Exploring students’ views about basic concepts in introductory quantum mechanics

Literature in physics education shows that students still experience difficulties learning quantum mechanics, although it is now part of the high school curriculum and many research-based proposals are available. Prior works mostly focused on specific misconceptions and a clearer picture of students’ ideas on general quantum concepts is still lacking. We addressed these issues by inspecting, through multiple correspondence analysis and cluster analysis, the responses given by 408 Italian high school and undergraduate students to a Likert scale questionnaire on quantum physics. From our preliminary results, we can conclude that the majority of students leave high school and enter university without a sound model of the quantum world.


Introduction and aims
Literature in physics education has shown that students have difficulty learning quantum physics (QP), in particular wave-particle duality, wave function and atoms [1].The abstractness and counter-intuitiveness of QP are at the origin of students' conceptual difficulties [2].Other reasons for such difficulties are: shifting from a deterministic to a probabilistic worldview, and relating the mathematical formalism of QP to experiences in the physical world [3].
Recently, also in Italy, QP has been introduced into the high school curriculum [4].Targeted concepts include, e.g., the quantum model of light, Planck's hypothesis, the discrete energy levels, de Broglie wavelength, the uncertainty principle and the photoelectric effect.Some of these topics are also addressed in the chemistry course at an earlier stage.However, curricular teaching mostly relies on a traditional lecture-based approach and there is limited evidence of the effectiveness of these reforms.Moreover, physics teaching in Italy differs across high school strands (science focused, math and physics oriented, humanities, . . .).Such inhomogeneity leads more interested students to attend extracurricular physics activities focused on quantum topics.It also impacts on students' initial preparation for undergraduate courses.Hence, it is possible that, also among STEM freshmen, knowledge about quantum topics widely differs.Finally, while prior research work mostly focused on specific misconceptions about, e.g., the uncertainty principle, Schrödinger equation and wave function at undergraduate level, research still has to provide a clearer picture of how students combine concepts such as stability of atoms, behaviour and properties of photons and electrons, probabilistic vs. deterministic viewpoint, into coherent mental models.Therefore, the specific research questions that guided this study were: what are the students' ideas about properties and behaviour of atoms, electrons and photons?What are the students' profiles that can be identified from these ideas?How these profiles are associated to different groups of students?

Sample and methods
To address the above research questions, we involved a convenience sample composed by 408 students that can be divided in four groups: 148 students from math and physics oriented high school, 59 from the same high school stream, who were attending extracurricular activities in physics, 127 freshmen in engineering, and 74 freshmen in biology.We considered those students, who participated in extracurricular activities, as a separate group, because they can be regarded as more interested in physics [5].Prior to our study, all high school students received instruction on quantum physics during their regular physics lessons.No intervention was included in our study.The reason for involving engineering and biology students is that they represent STEM undergraduates.Moreover, biology students are exposed to basic quantum concepts since their first year in the chemistry exam.
Our research instrument was a questionnaire about students' views on quantum mechanics designed from prior instruments proposed in [6,7,8] and [9].The first version of the instrument, featuring 54 statements on a 5-point Likert scale, was developed by Mashhadi and Woolnough [6].Their questionnaire was administered to 319 pre-university A-level physics students to gain a qualitative insight into students' understanding of quantum mechanics.They found that their sample could be divided into three clusters: Mechanistic, Intermediate and Quantum, respectively.The instrument was then adopted by Ireson [7], who administered a refined scale of 29 items to 338 first-and second-year undergraduate students in UK.Students' responses were factorized into two latent dimensions of conceptual understanding: 'Absolute thinking -Dual thinking' and 'Simple atom/deterministic mechanics -Complex atom/indeterministic mechanics'.Three clusters were formed using these factorial dimensions: Mechanistic thinking, Intermediate thinking and Quantum thinking.The questionnaire was also administered to 342 pre-university A-level physics students, of which only about 50% had already studied quantum physics [8].For the group that had not yet been exposed to quantum phenomena, four clusters were found: structure and mental image of entities, mechanistic thinking, quantum thinking and conflicting mechanistic thinking.For the group that had been exposed to quantum physics, three clusters were found: quantum thinking, conflicting quantum thinking and conflicting mechanistic thinking.By comparing the two groups, the author concluded that, even if there were differences between them, "only a minority of the significant changes can be traced back to statements in the syllabus.The majority of changes must, therefore, be due to factors outside the direct teaching of the syllabus material."[8, p. 20].In a follow-up study [9], the same author analysed all the data obtained in one of the previous studies [6], now also including 11 statements on the conceptual understanding of models (e.g., "models are constructions of human minds").While the same two factors were obtained from the analysis, the clusters' interpretation was refined as it follows: quantum thinking/descriptive models, conflicting thinking/conflicting models and mechanistic thinking/complete models.
For the present study, we kept the original formulation of most of the statements used in [6,7,8] and [9].However, we rephrased some items to eliminate ambiguities (e.g., "The energy of an atom can have any value" was changed in "The energy emitted or absorbed by an atom can take on any value") and designed new items to account for other students' misconceptions (e.g., "Electrons move at the speed of light" or "The electron spins around its axis").Therefore, the final version of the questionnaire featured 36 items on a 4-point Likert scale (see Appendix A for the complete list).

Data analysis
We first calculated Cronbach's alpha (α = 0.50) to measure the internal consistency of the instrument.This value can be considered acceptable, because the items concern different quantum entities.Hence, some problems of internal consistency are foreseeable [10].Descriptive statistics of students' answers was used to identify the most frequent misconceptions.
To answer the research questions, a multiple correspondence analysis (MCA), a hierarchical cluster analysis and a chi-square analysis were performed.In agreement with the studies described in [7] and [9], we used MCA to obtain the latent factors that describe students' views about QP.MCA can be thought as an extension to categorical variables of principal component analysis, which allows to identify one or more latent dimensions (factors) that explain most of the variance in the data.In our case the variables are represented by the degree of agreement (modality) with a certain item.From the mathematical viewpoint, instead of the correlation matrix used in factor analysis, MCA uses a contingency table of frequencies, with entries equal to 1 or 0, depending on whether a certain modality has been selected in one or more questions (for more details, see, for instance, [11]).As in factor analysis, the interpretation of the emerging factors takes into account the loading of the item in the factor (called weight in MCA).However, differently from factor analysis, the interpretation of the retained factors has to take into account also the observed modalities, since the items were not scored as in factor analysis.Therefore, in our case, to interpret the retained factors we took into account first the specific topic targeted in the item (e.g., photon) and then the modality, with which the item weighs in the factor.In general, different researchers could give different interpretations of the retained factors.For this reason, to label each factor from MCA, we interpreted the output of the analysis independently and separately, and, when necessary, we discussed the results to reach consensus.
Then, we used the factorial scores obtained from MCA to build students' profiles using hierarchical cluster analysis.This analysis involves an iterative procedure to obtain the best partition of the data set.Clusters were identified following a hierarchical-divisive procedure, which started from a subdivision of the sample into two main groups and then proceeded in successive steps, according to a hierarchical tree structure, which was cut to obtain a final solution that explains a reasonable percentage of data variance.The elements belonging to each cluster were classified in order of importance with the aid of a statistical criterion (test-value) to which a probability is associated: the larger the test-value, the lower the probability of exclusion of the element, the better the cluster is defined by that element.Finally, to better describe the obtained clusters, with a process of independent interpretation and successive comparison among the authors, only the elements with higher test-value were considered.
To evaluate the association between each cluster and the groups of involved students the chi-square test was performed and the corresponding p-value calculated.In particular, the null hypothesis is rejected if the p-value is less than or equal to a predefined threshold value set to 0.05.The software SPAD and SPSS were used to perform statistical analysis.

Preliminary descriptive statistical analysis
The analysis of the frequencies of wrong answers signaled common students' misconceptions (cf.Table 1).An answer is considered wrong, when the respondent chooses strongly disagree or disagree for a right statement, i.e., "A free electron can assume any energy value", or chooses strongly agree or agree for a false statement, i.e., "The electron moves at the speed of light."

Emerging factors and their interpretation
Three factors were retained from MCA.Indeed, in most cases, three factors are enough to describe the variance of the data [12].They are reported in Table 2 with the most significant statements and related modality that guided our interpretation of the factors.These latent factors could be regarded as the dimensions of the euclidean space in which to represent the data set (see Figure 1).In order of importance, the three extracted factors are: • Factor 1, labelled Level of agreement, from partially agree or disagree to completely agree or disagree.It can be interpreted as a metacognitive factor, because it identifies the extent to which the student agrees with the statement, regardless of the topic targeted by the item; • Factor 2: labelled Photon vs electron, from a quantum view about the electrons but a deterministic view about the photons to a quantum view about the photons but a deterministic view about the electrons.It considers the different views (deterministic vs quantum view) about electron and photon behaviour; • Factor 3: labelled Deterministic vs quantum view, from a classical mechanistic and deterministic view to a completely quantum view about all quantum entities.A photon has neither mass nor charge to a great extent Photon energy depends on the colour of the electromagnetic wave completely Nobody knows the position accurately of an electron in orbit around the nucleus because it is very small and moves very fast to a great extent As light is emitted by an atom, electron jumps from one orbit to another, i.e., it is not anywhere in between the two orbits completely . . . . . .
If we perform an experiment with a double slit and know quite precisely the initial conditions, then we can predict where the electron will hit the screen not at all The electron spins around its axis not at all How one thinks of the nature of light depends on the experiment being carried out not at all A photon moves at the speed of light not at all 3 Nobody knows the position accurately of an electron in orbit around the nucleus because it is very small and moves very fast completely The atom is more or less like a small sphere completely The structure of the atom is similar to the way planets orbit the Sun completely Light travels as a wave but is absorbed as a packet of energy or photon not at all Electrons move in a non-deterministic way around the nucleus within a certain region or at certain distance not at all Electromagnetism and Newtonian mechanics cannot explain why atoms are stable not at all . . . . . .

Nobody knows the position accurately of an electron in orbit around the nucleus because it is very small and moves very fast not at all
The atom is more or less like a small sphere not at all The structure of the atom is similar to the way planets orbit the Sun not at all Light travels as a wave but is absorbed as a packet of energy or photon completely Electrons move in a non-deterministic way around the nucleus within a certain region or at certain distance completely Electromagnetism and Newtonian mechanics cannot explain why atoms are stable completely a For this factor we do not report the statements corresponding to the modalities.See text for further details.

Clusters and their interpretation
A 5-cluster solution has been adopted.Two quantitative criteria were used for choosing the final number of clusters: (i) a subsequent subdivision of the clusters produces a limited increase of the explained data variance; (ii) a subsequent subdivision in the dendrogram sequence produces, at least, one cluster with less than 5% of cases of the sample.This choice was aimed to avoid the identification of clusters with low face validity and, hence, harder to interpret.Table 3 reports the statements and related modality with the highest test-value for each cluster.The emerging clusters can be described as it follows: • Cluster 1: Students adopting a full quantum view about all entities (17.2%); • Cluster 2: Students adopting a deeply deterministic view about all entities (17.0%); • Cluster 3: Students adopting a partial quantum view on specific entities, like electrons and atoms (30.1%);Table 3: Elements with the higher test-value for each cluster.

Cluster Item Modality 1
Electron is always a particle not at all Electrons move in a non-deterministic way around the nucleus within a certain region or at certain distance completely The wave or particle nature of the electron depends on the particular experiment carried out not at all How one thinks of the nature of light depends on the experiment being carried out completely A photon moves at the speed of light completely Electrons move along well determined orbits around the nucleus not at all 2 Electrons move in a non-deterministic way around the nucleus within a certain region or at certain distance not at all Electron is always a particle completely Electrons move along well determined orbits around the nucleus completely Nobody knows the position accurately of an electron in orbit around the nucleus because it is very small and moves very fast to a great extent The atom is more or less like a small sphere completely The structure of the atom is similar to the way planets orbit the Sun completely The atom is stable due to a balance between the attractive electric force and the centrifugal force  Clusters' representation in the factorial space (see Figure 2) shows that the labels of factors and clusters are coherent.As far as factor 1 [see Figure 2(a)], on the right side of the horizontal axis we find two clusters that group students, who completely agree or disagree with the statements (and, therefore, with a quantum or deterministic view).On the left side, we find the other clusters that are more or less shifted according to the level of agreement expressed by students they group.As far as factor 2 [see Figure 2(a)], clusters 1 and 2 are practically independent of this factor.On the contrary, clusters 4 and 3 are shifted along the vertical axis according to the students' quantum view about photons and electrons, respectively.Lastly, regarding factor 3 [see Figure 2(b)], it is evident that clusters 3, 4 and 5 are independent of it.On the contrary, it clearly discriminates between clusters 1 and 2, which are very far apart along this axis.

Chi-square test
The association between the four groups and the emerging clusters is statistically significant: χ 2 = 45.410,d.f.= 12, p < 0.001.In particular, as shown in Figure 3, it has been found that among the students, interested in extracurricular activities in physics, about 30% have developed a full quantum view on all entities and 34% a quantum view at least about photons.On average, this happens only for one third of the students in the other groups.

Discussion and conclusions
First, we note that our results concur with previous studies [13] showing that curricular activities do not allow students to achieve a full quantum view.Only more motivated students seem to achieve a more complete quantum view about all entities.Differently from previous ones, our analysis shows that conflicting views (e.g., probabilistic vs deterministic) about different quantum entities (e.g., electrons vs photons) may coexist, perhaps because of how the waveparticle duality is taught.In addition, it is likely that classical models, used by teachers to simplify some explanations or to apply classical formulas to different contexts or to follow a quasihistorical framework [4,14,15] (i.e., application of balance between electric and centrifugal force at Bohr's atom model), as suggested by several textbooks (e.g.see [16,17]), prevent students from acquiring a correct quantum forma mentis.However, further research is needed to support our interpretation.Moreover, as also pointed out in the context of photons and quantum optics [18,19], students often seem unable to catch the difference between the nature and the description of a quantum object.Indeed, also students belonging to the quantum cluster expressed conflicting views on these two statements.Actually, it would be interesting to see whether our results could be framed within the theoretical perspective proposed in [18].Future developments include the analysis of students' confidence, the analysis of teachers' views and the design of a teaching-learning sequence about the emerged issues.

Figure 2 :
Figure 2: Cluster representation in factorial dimensions: (a) Factor 2 vs 1 and (b) Factor 3 vs 1.The point size is proportional to the cluster size.

completely 3 Electron
is always a wave not at all A photon has neither mass nor charge not at all Light travels as a wave but is absorbed as a packet of energy or photon to a small extent Electrons move at the speed of light not at all A photon moves at the speed of light to a great extent 4Electrons move at the speed of light to a great extent Photon energy depends on the colour of the light to a great extent Photons can assume any energy value to a great extent The electron spins around its axis completely Electron is always a wave to a great extent How one thinks of the nature of light depends on the experiment being carried out to a great extent 5Electron is always a wave to a small extent Electrons move at the speed of light to a small extent Nobody knows the position accurately of an electron in orbit around the nucleus because it is very small and moves very fast to a small extent How one thinks of the nature of light depends on the experiment being carried out to a great extent During the emission of light from an atom, the electrons follow a definite path as they move from an energy level to another to a small extent • Cluster 4: Students adopting a quantum view only about photons (18.9%); • Cluster 5: Students adopting a partial deterministic view about all entities (16.8%).

Figure 3 :
Figure 3: Association between students' curriculum and emerging clusters.

Table 1 :
Common students' misconceptions found in the present study.

Table 2 :
Factors retained from multiple correspondence analysis (MCA).