unified atomic mass unit is we get
Let us see which are the values that takes for the Cornell experiment with rubidium and for the MIT experiment with sodium at the temperatures of 270 nanokelvin and 2 microkelvin respectively where the Bose-Einstein condensates started to form. We have
Often one can read that in states of the Bose gas that are ‘nearing the Bose-Einstein condensate’, the ’de Broglie waves’ of the particles start to ‘overlap’, and that this is the reason that quantum effect become dominant. There is an interesting measure to express in a quantitative way this notion of ‘overlapping de Broglie waves’ and it is called the ‘phase space density’ of the boson gas.
If we indeed multiply the atom density of the gas with the third power if the thermal de Broglie wave length we find the density of the atoms on a region of space of ‘de Broglie wave’ cube size. If the dimensionless number that is found in this way is much smaller than 1, this means that the de Broglie wave length is much smaller than the distance between the atoms, and hence there will be no overlapping, and the gas will behave classically. The more this number is bigger than 1, the more overlapping the de Broglie waves of the atoms are, and hence quantum behavior will increase. It has been shown (Bagnato, Pritchard & Kleppner, 1987) that independent of the trapping devise for the atoms, a box, or a magnetic trap – which is the one used in actually realized Bose-Einstein condensate – the condensate starts to form whenever the value of is such that
The value of in the process of realizing a Bose-Einstein condensate is determined by temperature and number density of the atom gas. In the last stage of the realization the temperature is lowered by a technique called ‘evaporative cooling under influence of a radio frequency field’. The effect is that also the number density becomes less, and hence to attain the quantum regime of overlapping de Broglie wave lengths of the atoms it is necessary to lower the temperature faster than diluting the atom gas. The group at MIT mentions explicitly the number density that they have reached in the gas when the Bose-Einstein condensate is formed, namely, between and atoms per cubic centimeter (Davis et al., 1995). The Boulder group, since they identified the formation of their rubidium Bose-Einstein condensate at a temperature of , taking into account (27), we can calculate that the number density of the rubidium gas must have been around atoms per cubic centimeter.
We give in Table 2 an overview on the energies and lengths that are characteristic for the realizations of the sodium condensate in MIT (Ketterle, Durfee & Stamper-Kum, 1999). Because the atom gas is very dilute and the temperature is very low, the size of the atoms is very small compared to the distance between the atoms, while the thermal de Broglie wave lengths are large, such that they are overlapping. Hence with each length scale there is an associated energy scale which is the kinetic energy of a particle with a de Broglie wavelength , hence
gives a good indication of the relation between sizes and energies.
A good measure for a size of atoms who are diluted like in the considered boson gas is the elastic s-scattering length . For sodium this has been measured to be 3 nanometers which hence, using formula (28) corresponds to an energy of 1 millikelvin in temperature (Marte et al., 2002). This means that around this temperature elastic s-wave scattering, the quantum of elastic one on one collisions, between the atoms are dominant.
|Energy scale E||Length scale l|
|limiting temperature for s-wave scattering||scattering length|
|Bose-Einstein condensate transition temperature||separation between atoms|
|Temperature T||thermal de broglie wave length|
|harmonic oscillator level spacing||oscillator length = 10Hz|
The separation between the atoms in the gas can be estimated by considering the number density and then gives us the number of atoms spread out over 1 centimeter. For sodium, with a number density higher than atoms per cubic centimeter, this gives rise to a spacing between the atoms of around 200 nanometer. The length can be calculated by making use of (27) which gives us
an estimate for , and hence, by making use of (28) we find that is around .
A temperature of around gives rise to a thermal de Broglie wavelength of around .
The largest length scale is related to the confinement characterized by the size of the box potential or by the oscillator length being the size of the ground state wave function in a harmonic oscillator potential with frequency (Appendix B). With we get a value for around . The energy scale related to the confinement is characterized by the harmonic oscillator energy level spacing given by . Again, for we get an energy value for the spacing around .
In Table 3 we made the calculations of length and energy scales for the rubidium 78 Bose-Einstein condensate, taking into account that a density of around atoms was realized within the condensate of 2000 atoms.
|Energy scale E||Length scale l|
|limiting temperature for s-wave scattering||scattering length|
|Bose-Einstein condensate transition temperature||separation between atoms|
|Temperature T||thermal de broglie wave length|
|harmonic oscillator level spacing||oscillator length|
We want to show now that our Bose-Einstein distribution model of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’ is well modeled by a Bose gas close to the Bose-Einstein condensate of this gas, and will take the rubidium and sodium gases that we described in detail above as inspiration. What is important to notice is the order of difference between the energy level spacings of the harmonic trap oscillator, they are of the order of , and the energies involved with the gas itself, of the order of . The story of the Winnie the Pooh story ‘In Which Piglet Meets a Haffalump’ is not in a Bose-Einstein condensate state, because then all the words of the story should be And, contained in the zero energy level. So, it is in a state which is close to a Bose-Einstein condensate.
We have not yet explained what the parameters and of (9) are for the situation of a physical boson gas. Applied to physics and more specifically to the situation of a Bose gas, the Bose-Einstein distribution is often written as follows
where is called the ‘chemical potentual’, and is called the ‘multiplicity’. With multiplicity of a specific energy level is meant the number of states that are different but have this same energy . That different states can have the same energy is connected to symmetries of the configuration, often symmetries in space. For example for the most simple model of the harmonic trap, the one of a quantum harmonic oscillator, the multiplicity in dimensions equals
in 3 dimensions, in 2 dimensions, and which is in 1 dimension. The different dimensions are relevant for the Bose-Einstein condensates realized in laboratoria, because, although the boson gas exists always in 3 dimensions, often the harmonic traps give rise to very elongated sigar-like configurations, such that the quantum description of a one dimensional harmonic oscillator is a better model. Anyhow, for the text of the Winnie the Pooh story we do not have to hesitate about its dimension, pronouncing a text while reading it, and even the reading itself, are 1 dimensional. Also a written text, although materialized on a page which is 2 dimensional, is a 1 dimensional structure. This means that in the formula for the Bose-Einstein distribution we have rightly taken for every energy level .
What about the ‘chemical potential’ ? There is another quantity which is introduced with respect to this chemical potential, and it is called the ‘fugacity’
If we look at (30) taking into account that and we get
which means that the chemical potential and the fugacity are determined by the number of particles that are in the lowest energy state, hence the number of particles that are in the condensate state. More specifically for the Winnie the Pooh story we find
Let us note that from (35) follows that the fugacity is a number contained between 1/2 and 1 in case we have at least one particle in the condensate state and the chemical potential is a negative number, they respectively approach 1 and 0 when the condensate grows in number of particles in the lowest energy level. For what concerns the second constant we have
which means that the second constant is given by the temperature of the Bose gas.
The rubidium condensate is a better example for the Winnie the Pooh story, also the number of atoms, 2000, is of the same order of magnitude as the number of words, 2655, of the Winnie the Pooh story. We will see in section 4, where we consider the text of the novel ‘Gulliver’s Travels’ of Johnathan Swift (Swift, 1726) as the piece of text we analyse, that the sodium condensate is a better example for this text. Indeed, the energy levels of the trap for the rubidium condensate are of the order of , while the temperature of the gas is (Table 3), which is 270 times bigger. We see for the Winnie the Pooh story that if we take as energy unit 1, we have following (15), and hence , being a good estimate for the temperature of the gas in one dimension, gives , which means that we are in this respect also in the same order of magnitude with the Winnie the Pooh story and the rubidium condensate. Hence concluding we can say that the Winnie the Pooh story can be looked at as behaving similarly to a Bose gas of rubidium 87 atoms in one dimension at a temperature of .
Let us introduce a second piece of text in Table 4, namely a story entitled ‘The magic shop’ written by Herbert George Wells (Wells, 1903), with which we want to illustrate an aspect of our ‘Bose gas representation of human language’ that we have not yet touched upon. If we look at Figure 2 and Table 1, we can see that the ‘energy spectrum’ does not cover the whole range of possible frequencies. More specifically, the amount of radiation increases starting from zero radiation for energy level , hence for the words that are captured in the zero energy level of the Bose-Einstein condensate, there is no radiation emerging from them – for the case of the Winnie the Pooh story the zero level energy state puts the cogniton in state And – and then the amount of radiation increases steeply – we have already a radiation of 111 energy units (and 105.84 in the Bose-Einstein model) for for the Winnie the Pooh story and the cogniton in state He. The energy radiation keeps increasing steeply – 182 for (179.36 for the Bose-Einstein model) for the cogniton in state The, 255 for (233.36 for the Bose-Einstein model) for the cogniton in state It, 280 for (274.65 for the Bose-Einstein model) for the cogniton in state A, 345 for (307.23 for the Bose-Einstein model) for the cogniton in state To, etc. – to reach a maximum at with a radiation level of 522.79 energy units for the cognition in state First. Then the radiation starts to decrease slowly. But, remark that at energy level with the cognition in state You’ve, which is the highest energy level of Table 1, we still have a radiation of 385.55 energy levels, which is still more than half of the maximum radiation reached at energy level for the cogniton in state First.
This can also be seen very well on the red graph of Figure 2, where the right hand side of the graph has still a substantial value, and is not at all close to zero. How can we understand this, because we have in Table 1 exhausted all the words of the Winnie the Pooh story and hence seemingly represented all possible energy levels. But is this true? To see clear in this we have to reflect about the difference of the numbers in the third and the fourth column of Table 1, respectively the frequencies of the specific words like they appear in the Winnie the Pooh story and the values of the Bose-Einstein distribution that we used to model these frequencies. This means that the numbers in the fourth column express probabilities for a story similar to the one of Winnie the Pooh with respect to the frequencies of appearances of the specific words while the numbers in the third column express real frequencies counted for one specific story. More concretely, with ‘similar’ is meant ‘containing the same total number of words, and containing the same total amount of energy’. Remember indeed that the Bose-Einstein distribution function only contains two parameters, which hence will be determined by the total number of words and the total amount of energy. Or to put it even more concretely, suppose we would collect a vast number of pieces of text all containing the same total number of words and the same amount of total energy , the Bose-Einstein distribution function (9) models the average values that will be obtained for all these texts, and the more numerous these texts the better their average values will correspond with the Bose-Einstein distribution function, because this function is the consequence of the limit process of all these texts with the same and (Bose, 1924; Einstein, 1924, 1925; Huang, 1987).
The above reasoning makes it clear that we should also introduce a place for ‘words’ that do no appear in the considered text but ‘could have appeared’. In the ranking of energy levels, they have to be classified by still ‘higher energy levels’ than the highest one we now identified with respect to the last alphabetically classified word that appears 1 time in the text. Again more concretely, let us consider the words that appear one time in the Winnie the Pooh story, and look for synonyms of these words, then the word that appears now ome time could not have appeared and instead its synonym could then have appeared. So, the synonyms can be listed in a new set of words to add with zero appearance, as ‘could have appeared’, and indeed, the Bose-Einstein distribution function will not be zero for them, which expresses exactly this ‘they could have appeared’.
To illustrate the above we consider the H. G. Wells story ‘The magic shop’ (Wells, 1903) for which we have classified its words in energy levels in Table 4. As we can see, the energy level corresponding to the state of the cogniton characterized by the word Youngster, would have been the highest energy level in case we would have stopped, like we did for the Winnie the Pooh story, to add energy levels at the ‘one word appearance frequency’. For this new story ‘The magic shop’ we have however added the ‘zero word appearance frequency’ explicitly, starting with Garden, which is a word that does not appear in the story, synonym of Yard of energy level and we attributed energy level to the cogniton in a state characterized by Garden. And indeed, in the third column in the row where Garden appears in Table 4 there is 0, indicating that Garden does not appear in the story ‘The magic shop’. In the fourth column in the row of Garden in Table 4 we however have 0.25, which is the value of the Bose-Einstein distribution function at energy level . This number is linked to the probability of Garden to appear in a similar story than the story of ‘The magic shop’.
|Words||Energy Levels||Frequencies||Bose-Einstein||Maxwell-Boltzmann||Energies||Energies BE||Energies MB|
And obviously there should not be a zero in that place because there is a definite probability that Garden would appear in such a similar story. We added the word Okay at energy level as synonym of Yes at energy level , as a new not appearing state of the cogniton, however potentially appearing in a similar story. We continued in the same way adding Junior as synonym of Youngster, but there are no synonyms of You’d and You’re, which gives us the occasion to mention that the added words that could appear in a similar story do not have to be synonyms. The only criterion is that ‘they could appear in a similar story’.
We added much more energy levels, namely till the cogniton being in energy level , all of the words corresponding to the newly added energy levels, we have shown the seven last ones in Table 4, namely Continued, Adding, Mention, Similar, Criterion, Obviously and Appearing, having zero frequency of appearance in the H. G. Wells story, but their Bose-Einstein value of the Bose-Einstein model not being zero.
In (a) and (b) of Figure 4 we have represented respectively, the frequencies of the appearing and not appearing words with respect to the energy levels, a graph very steeply going down and the graph of these frequencies of appearing, where we take the logarithm of both and , with the aim of being able to verify better the level of fitting with the data of the distribution functions attempted for fitting. In Figure 5 we have represented the amounts of radiated energy with respect to the energy levels – and we see that this time the graph, after steeply going up and reaching a maximum, goes slowly down to touch closely the zero level of amount of energy radiated for high energy level cognitons. We see again, like in Figure 3, that the Bose-Einstein distribution function, the red graph, gives an almost complete fit with the data, the blue graph, and gives definitely a much better fit with the data than the Maxwell-Boltzmann distribution function, the green graph, does. Let us look more carefully to the amount of energy graph in Figure 5. Also here we see that the blue graph, which is the Bose-Einstein distribution amount of energy, is a much better fit with the blue graph, which is the data of amount of energy, than the green graph, which is the Maxwell-Boltzmann distribution amount of energy. We see that the maximum amount of radiation is reached at energy level in the state of the cogniton characterized by Door and the amount is energy units. So the frequency of Door would be the color with which the story ‘The magic shop’ shines.
We have not yet revealed the parameters , , and for the story ‘The magic show’, they have the following values
Comparing with the Winnie the Pooh story we have a higher temperature, equals 722 instead of 593, a higher fugacity, equals 0.9951 instead of 0.9923 and a higher chemical potential, is -3.576 instead of -4.581. This will be generally so when we consider longer texts like again will be illustrated by the text of ‘Gulliver’s Travels’ considered in section 4. We mentioned already that the sodium condensate realized in MIT and which we described above in detail is a better model for the ‘magic shop’ story, and indeed, in Table 2 we can see that the harmonic oscillator level spacing for the sodium condensate is around while the temperature of the sodium gas is , which is a factor 2000 in difference of size. In Table 4 we see that we have 3500 energy levels for the story ‘The magic shop’, which is a of the same order of magnitude. The number of atoms in the MIT sodium condensate was estimated to be 500000, which is way more still than the number of words in the H. G. Wells story ‘The magic shop’, which is 3934. When we analyse larger texts that come closer to this size, such as the text of Gulliver’s Travels in section 4, we find even more correspondence in magnitudes with the data of the sodium condensate. But first we have to investigate more in depth another aspect of our modeling, namely the aspect related to the ‘global energy level structure’.
There are two quantum models that also in physics are used as an inspiration for the energy level structure of the trapped atoms, one is the ‘harmonic oscillator and its variations’ (Appendix B) and the other is the ‘particle in a box and its variations’ (Appendix A). From the harmonic oscillator model follows that the energy levels are linearly spaced, which is also the way we have modeled them for the two examples that we have considered, the Winnie the Pooh story and the H. G. Wells story. However, the energy levels of the particle in a box are quadratically spaced. We will see in the following of our analysis that following our experimental findings in analyzing numerous texts in all generality the energy levels of the cogniton, depending on the story considered, are spaced following a power law, with a power coefficient which is in principle between 0 and 2, but for all the stories that we investigated was between 0.75 and 1.25. This indicates that different energy situations on both sides of the ‘harmonic oscillator’ are at play, from the ‘anharmonic oscillator’, with converging spacings between energy levels to the ‘particle in a box’, with quadratic spacings between energy levels. We will show in next section how this generalization for the energy spacings strengthens the correspondence with Zipf’s law in human language.
4 Zipf’s law and the Bose-gas of human language
Zipf’s law is considered to be one of the mysterious structures encountered in language (Zipf, 1935, 1949). It was originally noted in its most simple form, namely if, after ranking words according to their frequency of appearance in a piece of text, one multiplies the rank with the frequency, one gets more or less a constant number. Hence Zipf’s law was originally stated mathematically as follows
where is the ranking, the frequency, and is a constant. We have presented in Figure 6 the products for the text of the Winnie the Pooh story that we have investigated in section 2 where is the th ranking in Zipf’s ranking and is the frequency corresponding to this ranking. The -coordinate of the graphs in Figure 6 represents the rankings , and the -coordinate represents the products for the blue graph, and the values of respectively the Bose-Einstein distribution, and the Maxwell-Boltzmann distribution for the red and green graphs.
It is not a coincidence that there is a striking resemblance between the graphs shown in Figure 6 and the energy distribution graph of the Winnie the Pooh story as a boson gas shown in Figure 2. Indeed, the energy levels that we introduced are very simply related to the Zipf rankings , the only difference between both being that we started with value zero for the lowest energy level, while Zipf started with value 1 for his first rank. Hence, more concretely, we have
Consulting Table 1 we can see that the biggest difference is at the zero point of the graph, where on the -axis and , hence between the product , which equals , hence and . This can not easily be seen as a difference between the graphs of Figure 6 and the graphs of Figure 2, since 133 is still little compared to the values the functions take at and . Again consulting Table 1 we indeed see that , while . This means that both the ‘product graph’ of Figure 6 and the ‘energy distribution graph’ of Figure 2 go steep up between and and between and , the first from value 113 to value 222, and the second from value 0 to value 111, which is almost with the same steepness. Both graphs will then remain increasing quite steeply and slowly flatten till they reach their maxima at Zipf rank and energy level . Then from this maximum on both the Zipf product and the energy distribution slowly decrease from their maxima to a lower value. Let us make this more concrete, the maximum value is in both cases, and at the last considered Zipf rank and energy level we find values 359.22 and 358.55 respectively. This shows that there is a decreasing for the Zipf products and not constancy like Zipf’s law states.
In the foregoing reasoning on Zipf’s law, we have always considered the two graphs, the blue and the red one, in both Figure 6 and Figure 2. Of course, Zipf did not know of the Bose-Einstein distribution that is represented by the red graph in both figures, and which we used to model the data, represented by the blue graph in both figures. Hence Zipf only had the blue graph in Figure 6 available to come up with the hypothesis that the product of rank and frequency is a constant. If one considers the blue graph in Figure 6, one could indeed imagine it to vary around a constant function, certainly in the middle part of the graph. The beginning part can then be considered as a deviation, which is also what Zipf did when noting that in the first ranks the law did not hold up well. It was also known to Zipf that the end part of the graph, as a consequence of how ranks and frequencies behave there, making the product go up and down heavily, did not behave very well with respect to his law, and the slight downward slope all at the end had been identified by Zipf, the one we see explicitly pictured by the red graph, representing the Bose-Einstein distribution modeling the data.
There is however another aspect of the situation which was overlooked by Zipf. It is self-evident that ‘if Zipf’s law is a law it has to be a probabilistic law’. Let us specify what we mean by this. Suppose we would have a large number of texts available with exactly the same number of different words in it, such that a Zipf analysis would lead to the same total number of ranks for each of the texts. Zipf’s graphs, including the ‘product graph’, i.e. the blue graph in Figure 6, will then show a statistical pattern for the set of texts where it is tested on. Suppose we make averages for the frequencies pertaining to the same rank over the available texts, then the function representing these averages of the frequencies of the different texts will be a distribution function with a steep upward slope in the first ranks going towards a maximum and then a slow downwards slope in the ranks after this maximum. It will be a function similar to the Bose-Einstein distribution we have used to model texts as Bose gases, i.e. the red graph. This will be even more so when we add the two constraints that in our case follow naturally from our modeling, namely that the different texts need to count the same total number of words, and the sum of the products, which in our interpretation of the Bose gas model is the total energy, needs to be the same for each one of the texts. What is however more important still is that ‘if Zipf’s law is a probabilistic law, we should also introduce rankings that represent words with frequency zero’, exactly like what we have done for the H. G. Wells story ‘The magic shop’ for which we have represented the data and the Bose-Einstein model in Table 4, and the graphs representing these data in (a) and (b) of Figure 4 and in Figure 5.
If we look carefully at the energy distribution graph in Figure 5, we can understand again somewhat better why Zipf came to believe that the products of the ranks and the frequencies are a constant. Indeed, having added the zero frequency till the energy distribution becomes close to zero in the high energy levels, like shown in Figure 5, we can see how the blue graph goes first far up where the one word frequency cases are, to compensate the long row of zero frequency cases that take a great part of the -axis. So, if one leaves out the frequency zero part, one easily can get the impression that the blue graph represents a constant on average, at least when neglecting the low energy levels at the start, where it goes steep up.
Most of the investigations of Zipf’s findings afterwards concentrated on the graph representation, where the is taken for the rank as well as for the frequency, hence the Zipf equivalents for the graphs we considered for our Bose gas modeling represented in Figure 3 and in (b) of Figure 4. For what concerns Zipf’s law expressed in (42), the graph of the Zipf product gives rise to a straight line with gradient equal to -1. Indeed, when we take the of both sides of (42) we get
which graph, with on the -axis and on the -axis, is a straight line with gradient equal to -1. It is indeed much more easy to see by the naked eye that such a graph like the ones in Figure 3 and in (b) of Figure 4 for the two pieces of text we considered in this article can be approximated well by a straight line as compared to seeing the constancy of the Zipf’s products in a graph like the one in Figure 6, where the constancy needs to be approximated to the up and down moving blue graph. However, the focus of all Zipf’s investigations on the graphs also has its down side, in the sense that the upper and lower parts of the graph will be more easily considered as slight deviations of the straight line, while, as we see with our Bose-Einstein distribution modeling in its energy graph version, they really represent essential and significant deviations from Zipf’s original product law (42). That in both Figure 3 and in (b) of Figure 4 the graphs are slightly bend towards a concave form is the expression of Zipf’s law essentially not being satisfied for low ranks and high ranks.
The foregoing analysis is meant to provide evidence to the Bose-Einstein distribution being a better model for the Zipf data than a constant, or also still than later more complex versions of Zipf’s law along the lines of still believing that the product graph is in good approximation a constant, and the version in good approximation a straight line. There is however another aspect of Zipf’s finding that we want to put forward here, since it will be important for our model of a Bose gas for human language.
In Figure 7 we represented the graphs of the Zipf data (blue graph) and the Bose-Einstein (red graph) and Maxwell-Boltzmann (green graph) distributions which we used to model them, and we added a straight line (purple graph) that approximates the other graphs as good as possible. We can see that the gradient of the straight line is not equal to -1, the straight line is indeed inclined more sharply, with a gradient of -0.94. Although Zipf himself kept focusing on the straight line with gradient -1, it was noted by many that studied Zipf’s law that a generalization was needed to take into account the gradient of the straight line usually being smaller than -1, and hence the version of law was generalized to
which made the original product of rank and frequency be generalized to
where is called the ‘power coefficient’ of Zipf’s law.
We will apply this introduction of a ‘power coefficient’ in Zipf’s law also in our modeling. Let us explain why and how we will do so. First of all, there is no apriori reason why the energy levels would be a as simple as we presented it in the two examples that we considered, namely such that
where is the unit of energy that we introduce. Of course, we have systematically taken , which makes the energy levels we have introduced in both stories even more simple, but it is not necessarily so that as a rule, which is why we formulate the ‘linear system of energy levels’ as in (47). This simple linear system for the energy levels is inspired by the energy levels of the quantum harmonic oscillator (Appendix B), where we have
with being the frequency of the oscillator. But that energy spacings between consecutive energy levels are the same, like in the case of the harmonic oscillator, is a very exceptional situation of quantization. For general quantized systems the spacings between consecutive energy levels of the quantization will not be the same, and both cases exist, for not confined quantized situations the spacings will decrease, while for confined situations the spacings with increase. For example, for the quantized energy levels of the ‘particle in a box’ (Appendix A)’, we have
which means that the energy levels change quadratically in function of the unit of energy, i. e. the difference between the first and zero’s level
Remark that in Appendices B and A we have used to indicate the ‘quantum numbers’, because that is the traditional naming used for quantum numbers within standard quantum theory. In the approach we followed we have used to indicate the ‘energy levels’, because we do not want to make a direct and exclusive reference to standard quantum theory alone, since our aim is to also make a connection with Zipf’s law in language, and more generally, we want to elaborate a ‘quantum cognition theory for human language’ from basic principles on a more foundational level than the one where standard quantum theory is situated, e.g. there is yet no space realm for human language, which means that we will have to built a ‘quantum cognition’ without reference to space.
The ‘harmonic oscillator’ and the ‘particle in a box’ are both special cases of standard quantum theory, for both the one dimensional Schrödinger equation can be solved analytically, but for boson gases power law potentials have been studied as more general models (Bagnato, Pritchard & Kleppner, 1987), and hence we will introduce in our approach, and inspired by standard quantum theory for general types of potential energies, a more general variation of the energy levels than the linear one, namely one of a ‘power law change’
|(a) Gulliver’s Travels without power coefficient|
|Cogniton state||Energy level||Frequencies||Bose-Einstein|
|(b) Gulliver’s Travels with power coefficient|
|Cogniton state||Energy level||Frequencies||Bose-Einstein|
Let us show right away how the introduction of a power for the energy level spacings gives extra strength to the Bose-Einstein modeling of the texts of stories expressed in human language. This time we choose a much larger text than the two ones we investigated before, namely the text of the satirical work Gulliver’s Travels by Jonathan Swift (Swift, 1726), which contains in total 103184 words, hence of the order of 40 times more than the Winnie the Pooh story and 25 times more than the H. G. Wells story. When analyzed in the same way as the Winnie the Pooh and the H. G. Wells story, we find a total of 8294 energy levels without adding the zero frequency levels, and the ten highest frequencies and their corresponding words are The, 5838, Of, 3791, And, 3633, To, 3400, I, 2852, A, 2442, In, 1976, My, 1593, That, 1280 and Was, 1263.
In (a) of Figure 8 we represented the version of the ‘frequency of appearance’ graphs for the Gulliver’s Travels story, the blue graph representing the data, the red graph the Bose-Einstein model, and the green graph the Maxwell-Boltzmann model. We can see right away that again the Bose-Einstein model is a much better representation of the data than the Maxwell-Boltzmann model, but we also can see that it is a less good representation of the data than this was the case for the two foregoing pieces of text, the Winnie the Pooh story and the H. G. Wells story. Indeed, the red graph indicates noticeably too high values in the low energy levels as compared to the data represented in the blue graph and in a long region in the middle energy levels the red graph has values that are too low as compared to the values of the blue graph. In (a) of Table 5 we give the eleven lowest energy levels values of the Bose-Einstein distribution model corresponding to the states of the cogniton, i.e. the corresponding words, and compare with the data, and see that the first ones are too high, while then the following ones are too low.
For the lowest energy level, with cogniton in state The, we find the Bose-Einstein distribution to have a value of 16454.07 while the data frequency value is 5838. This is indeed a big difference, the Bose-Einstein is more than three times the experimental frequency value. We find a similar too high value for the Bose-Einstein distribution for the two next states of the cognition, i.e. Of, Bose-Einstein distribution value gives 6297.00, experimental frequency value gives 3791, And, Bose-Einstein distribution value gives 3893.39, experimental frequency value gives 3633. For the next states of the cognition the Bose-Einstein model however gives values too low with respect to the experimental values. To, Bose-Einstein distribution value gives 2817.73, experimental frequency value gives 3400, I, Bose-Einstein distribution value gives 2207.73, experimental frequency value gives 2852, A, Bose-Einstein distribution value gives 1814.80, experimental frequency value gives 2442, In, Bose-Einstein distribution value gives 1540.59, experimental frequency value gives 1976, My, Bose-Einstein distribution value gives 1338.35, experimental frequency value gives 1593, That, Bose-Einstein distribution value gives 1183.03, experimental frequency value gives 1280, Was, Bose-Einstein distribution value gives 1060.00, experimental frequency value gives 1263 and Me, Bose-Einstein distribution value gives 960.14, experimental frequency value gives 991.
We will now apply a ‘power’ to the spacings between the energy levels, following (51) and what is explained above and we will see that we can come to a much better match of the Bose-Einstein distribution with the data. Indeed, after applying the power to the energy spacings between the energy intervals we found a perfect match and represented the version of the graphs in (b) of Figure 8. The values for the eleven lowest energy levels comparing the data with the Bose-Einstein model with power coefficient 1.08 are given in (b) of Table 5.
We have tested the Bose-Einstein model on a large number of stories, short stories and long stories of the size of novels, and when we allow the energy spacings between different energy levels to vary according to a power law, we have been able to construct a perfectly matching Bose-Einstein model for the data for all of the considered stories. The power that we have needed is situated between 0.75 and 1.25.
We want to emphasize that it is remarkable how the application of the power 1.08 to the linear version of the text of the novel of Gulliver’s Travels makes the Bose-Einstein model fit the data, and we observed the effect of the introduction of a power on an original linear version of the model for many of the other example texts that we investigated working in exactly the same way. We mentioned already how those who studied Zipf’s law came to add a power to take into account that the gradient of the best fitting straight line in the version of the graphs was not equal to -1. However, also the concave slightly curbed nature of the lowest energy level rankings was noticed and tried to be remedied by making the law more general still, however in purely ad hoc ways with the only aim to fit the data (Mandelbrot, 1953, 1954; Edmundson, 1972). That this slight concave curb appears in the Bose-Einstein distribution as a consequence of adding a power to the spacings between energy levels in exactly a way to made it fit with the data is in this sense remarkable, and since we saw it happening in many of the other examples for different values of the power, it is a strong indication of the Bose-Einstein model touching onto a fundamental property of human language.
In Figure 9 we have represented the low energy part of the ‘energy distribution’ of the story of Gulliver’s Travels (Swift, 1726). The blue graph represents the energy radiated by the story per energy level, the red graph represents the energy radiated by the Bose-Einstein model of the story per energy level, and the green graph represents the energy radiated by the Maxwell-Boltzmann model of the story per energy level. We have not added the higher energy levels radiation because we wanted to show the detail of the low energy distribution, the one where the Bose-Einstein condensate dynamics of the text plays out. The maximum with a value of 18377.11 is reached at energy level 43.65 at quantum number 33, hence very close to the low level energies. The parameters , , and of the Bose-Einstein and Maxwell-Boltzmann models are
Comparing with the Winnie the Pooh story and with the H. G. Wells story we have a higher temperature, equals 19356 instead of 722 or 593, a higher fugacity, equals 0.9998 instead of 0.9951 or 0.9923 and a higher chemical potential, equals -3.648 instead of -3.576 or -4.581. As we remarked already when comparing the parameters for the two stories, the Winnie the Pooh story and the H. G. Wells story, this is generally so for longer texts.
5 Identity and Indistinguishability
What can the obtained results teach us about the notions of ‘identity and indistinguishability’ also with respect to how they are used in human language and in quantum theory. We also want to reflect about the way in which the obtained results support the ‘conceptuality interpretation of quantum theory’ (Aerts, 2009a, 2010a, 2013, 2014; Aerts et al., 2018c, 2019c). Before we start our analysis, we repeat that all the words appearing in the stories that we considered are ‘states’ of the ‘cognition’, which is the entity which is for human language what a ‘photon’ is for light, or what a ‘rubidium 87 atom’ is for the rubidium gas used to fabricate the Bose-Einstein condensate in Anderson et al. (1995).
Let us first analyse what a quantum description of identical and indistinguishable particles is. In principle ‘indistinguishibility’ as conceived in quantum theory is a property independent of the state in which a specific quantum entity encounters itself. This is a consequence of the generally adopted mathematical rule that wave functions should be symmetrized or anti-symmetrized, depending of whether the quantum particles in question are bosons or fermions. This entails that a multi-particle wave functions is always a superposition of products of the single particle building blocks of the multi-particle wave function such that the different product pieces are chosen in such a way that the total wave function is symmetric or anti-symmetric depending on whether the composed quantum entity is a boson or a fermion. Let us make concrete what this means in case we apply a quantum model to the text of the Winnie the Pooh story. The set of energy levels shown in Table 1
are in principle the energy levels for a one particle situation in quantum theory, and the many particle situation of a text, or a gas, or light, is described in a Hilbert space which is the tensor product of, in the case of the Winnie the Pooh story, 2655 Hilbert spaces of which each one describes a one particle situation, and for which this product is made symmetrical by superposing with permutations of the original products in case of the composed system being a boson with a renormalization of the superposition to make it a unit vector of the tensor product Hilbert space. Let us consider the very simple version of this symmetrization and renormalization procedure for two identical and indistinguishable boson quantum particles which we calland to see how challenging it is to try to understand its meaning. Both particles are described by their wave functions and , where and are variables we considered for respectively particle and particle , and hence the joint entity consisting of both particles is described by
where is a renormalization constant. To see to what type of problems of understanding this symmetrization procedure leads, suppose for a moment that and are position variables pertaining to separated regions of space and , such that for both particles and we can understand and as being the wave function representing one particle mainly present in this region of space , and another particle mainly present in this region of space – and are for example wave packets of which their squared absolute values as Gaussians have negligible values outside respectively regions and of space. The symmetrized wave function describes then a quantum entity and the entity which it describes pertains to both regions of space and , but does not consist of one particle pertaining to the region and another particle pertaining to the region , because it predicts the presence of entanglement correlations between measurements performed in both regions and . This entanglement was put into evidence originally by Einstein and two of his students Boris Podolsky and Nathan Rosen and the correlations it produces are now called EPR correlations (Einstein, Podolsky & Rosen, 1935). The theoretical and experimental study of the EPR type of correlations has been one of the major subjects of quantum theory investigation for the last decades and resulted in showing that these correlations are non local. There is no longer any doubt in the physics community that the EPR type of correlations predicted by the entanglement carried in symmetrized states such as (55) constitute an intrinsic reality in the quantum world even if there is still an ongoing debate about how to understand them (Bohm, 1951; Bell, 1964, 1987; Aerts et al., 2019a).
Such a symmetrization for bosons and anti-symmetrization for fermions, following quantum theory, exist for all bosons and all fermions, which literally means that all identical quantum particles are entangled in this strong way giving rise to non local correlations of the EPR type. This state of affairs following from the theoretical mathematical model of quantum theory is still nowadays a serious unsolved and not understood conundrum for theoretical physics and philosophy of physics (Black, 1952; Van Fraassen, 1984; French & Redhead, 1988; Saunders, 2003, 2006; Muller & Seevinck, 2009; Krause, 2010; Dieks & Lubberdink, 2011, 2019), and this stands in great contrast with how experimentalists go along with it, for example, photons pertaining to different energy levels, hence carrying different frequencies, are treated by experimentalists as not identical and distinguishable (Hong, Ou & Mandel, 1987; Knill, Laflamme & Milburn, 2001; Zhao et al., 2014). The way in which experimentalists look at the ‘identity and indistinguishability’ of photons has become very explicitly written about in many articles because of the actual importance of the creation of entangled photons for different reasons, e.g. their importance for the fabrication of optically based quantum computers, and hence the focus in quantum optics on how to achieve this. Spontaneous parametric down conversion, which is a nonlinear optical process that converts one photon of higher energy into a pair of photons of lower energy has been historically the process for the generation of entangled photon pairs for the well-known Bell’s inequality tests (Aspect, Dalibard & Roger, 1982; Weihs et al., 1998)
. Parametric down conversion is however an inefficient process because it has a low probability and hence physicist looked for other ways to produce entangled photons. Hence, when a scheme for using linear optics in function of the needs of the production of qubits was presented(Knill, Laflamme & Milburn, 2001), this made arise an abundance of new research. Most of the applications of this new research rely on the two-photon interference effect with two ‘identical photons’ entering on different sides of a beam splitter and leaving in the same direction after undergoing the so called Hong-Ou-Mandel interference effect (Hong, Ou & Mandel, 1987)
. The crucial aspect of Hong-Ou-Mandel interference is the ‘indistinguishability of the two photons in the spectral, temporal and polarization degrees of freedom’.
This stimulated the direct study of the ‘indistinguishability of photons from different sources’ with the finding that ‘for photons to behave as indistinguishable bosons neither their frequencies nor their arrival times at the beam splitter cannot be too different, if not they behave as distinguishable quantum particles’ (Lettow et al., 2010). What is however most significant for what concerns our take on this and its value as support of our conceptuality interpretation of quantum theory (Aerts, 2009a, 2010a, 2010b, 2013, 2014; Aerts et al., 2018c, 2019c), is the result of an amazing experiment that was performed in the series of attempts of quantum opticians to create entanglement within linear optics by making use of the interference due to two photon indistinguishability. In this experiment, photons of different frequencies are used to enter the beam splitter, hence given earlier experiments (Lettow et al., 2010), these photons should not behave as indistinguishable bosons, but on the outcoming part of the beam splitter a setup is realized that ‘erases’ the information about the different frequencies of the incoming photons. The result of the experiment is that this erasing makes the photons of different frequencies behave as indistinguishable bosons (Zhao et al., 2014). This experiment shows that it is sufficient for the photons to be indistinguishable when they are measured for them to behave as indistinguishable bosons. We should actually not be amazed by this result because this is what the so called ‘quantum eraser experiments’ are all about (Scully & Druhl, 1982; Kim et al., 2000; Walborn et al., 2002), and if we carefully read the famous analysis of the double slit experiment by Richard Feynman (Feynman, Leighton & Sands, 1963; Feynman, 1965), the dependence of interference on the possibility of the measurement apparatus to ‘know about it’, was already in the center of his analysis. Hence, given the above analysis and given our conceptuality interpretation of quantum theory, we want to put forward now our view on the issue of ‘identity and indistinguishability’ as follows
The way in which we see and understand in a straight forward way ‘what identity and indistinguishability are with respect to human language and the human mind’ teaches us ‘what identity and indistinguishability are in quantum theory’.
Let us formulate the reason why it makes good sense to state our view as just expressed given the conceptuality interpretation of quantum theory. The main hypothesis of the conceptuality interpretation is that ‘the role played by the human mind in relation with language is the same as the role played by a measuring apparatus (but also a heat bath and also a context that is perhaps not willingly used by a human being to measure) in relation with a collection of quantum entities’. The statement above in italic follows directly from this hypothesis.
Let us become more concrete and consider the text of the Winnie the Pooh story of which the words can be found in Table 1. We see that – and the reasoning we develop now can be made for any other of the considered words – the word Piglet corresponds to the cogniton being with energy , and it appears 47 times in the text of the story. In the quantum wave function that represent the story, which has 2655 slots, namely the total number of words, i.e. states of the cogniton, that appear in the story, Piglet is the state on 47 of these slots. It is straight forward that each of the Piglets in each of the slots can be interchanged with each other of the Piglets in each other of the slots without the story being changed even in the slightest way. This means, in physics jargon, that the wave function is symmetric (or anti-symmetric) with respect to the interchange of all these Piglets in the slots where they appear. And, the symmetry (or anti-symmetry) is a consequence of the ‘absolute identity and indistinguishability’ of all these Piglets. The different Piglets also have no individuality whatsoever, they just indicate that in that specific slot the state of the cognition is a state with energy level equal to , and this is an energy level that our human mind identifies as Piglet. It is also easy to understand that this ‘absolute identity and indistinguishability and lack of individuality’ is due to Piglet being a concept, and not an object. Indeed, let us imagine for a moment, just to make the above more clear still, that the scenery of the story would be pictured in some physical theatrical form with real piglets on the slots where now the concept Piglet appears in the text. If we would interchange these real piglets, of course this would influence the physical scenery of the story. It is indeed not possible to ‘interchange a real physical piglet with another real physical piglet without changing the whole of the physical scenery’. That is why real piglets when put in baskets will follow a Maxwell-Boltzmann statistics and not a Bose-Einstein statistics what Piglets do when they are concepts. The ‘interchanging of concepts in a piece of text’, hence in the slots of the wave function representing this piece of text, is an intrinsically different operation than the ‘interchange of objects in space’, and the basic hypothesis of the conceptuality interpretation of quantum theory consists in believing that quantum particles are like concepts, and that the reason why we find their behavior not understandable is because we think of them as objects. One of the crucial difficulties when thinking of quantum particles as objects comes to the surface exactly in their behavior as identical indistinguishable entities, for objects this is something impossible to understand, while for concepts it is something straightforward and natural.
Let us show now how we can also easily understand the difference we indicated above between theoretical physicists who are struggling with the issue that following quantum theory all photons should be identical in contrast with experimental physicists who pragmatically consider photons of different frequency as distinguishable and hence not identical. Consider again the Winnie the Pooh story, although we all understand right away that all Piglets as concepts are ‘absolutely identical and indistinguishable’, we also are convinced that two different energy states of the cogniton are not identical and are distinguishable. For example energy state , which is the concept Robin, appearing 12 times in the text, is not identical and is distinguishable with Piglet. It is even very important for the meaning carried by the story that these two states are different and distinguishable. In a very similar way for any measuring apparatus that is sensitive to the frequency of light, it is very important that a red photon is different and distinguishable from a blue photon, e.g. for our eyes, but also, we suppose, for plants practising photosynthesis. It is even the ‘essence of the measuring apparatus’ to ‘distinguish and identity the difference’, e.g. between a red photon and a blue photon. However, when a special purpose apparatus is fabricated that, when we would read the Winnie the Pooh story, the slots where Piglet appears are made not distinguishable any longer with the slots where Robin appears – and there is a multitude of ways we can imagine this to be done – the two cognitons that are still read by us, will be identical and indistinguishable. Again, such an operation consisting of completely erasing the Piglet nature and Robin nature of both concepts, can only work ‘because both are concepts and not objects’. Underneath all of the words of the Winnie the Pooh text is indeed the notion of Concept, and hence we can make all words into this underlying notion, which would make all of them identical and indistinguishable. There are different ways of ‘erasing’, some ways more close to the ontology of the concepts, other more close to the measuring itself, and that is also why the quantum eraser effect can be understood very well within the conceptuality interpretation (see Aerts (2009a) section 4.4).
Does the above mean that ‘words in different states are different and are distinguishable’ and ‘words in the same state are identical and indistinguishable’? Let us proceed in refining our answer to this question. It certainly does not mean that ‘words in different states are objects’, they are concepts, and hence behave like concepts, and not like objects. And since they are concepts, they carry an intrinsic potential trace of ‘identity and indistinguishability’ inside, so let us explain this now such that the seemingly opposite attitude between theoreticians and experimentalists when it comes to ‘photons of different frequency’ can be resolved.
What about the concept Piglet or Christopher Robin? It has lost its semi-identity of indicating Piglet and also its semi-identity of indicating Christopher Robin. But more importantly, it has lost the defining feature that both of them apart had, Piglet and Christopher Robin. Can it appear in a story? Of course, it does not appear in the story that we considered, but it is very easy to imagine that a sentence would contain the disjunction of Piglet and Christopher Robin, let us give an example.
Piglet or Christopher Robin are the ones that could have moved the stone in front of the cave, but who of both did it? Pooh wondered.
In the above sentence Piglet and Christopher Robin although different and distinguishable as concepts come to be in a superposition as a consequence of the ‘disjunction’. We have studied the superposition brought about by the disjunction in earlier work and shown how it can be modeled making use of the Hilbert space formalism of quantum theory (Aerts, 2009b, 2011). Our proposal in accordance with the conceptuality interpretation of quantum theory is that a completely similar situation exists for two photons of different frequencies, they are different and distinguishable as concepts and not as objects, which means that their difference can easily be erased by an apparatus which works for the photons like the disjunction works for the human mind. If a Winnie the Pooh story is written wherein Piglet is each time in disjunction with Christopher Robins, hence the story hides whom of both is involved each time one of them is involved, both concepts Piglet and Christopher Robins will join into the concept Piglet or Christopher Robins, all of these being ‘identical and indistinguishable’ in an absolute way. Hence, theoreticians have a point, all photons are potentially ‘identical and indistinguishable’, because they are concepts, but when they are in stable states, of specific frequencies, they becomes different and distinguishable, however still ready to ‘loose this differentiation and distinguishability’ whenever encountering a material apparatus that takes away the material knowledge about their difference and distinguishability. And measuring apparatuses often are even constructed exactly to be make it possible to distinguish and identity, which means that ‘identity and indistinguishability’ are contextual properties of quantum entities, and it is the way in which theoreticians have the tendency to consider them as not contextual that confronts us with the impossibility to understand them.
However, what we can understand about the nature of reality goes further than what we have formulated till now, in case we interpret quantum theory following the conceptuality interpretation. We showed in earlier work that not only disjunctions of concepts give rise to quantum superposition, but also ‘conjunctions of concepts’ do, a phenomenon we have studied in detail in our Brussels group (Aerts & Gabora, 2005b; Aerts et al., 2010, 2012; Sozzo, 2014; Aerts, Sozzo & Veloz, 2015; Sozzo, 2015; Aerts et al., 2017). We used Fock space in our quantum models of disjunction and conjunction, and could prove that disjunction as well as conjunction are modeled in the second sector of Fock space, and also that ‘any combination of concepts’, independent of whether it is a disjunction or a conjunction, gives rise to superposition in the first sector of Fock space (Aerts, Sozzo & Veloz, 2015). Every sentence in a text is a combination of concepts. Also every paragraph in a text is a combination of concepts, since sentences, as combinations of concepts, combine amongst each others to form the paragraphs. Depending on the nature of the text, this process, of ever larger pieces of the text being essentially ‘combinations of concepts’, keeps going on, certainly for stories, where the meaning of the content of the story glues all the concepts together in combinations. This implies that first sector Fock space superpositions will also form in pieces of texts amongst a certain, often large subset of combined concepts. We believe that this is exactly the mechanism which we call ‘understanding’ when the human mind is engaging in these pieces of text. More concretely, suppose the human mind reads a piece of text. When reading there is no direct focus on single words as a collection, on te contrary, when the words are read, after they have formed a combination of several concepts, a ‘new state is being formed’, which contains ‘the meaning carried by the combination of all the concerned concepts’. This new state carrying the meaning of the piece of text formed by the combination of these words is exactly the superposition state which we identified already in earlier work as being a state of the first sector of Fock space (Aerts, Sozzo & Veloz, 2015), and it are these superposition states that form again and again by combining concepts of sentences or paragraphs that again superpose in the course of the reading of the whole text, and lead to understanding of the whole piece of text. A similar process takes place when talking, thinking or writing, albeit in general more discontinuous and complex than the one by reading. We believe that what happens with a physical Bose gas close to its Bose-Einstein condensate can be understood similarly. The role played by the human mind with respect to the text is now played by the heat bath and the measuring apparatuses applied with respect to the Bose gas. When the temperature is low enough and the diluteness of the gas is such that the phase space density (26) satisfies (27) and hence the thermal de Broglie wave length (18) is larger than the distance between the atoms, this process of superposition formation starts to happen. Indeed, the de Broglie waves of the different atoms will overlap heavily and give rise to these superpositions, which means that the process which we call ‘understanding’ when the human mind and text are involved takes place in the Bose gas with the heat bath. These superpositions are new emergent states that do not pertain to one of the atoms any longer, but represent a several atoms joined new entity, just like the several combined concepts represent an emergent meaning. The more the temperature is lowered and the density of the gas is kept such that the de Broglie waves overlap on larger and larger regions of the gas, the more new states are formed containing synthesized material reality different from single atoms. The Bose-Einstein condensate is an ultimate state where all the atoms have been gathered in the lowest energy state and for the whole gas has emerged one new state. This corresponds to the meaning state after understanding the whole text where indeed no single concept of the text plays still any role with respect to this synthesized meaning.
Appendix A The Particle in a Box
Schrödinger’s equation is the fundamental equation of quantum theory and we are specifically interested in its time independent form, because that is the form which gives rise to the quantum eigenstates of the energy, hence states with a predictable fixed energy for a specific energetic situation. How this energetic situation is, we can take inspiration of what we know from classical physics, hence constituting the situation with the energy equal to a part of kinetic energy plus a part of potential energy , and hence the total energy is the sum of both
For the specific energetic situation of a ‘particle in a box’, we treat the particle as a free particle as long as it is inside the box, which means that its kinetic energy equals and the potential energy is a potential which is zero inside the box, and infinite in the region outside of the box. The Schrödinger equation ‘inside the box’, where the potential equals zero, becomes the equation for a free particle with mass , hence
which is equivalent to the equation
When we put
the Schrödinger equation becomes
which is a second order differential equation of which the general solution is well known
where and are constants, which can be complex numbers, that can be chosen depending on extra conditions to be satisfied. Remark that (61) is the wave function representing a free quantum particle in one dimension because we have not yet expressed in any way the presence of the infinite potential representing the box. Suppose we place the box between and , where is the width of the box as we have shown in Figure 10. Hence, this means that at and we need to have , expressing that the walls of the potential representing the box are infinite. Hence, making use of (61) we have
We also have
where we have used that . This means that is quantized, and the wave functions which are solutions of the Schrödinger equation for different quantum numbers are given by
We still have to calculate the value of by expressing that the probabilities to find the particle at a specific point , given by sums up to 1, hence
where we make use of . This gives us the final solution of the Schrödinger equation for the particle in the box
If we use (59) we can calculate the energy of the particle, and see that it is also quantized
We remark that for , hence the lowest energy level, corresponding to the ground state wave function, we have
which means that the energy of the particle is different from zero even in the ground state. This energy is called the ‘zero point energy’, it means that quantum mechanically the particle is unable to ‘not move’, complete lack of motion would indeed violate the Heisenberg uncertainty relations. In Figure 10 we have represented the energetic situation of the box described by an infinite potential well and drawn the wave functions corresponding to the first four quantum numbers and . We can see that the wave functions are ‘standing waves’ that can be imagined to be the wave modes in a string which outer ends are fixed to the walls of the potential well. Remark that the wave lengths and energies are inversely proportional to the width of the box, i.e. smaller boxes give rise to larger wave lengths and higher energies. This explains some of the differences between the macro-world, where is large, and hence energies and wave lengths are small, such that no overlapping exists, and typical quantum superposition effects are absent, and the micro-world where energies and wave lengths are large with substantial overlapping such that quantum superposition effects can be abundant (Aerts, 2014).
Appendix B The Quantum Harmonic Oscillator
The potential energy of a harmonic oscillator is traditionally written as follws where is the force constant, which is is a measure of the stiffness of the spring, in case we realize the harmonic oscillator by means of a particle with a mass attached to a spring. We also can write the potential energy in function of the frequency of the oscillator and the mass of the particle by using that , and hence the potential energy becomes then . This gives rise to the following Schrödinger equation
The ‘particle in a box’ Schrödinger equation’ which we considered in Appendix A was easy to solve, and hence we constructed explicitly its solution. The ‘quantum harmonic oscillator Schrödinger equation’ is less straight forward to solve and hence we will give its solutions directly. They are again quantized and to write them in a more simple form we introduce
The general normalized solutions of the Schrödinger equation are then
where is the Hermite polynomials of grade
and hence for the seven lowest energy levels, the ones illustrated in Figure 11, these polynomials are the following