Investigating Quantum Mechanical Tunneling at the Nanoscale via

Article pubs.acs.org/jchemeduc

Investigating Quantum Mechanical Tunneling at the Nanoscale via Analogy: Development and Assessment of a Teaching Tool for Upper-Division Chemistry Marc N. Muniz* and Maria T. Oliver-Hoyo Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, United States S Supporting Information *

ABSTRACT: We report a novel educational activity designed to teach quantum mechanical tunneling to upper-division undergraduate students in the context of nanochemistry. The activity is based on a theoretical framework for analogy and is split into three parts that are linked pedagogically through the framework: classical ball-and-ramp system, tunneling involving a familiar substance (NH3 inversion), and tunneling in core/shell quantum dots. Students first begin in the classical worldthe world within which they are most familiar, explore tunneling in the NH3 inversion paradigm to gain exposure to the stark differences between classical and quantum behavior, and finally extend the concept of tunneling to the nanoscale through the examination of spectroscopic results in the literature of CdSe/ZnS core/shell quantum dots. Additionally, the activity has been assessed via a mixed-methods approach using qualitative analysis of individual student interviews (pre- and postactivity) and recordings of students’ group discourse, along with small-sample statistical techniques when appropriate. Our findings suggest that students are able to successfully incorporate the language of quantum mechanical tunneling into their scientific language, and there is evidence for analogical transfer between relevant concepts within the activity. KEYWORDS: Upper-Division Undergraduate, Chemical Education Research, Physical Chemistry, Analogies/Transfer, Quantum Chemistry, Nanotechnology FEATURE: Chemical Education Research

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potential energy being on the “y” axis in 2-dimensional diagrams. In order to understand the tunneling model and apply it to chemical systems, these fundamental conceptual issues cannot be ignored. It also cannot be ignored that students will come to the table with experience living in the classical worlda world from which quantum chemistry is remote. This article describes an activity in which students are exposed to quantum mechanical tunneling in the context of nanochemistry. Students progress through the activity, which takes them from the classical world to the manifestation of tunneling in the nanoworld. The activity has been assessed to include pre- and postactivity individual interviews with students and analysis of students’ discourse in small groups during the activity itself.

INTRODUCTION The concept of quantum mechanical tunneling is relevant in several important areas in chemistry, from hydrogen bonding to alpha decay in nuclei,1,2 isotope effects,3 and biological systems.4−6 As such, it is arguably an important extension of wave-particle duality for undergraduate physical chemistry courses. Several articles have appeared in the educational literature addressing this topic;7−11 however, few have explicitly linked the concept of tunneling to nanochemistry and none report on instructional tools where students may explore the phenomenon in multiple chemistry contexts. There is also a lack of educational activities that make a transition from the classical world to the quantum world in the context of tunneling. Although work has been done in physics education to (a) develop modules for a broad array of topics from energy bands to tunneling,12 (b) assess students’ learning of tunneling in modern physics courses,13,14 and (c) to confront students’ understanding of energy conservation in tunneling,15,16 there remain fundamental misconceptions about the pillars of quantum mechanics. A prime display of such misconceptions among students may be found within Singh and Zhu’s17 work in which students with robust mathematical backgrounds (senior-level undergraduate physics students) make such conceptual errors as drawing discontinuity in wave functions and ignoring the implications of © 2014 American Chemical Society and Division of Chemical Education, Inc.

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THEORETICAL FRAMEWORK AND RESEARCH QUESTIONS The use of analogies in a chemistry context has been found by Sarantopoulos and Tsaparlis to be an effective pedagogical method.18 In their longitudinal study of secondary-level students, they found that analogies incorporated into basic chemistry instruction were highly effective at teaching students who are at lower cognitive levels in terms of Piaget’s theory of cognitive Published: July 25, 2014 1546

dx.doi.org/10.1021/ed400761q | J. Chem. Educ. 2014, 91, 1546−1556

Journal of Chemical Education

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• Is there evidence that the experience of core and nanochemistry concepts in the context of familiar objects leads to a more scientifically normative viewpoint, on the part of students, for each knowledge domain? We have carried out an assessment of the activity, which will be detailed in the Assessment and Results section, to move toward addressing these questions and to provide instructors with details as to what students’ tendencies may be while partaking in the activity.

development (specifically concrete operational level). Although the effects of analogies were, statistically, more beneficial for students at the concrete operational level, there were also positive attitudinal results for higher level students (transitional and formal operational). This provides encouraging evidence for the use of an analogical framework to inform the development of instructional materials and to set the basis for possible future work in characterizing the efficacy of these materials in longitudinal studies. We utilize a theoretical framework for analogy upon which the pedagogical goals were built.19,20 The framework itself is designed to focus on the connectivity between attributes (features, such as color, that involve a single argument) and relations (interactions between objects, such as magnets repelling one another, that involve two arguments) in one domain to another. The domain from which core knowledge is to be drawn is referred to as the base and the one to which the comparisons are to be made is called the target. Through the use of a bridging topic, one that serves as a target to the core concept but a base to the nano concept, we aim to better address the disparities between the classical and nanodomains.21,22 Ultimately, the goal is to capitalize on the relations that are connected, or map, from core to bridge to nano, constituting the formulation of an activity based on analogy. Bridging analogies have been described in the physics education literature for several decades.23−26 Of particular note is the work of Minstrell,23 as well as Clement and co-workers,24−26 on students’ conceptions about objects at rest (statics). Minstrell, for example, characterized high school physics students’ conceptions about the role of the normal force in the analysis of a book on a table.23 It was found that by introducing multiple bridging analogies, only 1 of 27 students still had the misconception that there was only a force downward on the book. Further, Brown and Clement,25 through the analysis of oneon-one “tutoring interviews” with high school physics students, described both successful (i.e., “shuffleboard puck” and “intermeshed hairbrushes”) and unsuccessful (i.e., “colliding billiard balls”) use of bridging analogies to address students’ conceptions of force in classical physics. Presence of some historical success of bridging analogies in directing students’ conceptions toward more normative understandings of classical physics is encouraging and provides evidence toward justifying the use of a bridging concept in the development of activities like the one described in this article. However, it is important to note that the role of the bridging analogy in this work, unlike the work of Minstrell and Clement and co-workers, is not to overcome misconceptions but to induce understanding of a concept (quantum tunneling) and to bridge across to this novel domain, as will be further elaborated on below. Within this construct, we initially center students’ frames of reference on a classical system and encourage them to think deeply about this seemingly trivial case. We then allow the object relations and attributes of the classical system to be related to those in the chemical world via the spectroscopic studies of familiar compounds. Finally, an application of the tunneling model to nanochemistry is employed to help describe the effects of quantum confinementa phenomenon largely responsible for the useful optical properties observed at the nanoscale. The principal research questions for this work are: • Is there evidence that students are able to effectively map core physical concepts (i.e., potential energy) to the novel domain of quantum mechanical tunneling in the nanochemistry context?

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THE ACTIVITY In its entirety, the activity takes approximately 3 h to complete. The following is a brief description of each part of the activity, with more specific details to follow in subsequent sections. Initially, students conduct an experiment with a macro-scale ball-and-ramp apparatus. This is employed as way to center students’ focus on energy in the classical domainthe world familiar to them. Students confirm that their ability to drop the ball from any point on the ramp constitutes a continuum of available energies (distinctly classical feature) and that the only way to transcend the barrier is to drop the ball from a height greater than the height of the barrier itself. Therefore, they are asked to confirm that the potential energy can never be smaller than the kinetic energy required to have access to the other side of the physical apparatus. In the second part of the activity, students are provided with spectral data and study the inversion of NH3, which is a prime example of quantum mechanical tunneling. This inversion may be modeled with a double potential well whose minima represent the C3v symmetry of the molecule with the barrier representing the D3h intermediate. The energy required to go over this potential barrier in the classical sense would surpass the logical population of vibrational states at ambient temperatures. Yet, the splitting of bands shows that this inversion is indeed taking place at room temperature.7 We used NH3 inversion as a transitional concept into the nanodomain and to present to students compelling evidence that the laws of classical mechanics do not apply to this system. A rigorous treatment of this topic is outlined in an experiment by Halpern and may be used as complementary information.7 In the activity we developed, calculation of the relevant vibrational partition function is used to show students that the population of states is such that the tunneling model serves as a plausible explanatory device for why the inversion occurs. The first two portions of this activity aim at encouraging students to think critically about tunneling and the energetic rationale for it in the nuclear sense, and the third portion centers on students’ theoretical evaluation of bare and core/shell quantum dots to explore the use of the tunneling model at the nanoscale and its implications therein. There is an emphasis on expanding the particle-in-a-box model for the electrons in a bare quantum dot to apply a basic tunneling model to the core/shell type and to show that electron tunneling from the core material into the shell material is plausible. Having been provided with spectral data from the literature, students are then asked to discuss what some implications of such a phenomenon in these nanostructures might be. Part I: Physics ApparatusThe Classical Potential Barrier

Students directly investigate potential energy in the classical sense where the conservation of mechanical energy and conversion between kinetic (KE) and potential energies (PE) are explored within a common undergraduate physics approach. 1547



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significant amounts of AsH3 or NH3 molecules to go over their respective potential barriers. This may be shown using statistical thermodynamics. Students calculate the vibrational partition function for both AsH3 and NH3 to express the sum over states populated above the zero point energy at a given temperature using the following equation:

Students use a simple ramp to drop a steel ball from various heights and tabulate their observations including the initial height, calculated PE of the ball, and the PE of the barrier. The total energy expression for this system is27 Etotal = KEt + KEr + PEg

The gravitational potential energy (PEg) will be the total energy when the ball is initially at rest from its point of departure, and students are to (quantitatively) ignore the relatively small contribution from the rotational kinetic energy (KEr). KEt represents the translational kinetic energy. Students conclude that the ball will not transcend the barrier without having been dropped from at least the height of the barrier (in reality, a bit higher due to friction and rotational kinetic energy of the ball). Figure 1 shows the apparatus students use during the first part of the activity.

qv =

1 1−e

−(hcv)/ kT

Where q is the vibrational partition function, ν is the vibrational frequency of the mode, and k is Boltzmann’s constant. The expression shows series convergence form of the vibrational partition function.29 The zero point energy has been taken as the “0th” level, as is common in the convention of expressing partition functions. Students calculate and discuss the implications of the following: • The partition function for the single mode of AsH3 over the 200 to 800 K range. • The partition function for the relevant vibrational mode, ν2, of NH3 to inversion tunneling over the 200 to 800 K range. This may be further extended to the concept of symmetry in quantum mechanics in general. The dipole moment operator, which gives rise to vibrational transitions in the IR region of the electromagnetic spectrum, is itself antisymmetric.30 This leads to the observation of two prominent bands (∼930 and 965 cm−1) in the NH3 spectrum and is confirmatory of the splitting and therefore the tunneling effect. Students should also observe the spectrum of the structurally analogous AsH3, whose inversion tunneling period approaches two years31 and is not spectroscopically observed in the form of splitting and exhibits only a single band. Students mix the states qualitatively leading to positive and negative linear combinations for each state. They first do this in the context of simple H2 molecular orbitalsby which the same energetic configuration occurs (a positive linear combination is lower in energy than a negative combination). They also construct and fully label their own double-well potential plot and represent energy values on this diagram. Through this, students are probed to think about the following: • Relevant peaks in the FTIR spectra and how they relate to the tunneling splitting (in NH3) in the context of the barrier height. • How the linear combination of states factor in, and which states correspond to the positive and negative linear combinations (students must indicate this on double well potential diagram). • What leads to the observation of such mixing in the case of NH3. How and why the inversion does not take place via classical activation at room temperature, via interpretation of the partition function, and how quantum mechanical behavior plays a role in the observation of the inversion (via tunneling). • Justification as to why chiral phosphines (which have a barrier to inversion intermediate to amines and arsines) are more readily isolated than chiral amines. • How many events are taking place and being observed spectroscopically (many events), what is the relation of this to the measurement (probabilistic) and the nature of the problem, and how this is different from the classical case (ball and ramp = single event monitored each time). v

Figure 1. Image of the ramp used in this activity with a ball that may be released from different heights.

Students are probed to think about the following: • Based on your observations, what are the requirements for the ball to go over the barrier? Support your response with experimental evidence. • In this experiment, how many events were you monitoring at a given time? (Define an event as the release of the ball from height xh, where x is any of the factors in your table, until it comes to rest.) • You are designing a classic wooden roller coaster before the use of hydraulic launch is widespread. Can the height of the initial drop be lower than that of subsequent drops without extra chains to pull the cars up further again? What is the limit here? Thoroughly justify your response (adapted from Serway and Jewett).28 Part II: Tunneling with a Familiar SubstanceAmmonia Inversion

This part of the activity focuses students’ attention on the infrared spectra of ammonia (NH3) and arsine (AsH3). Depending on the resources available to the instructor, the spectrum may be taken by the students or may be provided to them. In assessing this activity, students were provided with the spectrum. In accordance with Halpern et al.’s7 thorough treatment of the subject for undergraduates, students view the ν2 vibrational mode of NH3 and observe the splitting. They are then asked to construct the potential scheme with the apical displacement coordinate centered at 0 at the D3h transition state1,7 and the average energy parameters, including the barrier height, and the first couple of vibrational levels (the fundamental transition). A key concept that must be discovered by the students at this point is that there is not enough energy in the system to allow 1548



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Figure 2. Students’ qualitative sketch of the lowest energy level wave function for a CdSe/ZnS core/shell quantum dot. Note the students’ normative incorporation of the term “tunneling” into the portions where the wave function decays into the classically forbidden region.

• In what limit there would be no tunneling observed by any means whatsoever (in the limit that the potential barrier to inversion is infinite). Making inferences from the fact that splitting is not observed in AsH3 and that its barrier to inversion is significantly higher than NH3.

made to simplify the system to a single-dimension.32 For example, the formation of the exciton is not considered, along with the effective masses of the electron and hole. This would complicate the transition into describing the tunneling, which will become relevant when students consider the CdSe/ZnS core/shell structure, as boundary conditions would then have to be considered for the electron and hole alike. The goal is to provide enough mathematical rigor to describe the system while conserving the ultimate aim of this work: connect concepts from more familiar paradigms to one at the nanoscale. The boundary conditions are, physically, extremely important as they fulfill the requirement that a wave function must be differentiable and continuous across all regions for which it is defined. In essence, the nano aspect of this activity is split up into two parts: treatment of the core alone, and treatment of the tunneling of the electronic wave function into the ZnS barrier in the core/shell structures. It is important for students to remember that there are on the order of 1022 particles in a typical spectroscopic sample. Thus, the seemingly small difference in the energy term becomes significant. This manifests itself spectroscopically with a slight red shift, as seen in Figure 1 of Dabbousi et al.’s work.33 The figure from Dabbousi et al. is provided to students for their analysis in this part of the activity. Another significant point to be made in this portion of the activity is for students to construct separate representative diagrams for the wave function extending into the classically forbidden region and the energy eigenvalues. This is important as a number of physical chemistry texts and educational resources group the wave function representation along with the energy levels.2,34,35 This sort of representation could cause confusion with students, especially with respect to what the y axis in the diagram represents.14−17 These entities, although coupled via the eigenfunction−eigenvalue relationship, are distinct: the wave function describes the behavior of the particle (electron), whereas the energy eigenvalue describes the allowed energy levels of the quantized system. The students constructed qualitative sketches of these plots from provided expressions (see Supporting Information). Figure 2 shows a student’s sketch for the CdSe/ZnS core/shell system. In bare quantum dots, the electron is confined by the entire nanoparticle itself and the energy of absorption/emission remains simply linked to the size of the particle. In core/shell particles, there is a probability that the electron will tunnel into the shell, and the wave function will be less confined than in the original core structure. Therefore, the energy levels will be closer together than a bare quantum dot of the same size. This effect may be observed by solving the Schrodinger equation for a

Part III: Tunneling in Quantum DotsEffects on Quantum Confinement

A major factor related to the surge in research related to metallic and semiconductor nanoparticles is the fact that they exhibit interesting and potentially useful optical features. The reasoning behind these unique effects is generally explained via quantum conf inement. As its name suggests, the confinement of the electronic wave functions as the physical dimensions of the material approach the nanoscale leads to a difference in energy levels that becomes significant at this scale. In the case of quantum dots, the simple particle in a box model may be utilized to show energy level differences. In this model, the energies become closer together as the size of the spherical “container” increases. The energy values move further apart when the “container” is compressed. As a result, the actual act of confinement leads to a greater amount of energy required in the form of electromagnetic radiation to induce an electronic transition an absorption event.

En =

n2π 2ℏ2 2meL2

This is the general energy eigenvalue expression for the simple 1-D particle-in-a-box model. En is the energy for the nth energy level, me is the mass of an electron, and L is the length of the box. This may be used as a simplified approximation of the spherical treatment, in which the radius is now replaced by the diameter (assuming an S state (l = 0), the particle-in-a-sphere eigenvalues reduce to the particle-in-a-1-D-box eigenvalues).27,28 Students study two types of quantum dots in this part of the activity. In the first portion of Part III, students initially deal with quantum dots of constant composition (CdSe only). Therefore, they work the particle-in-a-box Schrodinger equation and derive the expression for the energy eigenvalue (as seen above). They then utilize the expression to explain variation in the spectra of multiple sizes of quantum dots. This is important for establishing the relevance of the model. Additionally, students are asked to construct a qualitative visual representation of the problem; labeling each component and relating it back to the expressions. The treatment itself illustrated in Brus’ work is, arguably, beyond the mathematical sophistication of most undergraduate chemistry students. Therefore, a number of approximations were 1549



1550

Maps

Many events involving the CdSe and CdSe/ZnS quantum dots are being measured spectroscopically PE must approach infinity for no transcendence to occur Maps

Maps

Single events (steel ball at different heights)

PE of the hill is higher than KE of the ball; transcendence will not occur

Multiplicity of events

Limiting case: potential energy

Does not Does not

PE of the ball at the outset must always be = to or > than the potential of the barrier in order for it to cross Energy calculated with the classical energy expression Object Relations: Mechanisms for overcoming the barrier Measurable observations

Barrier

Gravitational potential as related to lowest point Physical hill Potential Energy

Partial

Only at high T’s does there exist the possibility for the NH3 system to classically transcend the barrier, as more vibrational states are populated; at ambient T, it does not occur Calculated partition function shows not enough energy to classically transcend. IR splitting gives barrier height of the planar inversion intermediate Tunneling observed spectroscopically as the result of many molecules and many events PE must approach infinity for no transcendence to occur

Does not Maps

Maps

Motion is quantized and the wave functions are able to tunnel through the potential barrier D3h intermediate barrier (a function nuclear potential on either side).

Quantized energy levels involving the tunneling of the electrons through the barrier at the CdSe/ZnS interface Rectangular potential barrier separating the electron in CdSe from the ZnS layer Hydrogen atom in the bridge and the electron in the target can tunnel through a potential barrier as their wavelike qualities allow for penetration into the classically forbidden region. Extension of the electronic wave function of the CdSe into the ZnS layer is seen via a slight redshift in the electronic absorption spectrum

Partial

Ammonia Inversion Bridge Mapping

Tunneling of a proton (particle in motion) across NH3 plane

Nanoscale World Target

The assessment of the activity involved 12 upper-division students, split into three groups (one of five, one of four, and one of three). A preactivity interview was conducted individually, with the goal of gauging students’ pertinent background knowledge, as the analysis was performed largely through a constructivist lens. Informed consent was obtained from each student prior to the collection of data. Students’ discourse within each group was recorded through the duration of the activity, and analysis focused on the nature of the exchange between students in the group as well as the investigator and the group. A postactivity interview was also conducted, with the aim of assessing if, individually, students were able to speak in a scientifically normative manner of the topics at hand. Codes were developed and applied to transcripts in congruence with the analogy framework for the activity and the theoretical perspectives to be discussed below. When relevant, an exact test for small-sample data36 was applied to quantify relationships

Does not Does not Does not Partial

Mapping

Overview of the Assessment

Classical World Base

ASSESSMENT AND RESULTS

Steel ball

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Table 1. Application of the Theoretical Framework to Concepts within the Activity

APPLICATION OF THEORETICAL FRAMEWORK The key element for an effective analogy is the emphasis on object relations mapping from base to target even when relatively few object attributes would map.19 In this activity, Part I is the classical macroscopic scenario (core) that serves as a base for Part II, the NH3 system. The NH3 system serves as a bridging concept21,22 that serves as a base for Part III, the nanoscale system. Table 1 summarizes the degree of mapping across the three systems. As should be noted from Table 1, the focus is on maximizing the number of object relations that map. Further, the role of the bridge is to overlap with the attributes of the base and the target, making two disparate domains more readily linked.

Object Attributes

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Particle

system with a potential well (the CdSe) surrounded on either side by a potential barrier of finite magnitude (the ZnS).2,32 This one-dimensional approximation is pertinent to the system insofar that it describes the layering of the core and shell and the behavior of the electronic wave function and resulting eigenvalues. Concluding this part of the activity, students are probed to consider the following: • The limit of an infinite barrier is needed with the bare core to model the absence of tunneling. • The particle under study (steel ball, vibrating atoms within NH3, and electrons in the quantum dots) and how it compares to those studied in previous parts. • How the construction of the wave function relates to the physical system, making the mathematical formalism relevant in terms of boundary conditions and the rationale for separating eigenfunction and eigenvalue. • How the wave function and corresponding eigenvalues change when considering bare and core/shell quantum dots. What the effects of confinement are on the energy levels in all cases (e.g., tunneling leads to a less confined electron and smaller energy level spacing). Discussion of the effects in the context of Parts I and II. • The splitting in the spectrum of NH3 from Part II, and the slight red shift of the spectrum of the core/shell quantum dots relative to the bare quantum dots in Part III provide spectroscopic evidence for the tunneling model’s use.

Electrons in motion. Elementary particles but different composition

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Table 2. Results of Fisher’s Exact Test Comparing Misconceptions and Valid Comparisons or Contrastive Statements to the Background Knowledge of Students Student Cohortsa Strong Background (n = 2) Medium/Weak Background (n = 3)

Misconceptions 7 8

Medium Background (n = 2) Weak Background (n = 1)

10 5

Strong Background (n = 1) Medium/Weak Background (n = 2)

4 17

Valid Comparison/Contrastive Statements Cohort 1 (N = 5) 6 6 Cohort 2 (N = 3) 9 0 Cohort 3 (N = 3) 2 13

p Values, Two-Tailedb

Significance of Results

1.000

Nonsignificant

0.118

Nonsignificant

1.000

Nonsignificant

a

N = number of students who took a preactivity interview in that particular cohort, and n = number of students in a given cohort with a given background knowledge rating. bEvaluated at the p = 0.005 level.

between certain codes and other variables involved in the study. A more comprehensive analysis of the qualitative data based on students’ group discussions is presented in the Supporting Information.

(4) Background knowledge statements: Any statement that invokes a group member’s prior exposure or knowledge of the topic(s) at hand. (5) Investigator intervention: Statements made by the investigator to cultivate students’ group discussion or assist in conceptual development. Simple procedural advice is not included in this category. The application of the codes to the transcripts was carried out using the analysis software Dedoose.41 Upon coding, the excerpts were sorted through, patterns were elucidated, and the mixedmethods analysis ensued.

Theoretical Perspectives

The main framework utilized for the qualitative portion of the analysis is based on a discourse analytical perspective, roughly adapted from the work of Duit and co-workers.37,38 The discourse analytical perspective is invoked in order to, as Duit and co-workers37,38 express it, analyze students’ situated responses to the learning or interviewing environments within which they find themselves. This allows emphasis to be placed upon the nature of students’ discussion that reconciles the fact that they are, at one point, in a group environment interacting with peers and the instructor and directly one-on-one with the researcher during the interviews. In addition, the learning condition as defined by Michelene Chi39 was also taken into consideration: “Students may have ideas that are in direct conflict with what is under study, and this constitutes the conceptual change learning condition.” Chi’s expression certainly applies in the case of quantum mechanical tunneling, as students largely come to the activity with conceptions and a language that is rooted in classical mechanics. By its very nature, the laws binding classical bodies directly conflict with the notion of quantum mechanical tunneling, for which there is no direct classical analogue.

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RESULTS Background knowledge of students, as assessed from the preactivity interviews, was classified as weak, medium, or strong. These classifications were determined by assigning numerical values (0, 1, or 2, with 2 being the most scientifically valid) to answers to each of the three preactivity interview questions. In total, there were three, five, and three students categorized as having weak, medium, and strong backgrounds, respectively. One student, pseudonym S3G3 (student 3, group 3), did not complete the pre- or postinterview and was, therefore, not included in this portion of the study. To examine how students’ background knowledge related to the number of misconceptions as opposed to valid comparison or contrastive statements, a 2 × 2 contingency table was set up within each cohort and Fisher’s exact test calculated.42 The results are shown in Table 2. Note in Table 2 that this analysis involves the number of statements that fall into a particular category, not the number of students, which will vary from cohort to cohort. For example, in cohort 1, there were seven misconception statements made by students with a strong background. The ultimate goal was to determine to what extent students’ background knowledge level related to their propensity toward making valid comparison/ contrastive statements or stating misconceptions. In Group 2, there were no students with a strong background knowledge level. Therefore, the separation needed to be between those with weak and medium backgrounds. Still, the p value (p = 0.118) is not considered statistically significant at confidence levels that are reasonable to such small data sets. It is important to note that the 2-tailed p value was utilized in determining statistical significance at the p = 0.005 level. The lack of statistical significance holds even when all students from all groups, within their background knowledge classifications, are considered. This is summarized in Table 3.

Coding

Coding components (in congruence with the epistemology developed by Quine40 and adopted by Duit et al.37,38) were motivated by the principal purpose of analyzing students’ expression of what they experience and how these experiences, in turn, are expressed verbally. The codes that were applied to the transcribed interview and group data are as follows: (1) Misconceptions: Erroneous or scientifically invalid statements about any particular component of the activity. (2) Comparison and contrastive statements: Statements that compare one component of the activity to another; can be either intra- or interdomain, or even external in the case of establishing spontaneous analogical transfer. External transfer is a separate subcode. (3) Reformulations: Corrections to misconceptions; such corrections may be group-induced, investigator-induced, or carried out by an individual who has expressed their own misconception. 1551



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S2G2: (Interjects) Right. S1G2: Even though the, uh.... S2G2: (Interjects) Barrier. S1G2: The potential barrier, was higher than the excited states that was energy for tunneling and the energy that we got from the one encapsulated in the shell was in between, even though it was the same diameter as 4 nm.... S2G2: (Interjects) The energy was lower than the box just by itself. Note that S1G2 first makes the physical observation statement that two peaks were seen from NH3, referring back to the manifestation of quantum mechanical tunneling in that domain. The student then refers to the fact that their calculated energy (based on the particle-in-a-box approximation) for the CdSe/ ZnS was intermediate relative to the bare CdSe 4 nm quantum dot and the larger 6 nm quantum dot. This moves toward a deeper understanding of the reduction in confinement associated with tunneling into the ZnS layer. A significant example of students utilizing spontaneous external analogical transfer can be found in the following exchange within group 1: S2G1: Well in the event of the infinite box, you know, just CdSe in a (sic) organic matrix, you have effectively an infinite box. In the case that you coat it with the ZnS you give it a squishy outer layer. You give the box...not so rigid sides. Instead of a steel box, you make it like a paper bag. S5G1: So it loses some energy. S2G1: Yeah so it can relax a little. S1G1: Pshhht. I give up on tunneling; I do not understand tunneling...I just.... S5G1: Do not think of it as tunneling, just think of it as like, ok, if you have like he was saying a metal box and you throw a bouncy ball in it, like it is going to hit, like you know. S1G1: I do not see how this relates to any of this energy. S5G1: Yeah, no, like that is what I’m saying. Like it has much more strict, it has a lot higher energy because it is bouncing off the walls...it is very confined. Whereas if you put it in a big plastic bag that is...blown up and you bounce it it is going to smash into the walls and you know it is going to have like, it is going to lose a lot of energy because it is not as confined. Student S1G1 is struggling with the quantum mechanical tunneling concept and is ready to give up on it altogether. S5G1 then utilizes a comparison and contrastive statement to build upon one already posed by S2G1 and is able to bring in knowledge from an external domain (bouncy ball in a plastic bag as opposed to a steel box) to describe the effect of quantum mechanical tunneling on confinement. This is highly significant, as it illustrates the potential of the instructional material to induce spontaneous external analogical transfer and it allows for

Table 3. Results of Fisher’s Exact Test Comparing Misconceptions and Valid Comparisons or Contrastive Statements to the Background Knowledge of Students (All Groups) Valid Comparison/ Contrastive Statements

Student Cohorts

Misconceptions

Strong Background (n = 2) Medium/Weak Background (n = 3)

11

8

40

28

p-Value, TwoTaileda


1.000

Nonsignificant

All Groups

a

Evaluated at the p = 0.005 level.

These results indicate that there is no discernible relationship between the students’ background and their tendency to make a valid comparison or contrastive statement as opposed to a misconception or vice versa. This suggests that nuances in the students’ discussions, as they struggle with the abstract concept of tunneling, will be most revealing when considering the overall implications of the activity. The three groups that performed the activity managed to discuss and adopt the language, in a scientifically normative way, required for describing tunneling effects both within the paradigm of a familiar molecular structure (NH3/AsH3) and in a novel nanodomain. For instance, consider the following group discourse excerpt: S4G3: Alright, so both sides of the barrier are interacting, that means that you would be able to make it over the potential without having that much energy. S1G3: Yeah, so you’d have to, um.... S4G3: (Interjects) Which does not make sense in the classical sense. Since, like, with the ball it sort of stops. Such discussions were meaningful, despite students’ representational challenges (see Supporting Information) pertaining energy diagrams and wave function representations in multiple parts of the activity. For example, in Part 3: S1G2: Yeah, I mean on the outside of the box, I mean but that is representative of the tunneling effect, which you know back in, when we’re doing...the ball, the steel ball thing.... S2G2: Yeah, the classical mechanics does not do it at all. S1G2: That did not occur. Students were able to move from the effect of quantum mechanical tunneling within the bridge domain and ultimately in the nano target. This was exemplified in the following excerpt from students in group 2: S1G2: So, part two we saw two peaks for NH3... S2G2: (Interjects) Right. S1G2: Which, which implied that there was two, that there was two excited states.

Table 4. Results of Yates Chi-Square Test Comparing Non-Comparative Physical Observation Statements and Spontaneous External Transfer to the Level of Investigator Intervention Student Cohorts

Physical Observation Statements (non comp.)

Spontaneous External Transfer

Chi-Square Value

p Value, Two-Taileda


10.418

0.0012

Nonsignificant

All Groups High Investigator Intervention (n = 2) Low Investigator Intervention (n = 3) a

232

1

75

6

Evaluated at the p = 0.005 level. 1552



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Most students, overall, were able to invoke the concept of quantum tunneling when asked about how potential energy manifested itself in each of the cases. For example: Investigator: Ok, so like what was the nature of the potential energy in each. So like starting with the classical and working your way to other parts? S3G2: Classical was sort of...rigid in the fact that it had to be a certain amount or it could not pass. And then in the ammonia, it looked like it had to be a certain amount but because of the quantum tunneling effect you could sort of get around that. And, I guess, sort of the same way for the electrons in the...third example. Here, S3G2’s comparison and contrastive statements clearly illustrate their understanding of the line of demarcation drawn between the base and the bridge (classical and quantum worlds). Further, they articulate the relationship between the bridge and the target insofar as the quantum mechanical tunneling applied in both of these domains: within the NH3 inversion and at the nanoscale with the electrons in the CdSe/ZnS core/shell quantum dots. Responses were generally normative concerning how events were measured or quantified. Take, for example, the following exchange involving student S4G3 in response to the fourth question (in each case, how were events observed, measured, or quantified?) of the postactivity interview: S4G3: Oh, so with the ball it was just like rolling and the spectroscopic one with the ammonia was just the transitions that were observed at some wavenumber, and then the particle in the box that is also spectroscopy with the nanoparticles. Further, S4G3 commented on the difference in the potential barrier limits in the classical and quantum worlds: Investigator: So how are the limits different in a classical system as opposed to a quantum system? S4G3: Um, in a quantum system you can, the limits are not, like, as well-defined since you can go over them but in a classical system you cannot just go over any barrier. The final content-related question of the postactivity interview asks students to describe the connections between each portion of the activity. The most developed of the responses to the overall connection question (“Describe, in the context of all you have experienced with this activity and what we have discussed, what the connections are between each segment of the activity.”) was given by student S1G2, who had a moderate level of background knowledge. The following is an excerpt from the response: S1G2: Um, so the first section kind of gave...a broad, um, a broad overview of what we were going to be looking at, uh, the particle being the steel ball, was dropped from a specific height which was found that the height had to be greater than or equal to the height of the potential barrier, which is basically the second hill. Um, and that kind of played into the next, um, section where we found that, um, when you go from this macroscopic world to the microscopic world that this potential barrier, um, although it exists and it is kind of restricting, that there are other ways, other, other things that are happening that res−, that kind of, um, do not necessarily play into the classical world. In this portion of the response, the student has immediately referred to, and distinguished, the energy requirements of the classical and quantum worlds by way of the base and bridge portions of the activity. Further within the response:

students’ development of scientifically normative concepts in both the base and target knowledge domains. One point of interest is that not all groups received the same amount of nonprocedural investigator facilitation. It was noted, through the code application and code co-occurrence tables obtained, that the number of spontaneous external transfer events seemed to be lower in groups with higher instances of investigator intervention. To test this quantitatively, another contingency table (Table 4) was set up to compare spontaneous transfer statements to noncomparative physical observation statements from students in groups with low investigator intervention and high investigator intervention alike. This time, a Pearson chi-squared approximation with Yates correction43 was utilized in order to account for data, as some of the numbers were large, yet over 20% of the contingency table consisted of values below 5. Note that, under the Pearson chi-square approximation with the Yates correction, the p value indicates high statistical significance (p < 0.005). From the perspective chosen for this analysis, it is plausible that the investigator intervention shifted the learning environment from one of individuals on the same power-structure level (peer to peer) to one with a slanted power-structure level (investigator to student). Therefore, the situated responses put forth by the students in the latter case may very well have been to please the investigator, thereby avoiding any commentary that drew from outside the frame of reference of the current activity. This effect is notable in the postactivity interviews, in which the environment is significantly different from that of the small student groups. Students in that situation tend to be much more concerned with stating a “correct answer” as opposed to stating directly what comes to mind. It is, therefore, recommended that instructors who may adopt this activity use limited, yet prudent intervention in matters that go beyond simple procedural instruction. Post-Activity Interview Implications

Individual postactivity interviews focused largely on probing students to identify the relevant components of the activity, connect the components together, and obtain feedback as to how their viewpoint of analogies in this context may have changed. The questions are structured to align with the components as defined within the learning goals of this activity. An interesting observation immediately at the beginning of the interviews is that students had a relatively high degree of difficulty in determining the “particle” in the third portion of the activity. On several occasions, students referred to the quantum dot structure itself as the particle, whereas, in reality, the electron was the particle under study. This may have occurred due to the fact that when one imagines the quantum dot, a spherical-type structure comes to mind. Another consideration is that students may not be familiar with quantum dots to begin with. Further, the fact that such structures are referred to as “nanoparticles” brings with it language that implies a particulate entity. Consider, for example, the response by S4G1: S4G1: Huh. Um, ok, so the classical it was, um, it was rolling down the thing, down the, uh, I do not know, ramp I guess. Um. In the ammonia, we were looking at vibrational states I think? So, it was just vibrating. And then, in the quantum dots, the um, I guess the particle was, uh, inside like a box almost and that is what gave it its quantum dot. Investigator: Wh−, what was the particle itself? S4G1: Umm. Like the dot itself?.... 1553



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• A classical ball-and-ramp apparatus to start in the world with which students are familiar. • The spectroscopic manifestation of tunneling in familiar molecules (NH3/AsH3). • The tunneling effect as observed at the nanoscale in core/ shell quantum dots. The use of an analogical theoretical framework provided the foundation for an activity whose components are centered about relevant attributes. Rather than isolating phenomena, our approach allows, simultaneously, for the discovery and critical analysis across knowledge domains, ultimately leading to exploration of concepts at the nanoscale. Upon the testing of the activity with groups of upper-division undergraduate chemistry students, a mixed-methods approach was taken to code and analyze interview and student group discourse data. Pertinent statistical tests were invoked (Fisher’s exact, chi-square with Yates’ correction) when their use was of value to the analysis itself. These tests revealed three important features of the data: (1) Inherent issues with translating mathematical language into visual representations were found to be congruent with existing literature in physics education research, particularly the problems involving proper labeling of plots and diagrams and the separation of eigenfunction and eigenvalue in representations.13−17 (2) Misconception statements were no more likely to be made by students with low or moderate background knowledge. (3) High investigator intervention was associated with higher levels of physical observation statements, but lower levels of spontaneous external analogical transfer within student group discourse. The findings lend evidence to support that students are able to struggle with, yet ultimately adopt, the normative language and conceptual framework of quantum mechanical tunneling in the context of the nanoworld. Further, there is evidence both within the interview data and student group discourse that a high degree of critical reflection occurred in the base domain, as there was significant discussion about the classical world in order to begin to describe an abstract model (tunneling) whose understanding requires direct conflict with the classical paradigm within which we experience everyday life. This was facilitated by questions in the activity itself and spontaneously through group interaction. Ultimately, this serves to positively support the two research questions. The evidence supports that students were able to connect their core physical and chemical concepts to quantum mechanical tunneling at the nanoscale and reflect upon their prior knowledge in a scientifically normative manner, particularly in cases of spontaneous analogical transfer arising from external knowledge domains. This work represents an explicit, assessed approach to confronting the abstract upper-division concept of quantum mechanical tunneling in multiple contexts, and evidence has been provided through the qualitative analysis to show that analogical transfer can occur with implementation in small group settings. A principal limitation of this study is that it was conducted within a relatively short amount of time and, consequently, did not explore the implications of students’ long-term retention of knowledge in the context of this analogy. Another is that the scale of the study was small: it occurred with students in a traditional upper-division laboratory course within a single research I university. Although small-sample statistics and qualitative results provided useful insight and evidence for analogical

S1G2: Um, we saw that there were tunneling effects even though the potential barrier, uh, was higher than, um, the actual, I do not know how to word this, I’m probably going to word it wrong, but like the excited states were. Um, and then that kind of played into...the third part where we saw that, um, the infinite potential on the outside of the box acted as a (sic) infinite barrier and we saw that, um, there were not any tunneling effects given that infinite potential. The student has incorporated scientifically normative language of quantum mechanical tunneling into the response at this point and invoked the particle-in-a-box model, thus transitioning from the bridge to the target domain at the nanoscale. S1G2 was initially apprehensive about stating their response with the concern that it may not have been normative, which may be interpreted, from the discourse perspective, as situationally appropriate to the implicit investigator-student relationship that took place during the interview. Finally: S1G2: And, we also saw after that we reduced the potential down to a finite number, that tunneling, uh, did happen, so I would kind of equate...the second part...of part three to part two and...the beginning of part three, to part one almost but not, not in like the, there’s still, part one is classical and then part one, and part three was still kind of quantum, but, they kind of related to each other more. Here, the student clearly states that the second part of Part III is like Part II, and that the beginning of Part III is like Part I, though the student was careful to emphasize that Part III considers a quantized particle and Part I considers a classical particle. This is an astute observation, considering the second portion of Part III involves the finite potential barrier, as does the entirety of Part II. Furthermore, the first portion of Part III involves an infinite barrier that is completely impenetrable, just as the ball is confined to the trough in the ramp apparatus when it is not given enough potential energy to cross the barrier. It is important to note that although analogical transfer occurred in all groups, not all students as individuals were readily able to transfer information between the knowledge domains in a scientifically normative fashion. This was the case of student S2G3: Investigator: So, uh, describe in the context of what you have experienced with the activity, and what we’ve discussed, what connections are made between each segment of the activity. S2G3: You know, take the next step of thinking about something that is very...esoteric and...theoretical, and we cannot ever put our hands on it or touch it, but at least we can kind of look at this, like you said earlier, there’s, this is an analogy.... Of particular note is the fact that S2G3 remarks that the tunneling effect is, in their view, an esoteric theory. This is unsurprising, given the conceptual change that is required in order for students to overcome their notions of the classical world and turn their attention to a phenomenon that they cannot observe outside of an instrumental measurement. It should be noted that all students, even those with higher background knowledge levels, struggled with the concept. In S1G1’s case, however, the group intervened with spontaneous external transfer and helped guide the student toward understanding the concept.

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CONCLUSIONS Students’ exploration of the effect of quantum mechanical tunneling in this activity occurs through a progression between three domains: 1554



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(11) Veguilla-Berdecia, L. A. Tunneling in Quartic, Symmetric, Double Well Potential: A Simple Solution Using a Hermite Basis. J. Chem. Educ. 1993, 70, 928−931. (12) Zollman, D. A.; Rebello, N. S.; Hogg, K. Quantum Mechanics for Everyone: Hands-on Activities Integrated with Technology. Am. J. Phys. 2002, 70, 252−259. (13) McKagan, S. B.; Wieman, C. E. Exploring Student Understanding of Energy through the Quantum Mechanics Conceptual Survey. AIP Conf. Proc. 2006, 818, 65−68. (14) McKagan, S. B.; Perkins, K. K.; Wieman, C. E. Deeper look at student learning of quantum mechanics: The case of tunneling. Phys. Rev. ST Phys. Educ. Res. 2008, 4, 020103−1−020103−18. (15) Wittman, M. C.; Morgan, J. T.; Bao, L. Addressing student models of energy loss in quantum tunnelling. Eur. J. Phys. 2005, 26, 939−950. (16) Morgan, J. T.; Wittman, M. C.; Thompson, J. R. Student Understanding of Tunneling in Quantum Mechanics: Examining Interview and Survey Results for Clues to Student Reasoning. AIP Conf. Proc. 2004, 720, 97−100. (17) Singh, C.; Zhu, G. Cognitive Issues in Learning Advanced Physics: An Example from Quantum Mechanics. AIP Conf. Proc. 2009, 1179, 63−66. (18) Sarantopoulos, P.; Tsaparlis, G. Analogies in Chemistry Teaching as a Means of Attainment of Cognitive and Affective Objectives: A Longitudinal Study in a Naturalistic Setting, Using Analogies with a Strong Social Content. Chem. Educ. Res. Pract. 2004, 5, 33−50. (19) Gentner, D.; Markman, A. B. Structure Mapping in Analogy and Similarity. Am. Psychol. 1997, 52, 45−56. (20) Gentner, D. Structure Mapping: A Theoretical Framework for Analogy. Cognitive Sci. 1983, 7, 155−170. (21) Greenberg, A. Integrating Nanoscience into the Classroom: Perspectives on Nanosscience Education Projects. ACS Nano 2009, 3, 762−769. (22) Palmer, E. A. The Nanoleap Project Evaluation Report 2007−2008; ASPEN Associates: Edina, MN, 2009. (23) Minstrell, J. Explaining the “at rest” Condition of an Object. Phys. Teach. 1982, 20, 10−14. (24) Clement, J.; Brown, D. E.; Zietsman, A. Not all Preconceptions are Misconceptions: Finding “Anchoring Conceptions” for Grounding Instruction on Students’ Intuitions. Int. J. Sci. Educ. 1989, 11, 554−565. (25) Brown, D. E.; Clement, J. Overcoming Misconceptions via Analogical Reasoning: Abstract Transfer Versus Explanatory Model Construction. J. Instr. Sci. 1989, 18, 237−261. (26) Clement, J. Using Bridging Analogies and Anchoring Intuitions to Deal With Students’ Preconceptions in Physics. J. Res. Sci. Teach. 1993, 30, 1241−1257. (27) Serway, R. A.; Vuille, C. College Physics; Brooks/Cole: Boston, MA, 2012; Vol. 1, pp 254. (28) Serway, R. A.; Jewett, J. W. Principles of Physics: A Caluclus-Based Text; Brooks/Cole: Belmont, CA, 2006; Vol. 1, pp 192. (29) Atkins, P. W. Physical Chemistry; Oxford University Press: New York, 1982; pp 699−700. (30) Adv. Photochem.; Neckers, D. C., Jenks, W. S., Wolff, T., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, Vol. 29, pp 128. (31) Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy; Dover Publications Inc.: New York, NY, 1975; pp 300−307. (32) Mitin, V. V.; Sementsov, D. I.; Vagidov, N. Z. Quantum Mechanics for Nanostructures; Cambridge University Press: New York, NY, 2010; pp 65−84. (33) Dabbousi, B. O.; Rodriguez-Viejo, J.; Mikulec, F. V.; Heine, J. R.; Mattoussi, H.; Ober, R.; Jensen, K. F.; Bawendi, M. G. (CdSe)ZnS CoreShell Quantum Dots: Synthesis and Characterization of a Size Series of Highly Luminescent Nanocrystallites. J. Phys. Chem. B 1997, 101, 9463− 9475. (34) Atkins, P.; dePaula, J. Physical Chemistry; Oxford University Press: New York, NY, 2006; pp 282, 288. (35) deSouza, R. T.; Iyengar, S. S. Using Quantum Mechanics To Facilitate the Introduction of a Broad Range of Chemical Concepts to First-Year Undergraduate Students. J. Chem. Educ. 2013, 90, 717−725.

transfer, it would be interesting to probe the efficacy of this instructional material in different pedagogical settings (e.g., inquiry-based laboratories) across different institutions. The limitations detailed above provide an opportunity to propose future studies. Possible future work would entail conducting a longitudinal study to gauge the long-term effects of the analogy on students’ quantum mechanical tunneling conceptions. Given the evidence provided by Sarantopoulos and Tsaparlis that analogies in chemistry can be meaningful longterm, it would be beneficial to provide evidence to determine whether or not this holds for this instructional material.18 Additionally, it would be beneficial to assess the efficacy of the analogical activity at different institutions and examine the effects of different pedagogical approaches on students’ transfer of information across domains. Further, refinements could be made to the nature of the questions asked in the activity based on evidence from this study in order to investigate whether or not such refinements improve transfer between knowledge domains.

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ASSOCIATED CONTENT

S Supporting Information *

A detailed analysis of the qualitative data, student handouts, and instructional guidelines based on the findings of the assessment. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*M. N. Muniz. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Jaap Folmer for his continued support in engaging his students in chemistry education research projects. We would also like to thank the National Science Foundation (NSF DUE-1043529) for their generous financial support of portions of this work.

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REFERENCES

(1) Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy; Dover Publications: Mineola, NY, 1989; pp 90−92, 168. (2) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997; pp 142−146. (3) Anslyn, E. V.; Dougherty, D. A. Modern Physical Organic Chemistry; University Science Books: Sausalito, CA 2006; pp 435−436. (4) Hopfield, J. J. Electron Transfer Between Biological Molecules by Thermally Activated Tunneling. Proc. Nat. Acad. Sci. U. S. A. 1974, 71, 3640−3644. (5) Kuhn, H.; Forsterling, H.; Waldeck, D. H. Principles of Physical Chemistry; John Wiley & Sons: Hoboken, NJ, 2009; pp 941−942. (6) McMahon, R. J. Chemical Reactions Involving Quantum Tunneling. Science 2009, 299, 833−834. (7) Halpern, A. M.; Ramachandran, B. R.; Glendening, E. D. The Inversion Potential of Ammonia: An Intrinsic Reaction Coordinate Calculation for Student Investigation. J. Chem. Educ. 2007, 84, 1067− 1072. (8) Cedillo, A. Quantum Mechanical Tunneling Through Barriers: A Spreadsheet Approach. J. Chem. Educ. 2000, 77, 528−531. (9) Marzzacco, C. J.; Baum, J. C. Computational Chemistry Study on the Carbene Hydroxymethylene. J. Chem. Educ. 2011, 88, 1667−1671. (10) van Staveren, M. N.; Edwards, K. D.; Apkarian, V. A. Playing with Light: Adventures in Optics and Spectroscopy for Honors and Majors General Chemistry. J. Chem. Educ. 2012, 89, 1447−1449. 1555



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(36) Hoyle, R. H. Statistical Strategies for Small Sample Research; Sage Publications Inc.: Thousand Oaks, CA, 1999; pp 133−149. (37) Duit, R.; Roth, W.; Komorek, M.; Wilbers, J. Fostering Conceptual Change by Analogies- Between Scylla and Charybdis. Learn. Instr. 2001, 11, 283−303. (38) Duit, R.; Roth, W.; Komorek, M.; Wilbers, J. Conceptual Change Cum Discourse Analysis to Understand Cognition in a Unit on Chaotic Systems: Towards an Integrative Perspective on Learning in Science. Int. J. Sci. Educ. 1998, 20, 1059−1073. (39) Chi, M. T. In Three Types of Conceptual Change: Belief Revision, Mental Model Transformation, and Categorical Shift; Vosniadou, S., Ed.; International Handbook of Research on Conceptual Change; Routledge: New York, NY, 2008; pp 61−82. (40) Quine, W. V. Pursuit of Truth; Harvard University Press: Cambridge, MA, 1990; pp 1−36, 61−76, 95−102. (41) Dedoose Version 4.5, Web application for managing, analyzing, and presenting qualitative and mixed method research data; SocioCultural Research Consultants, LLC: Los Angeles, CA, 2013; http://www. dedoose.com/ (accessed Jul 2014). (42) Preacher, K. J.; Briggs, N. E. Calculation for Fisher’s Exact Test: An Interactive Calculation Tool for Fisher’s Exact Probability Test for 2 × 2 Tables; Vanderbilt University: Nashville, TN, 2001; http://quantpsy. org/ (accessed Jul 2014). (43) Preacher, K. J. Calculation for the Chi-Square Test: An Interactive Calculation Tool for Chi-Square Tests of Goodness of Fit and Independence; Vanderbilt University: Nashville, TN, 2001; http://quantpsy.org/ (accessed Jul 2014).

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Investigating Quantum Mechanical Tunneling at the Nanoscale via

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