European Journal of Physics Eur. J. Phys. 35 (2014) 055024 (21pp)
doi:10.1088/0143-0807/35/5/055024
Teaching quantum physics by the sum over paths approach and GeoGebra simulations M Malgieri, P Onorato and A De Ambrosis Department of Physics, University of Pavia, Italy E-mail:
[email protected],
[email protected] and anna.
[email protected] Received 16 June 2014, revised 10 July 2014 Accepted for publication 11 July 2014 Published 8 August 2014 Abstract
We present a research-based teaching sequence in introductory quantum physics using the Feynman sum over paths approach. Our reconstruction avoids the historical pathway, and starts by reconsidering optics from the standpoint of the quantum nature of light, analysing both traditional and modern experiments. The core of our educational path lies in the treatment of conceptual and epistemological themes, peculiar of quantum theory, based on evidence from quantum optics, such as the single photon Mach–Zehnder and Zhou–Wang–Mandel experiments. The sequence is supported by a collection of interactive simulations, realized in the open source GeoGebra environment, which we used to assist students in learning the basics of the method, and help them explore the proposed experimental situations as modeled in the sum over paths perspective. We tested our approach in the context of a post-graduate training course for pre-service physics teachers; according to the data we collected, student teachers displayed a greatly improved understanding of conceptual issues, and acquired significant abilities in using the sum over path method for problem solving. S Online supplementary data available from stacks.iop.org/ejp/35/055024/ mmedia Keywords: teacher training, quantum physics, sum over paths, interactive simulations, Feynman paths, Mach–Zehnder, optics 1. Introduction In recent years a significant number of studies has addressed the problem of designing suitable approaches for conveying the fundamentals of quantum theory to an audience of students who are not physics specialists. Curriculum reforms enacted in several countries 0143-0807/14/055024+21$33.00 © 2014 IOP Publishing Ltd Printed in the UK
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have substantially increased the presence of modern physics in secondary school programmes, thus making the problem of preparing physics educators for the task of teaching quantum mechanics even more urgent and challenging. In this framework we designed a teaching sequence and we tested it with both pre-service teachers enrolled in a post-graduate university course, and in service teachers undergoing update training. Our proposal stands in a consolidated research line advocating the use of Feynmanʼs sum over paths method in teaching introductory quantum mechanics [1–8]. One of the main goals of our work was to extend the usual approach towards a treatment of recent interferometry experiments in quantum optics. These experiments can serve as a natural introduction to modern, information-based foundational views, which the language of Feynman paths is especially suited for expressing. Epistemological dilemmas peculiar to quantum physics may become clearer when seen from the sum over paths perspective. As an example, the issue of wave–particle duality has been identified in the literature as the source of several possible persistent misconceptions [9, 10]. However, in the sum over histories framework the behaviour of the quantum object can be made plausible by noting the reduction of the possible, and therefore interfering, paths, due to the acquisition of ‘which way’ information. One of the advantages of the Feynman perspective lies in the possibility for students to solve non-trivial problems, computing numerical results for detection probabilities, with very simple mathematical means. For more complicated examples the calculations can be computer assisted through interactive simulations. Simulations are also essential for visualizing the mathematical structure of the model, and for conducting discovery activities, which can potentially play an important role in the learning process [11]. For this purpose we adopted the software GeoGebra [12], which is widely used and praised in the community of teachers of mathematics and physics. In total, we have designed at the moment 12 simulations, which are available as supplementary material accompanying the paper (see stacks.iop.org/ejp/35/ 055024/mmedia).
2. Theoretical framework Based on the vast amount of literature evaluating the possible strategies for teaching quantum mechanics at an elementary level, and investigating the main sources of student difficulties, we identified some crucial points which we tried to address in our work. In particular: • A tension has been acknowledged between the perceived necessity of explaining the historical crisis that led to abandon classical ideas, and the urge to present the conceptual picture of quantum physics as radically new [13]. Research has shown that if semiclassical models are introduced too early they may crystallize in students minds [14, 15], hindering the formation of a fully quantum perspective. Our choice was not to commit to a historical approach, but rather to consider both traditional and modern experiments with the objective of sharply pinpointing the non-classical features of quantum theory. In this way, we aim to reduce the known issue of students’ production of hybrid quantum–classical models [16]. • Studies have reported a problem with the acceptability [17] of controversial quantum concepts on the students part. The Feynman perspective has the undeniable advantage of epistemological clarity; it allows to completely overcome some foundational difficulties, such as ‘wave–particle duality’, and to express others, such as the role of measurement, in particularly clear and comprehensible terms. Also, recent experiments provide more focused and unambiguous answers to doubts that could arise about the necessity of 2
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Figure 1. Schematic representation of the teaching sequence. In the central column, the
content presented is (to some extent artificially) divided in steps. In the left column, the experiments discussed. In the right column, the GeoGebra simulations more relevant to each subject. 3
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abandoning classical ideas, and therefore contribute to making quantum concepts more acceptable for students. • Several researchers have advocated the necessity of addressing conceptual and interpretative issues of quantum theory [18, 19]. It has been found that primarily conceptual knowledge of modern physics is retained by students almost entirely intact for very long times [20]. Therefore, we initially highlight the presentation of these aspects; in a later phase of the sequence, we propose exercises which include both conceptual issues and practical calculations. Whenever possible, we parallel the discussion of experiments with students’ exploration of GeoGebra simulations which model the relevant physical situation employing the sum over paths perspective. Simulations do not give any representation or visualization of the quantum objects themselves (which may be easily misunderstood), but make the formal characteristics of the model more accessible and understandable. We tried to improve on the existing software to teach the sum over paths approach to quantum theory [2, 5, 6] by making our simulations as interactive as possible, allowing the user to control all relevant parameters and introduce variations in the setup through sliders and checkboxes.
3. The sequence The teaching sequence is schematically represented in figure 1. 3.1. The sum over paths method in wave optics
The sum over paths method can initially be seen as a convenient way for describing interference phenomena in a classical wave perspective [21]. We discuss the Huygens’ principle, whose idea of a wave producing new wave sources at all points in space, has well-known analogies with the path integral formulation [22, 23]. Students learn to compute the value of the amplitude of a monochromatic wave at a given point P in space by summing the amplitude vectors resulting from all possible optical paths that the disturbance in the wave could have followed to reach P (provided it remained coherent) from the original source S. This source-to-detector [8] philosophy is central in the sum over paths perspective, and is often also an implicit assumption in the usual treatment of interference phenomena in wave optics. 3.2. The photon concept
The quantum discontinuity emerges in our approach from the photon idea. As anticipated, we do not commit to a historical reconstruction, but use experimental evidence gathered at different times, in the unifying perspective of providing a logically convincing, unambiguous construction of the idea of photon and of its fundamental properties. In particular, we discuss the following experiments: • The photoelectric effect, as basic evidence of the granularity of light interacting with matter, and introducing the Planck relation E = hν . • The Grangier et al 1986 experiment [24] on photon indivisibility. • Two-slit interference experiments with single photons [25] with a video [26] of the accumulation of individual photons, introducing the probabilistic interpretation. 4
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Our message to students can be summarized as follows: when light is thought of as a wave, it naturally has the property of being distributed in space so that, for example, it can pass through both of the two slits in the Young experiment, producing interference. But how can the phenomenon be explained if light is made of particles, and interference persists even if the intensity of light is so low that one at a time is emitted by the source? The Grangier experiment proves that the photon does not simply split in two at the slits. The Feynman idea of the photon following all possible paths is then presented as a logically consistent answer. 3.3. The Feynman photon model
In the single photon picture the basic problem consists in a source placed at point S, emitting a photon which may or may not be collected at a detector at point P. The phase ϕ = kx − ωt , previously associated to the classic monochromatic wave, must now be attributed to the photon itself, and computed over all its possible paths from the source to the detector. Then the corresponding phasors must be summed to find the ‘resulting arrow’. Since the time instant of arrival at the detector is the same for all paths, the temporal ωt term only produces an irrelevant overall phase, and only the path length term needs to be considered. A key difference with the classical perspective in wave optics lies in the probabilistic interpretation. The square modulus of the resulting arrow at point P must now be interpreted as (proportional to) the probability of detecting the photon at P. A clear and precise formulation of this point is very important, as students can be tempted to assign probabilities to individual paths; or, they may remain confused with the issues of normalization. The introduction of simulations regarding experiments in optics, carefully reworded in terms of photons and probabilities, is of great help in clarifying these points. Finally, the ‘wave function’ of a quantum object at a given time t can be defined (again, paying attention to the normalization problem) as the set of complex values corresponding, at the time instant t, to the ‘resulting arrows’ at all points in space [2]. 3.4. Using the Feynman approach for treating conceptual and foundational issues
The discussion of epistemological matters in terms of Feynman paths, which represents the core of our proposal, comprises three main elements: • The single photon Mach–Zehnder (MZ) interferometer [24] demonstrating the impossibility to attribute a single path to a quantum particle [27]. • Diffraction of a photon from a single slit with a variable width, introducing a discussion of the uncertainty principle [28]. • The Zhou–Wang–Mandel (ZWM) 1991 [29] experiment used to highlight the role of measurement and which way information. The MZ interferometer can be used to provide a convincing proof of the untenability of the classical trajectory concept. We do not here describe the experiment in detail, as it will be more thoroughly explained in section 4, where the related simulation is presented. Comparing the outcomes corresponding to different setups of the apparatus proves that the photon cannot be thought as taking either one or the other of two possible ways, but must be imagined as taking both of them simultaneously. In other words, it is impossible to associate a single path to a quantum object. The phenomenon of single slit diffraction with a variable slit width is discussed as a simple but effective introduction to the uncertainty principle [28]. The diffraction pattern can be easily related to the momentum probability distribution of the diffracted photon, and as the 5
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slit width is varied in a simulation, an intuitive understanding of the meaning of the principle h can be provided. Simple calculations lead to the approximate uncertainty relation ΔxΔpx ≳ 2 . We pay particular attention to making clear that the uncertainty principle should be interpreted as an intrinsic limitation to the information about complementary variables which quantum states can contain, and not as a perturbation of the system due by measurement, as in the old ‘Heisenberg microscope’ view. The ZWM apparatus is a two-way, single photon interference setup where ‘which way’ information is collected in a non-destructive manner through a clever use of nonlinear crystals. The main result of the ZWM experiment consists in proving that the modification in the final outcome of an experiment due to an intermediate measurement, which is peculiar to quantum physics, should not be thought in terms of a disturbance, but of information acquired or recorded about the system. The language of the Feynman approach is especially appropriate for summing up the lessons that can be drawn from the experiment. In particular it is sufficient to reinterpret the expression ‘all possible paths’ to mean ‘all paths compatible with the information about the system’, to reconstruct the idea of wave function collapse caused by a measurement, even if of non-destructive nature. The ZWM experiment also allows to introduce a generalization of the concept of path. In fact, what is to be summed in the sum over paths approach are not necessarily only the possible word-lines of a single particle, but more generally, all possible indistinguishable processes leading to the same experimental outcome: indistinguishable, in the sense that no information can be retrieved about which one of the processes has happened. An immediate consequence of this generalization is the possibility of introducing the schematic representation of processes in terms of Feynman diagrams. This part of the teaching sequence, devoted to the analysis of modern experiments, opens up the way for a collective discussion on the profound epistemological fracture between classical and quantum physics, and on current foundational views [30] in which information plays a crucial role. Advancing the foundational debate with students to the level of its present state of the art is one of the major themes our proposal. 3.5. Massive particles
Observational tests of interference using massive objects prove that the quantum behaviour cannot be relegated to some special property of light. Besides older setups using electron beams, such as the Davisson–Germer experiment, single particle interference effects have been demonstrated for electrons [31, 32], neutrons [33] and C60 molecules [34]. We base the extension of the sum over paths formalism to the treatment of massive particles on the following steps: • By reconsidering the photoelectric effect (leading to the photon energy E = ω ) and summarizing the main conclusions that can be drawn from the Compton scattering ( p ⃗ = k ⃗ ) we rewrite the photon phase ϕ = kx − ωt as
ϕ=
p E x − t.
(1)
We postulate this formula to be valid for the phases carried by both massless photons and massive particles alike. • From equation (1), proceeding by way of analogy, we derive the expression for the De h Broglie wavelength, λ = p . • In the case of a particle moving in a potential V(x) when the total energy E is a constant of motion we write the more general form 6
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ϕ=
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∫ p (x)dx − E t,
(2)
where S0 = ∫ pdx = ∫ 2 m [E − V (x ) ] dx is the abbreviated action and S = ϕ = ∫ (x, t )dt is the Hamilton action [35]. Similarly to the photon case, when the detection time has been fixed the time dependent term in equation (2) plays no role in determining the phase difference between paths. • Equation (2) can be compared with the analogous expression for the phase of monochromatic light entering an inhomogeneous medium with refractive index n(x), i.e.
ϕγ =
∫ k (x)dx − ωt = ∫ k 0 n (x)dx − ωt.
(3)
3.6. The limit of geometrical optics and the correspondence principle
When the wavelength of the quantum object becomes much smaller than the relevant length scales, the sum over paths approach reproduces the results of a classical theory. For the photon, the dominant path in this limit is the one predicted by the Fermat principle, thus retrieving the ray of geometrical optics. The behaviour of massive particles, on the other hand, approaches the one predicted by the stationarity of the Hamilton action, leading to the correspondence principle and explaining the emergence of classical mechanics as an approximate theory. Simulations in which the wavelength (or the mass) of the quantum object can be interactively varied are essential at this stage; examples include light refraction at an interface, parabolic mirror reflection, and diffraction of massive particles at a double slit (see section 4).
3.7. Quantization in the sum over paths approach
The sum over paths approach can be used in a simple way to find energy levels and eigenfunctions of quantum bound systems [36], reproducing the results obtained through the time independent Schrödinger equation for large classes of potentials [37]. In essence, the method consists in evaluating the probability of detecting at position xf a particle emitted by a source at xi by considering all possible paths with a fixed energy E, including those undergoing an arbitrary number of reflections at the potential walls. In this way, the detection probability is computed as a function of E. In the limit that the number of paths considered goes to infinity, only energies satisfying the condition that the vectors corresponding to different paths are in phase produce a non-vanishing probability. Thus, the quantization conditions for energy are recovered1. The same approach also allows one to compute the stationary state wavefunctions. The method is connected to both the Einstein–Brillouin–Keller and Wentzel–Kramers–Brillouin quantization schemes. We only introduce with this method some simple examples, such as the infinite and finite square potential wells, and a particle restricted to a circular orbit. In perspective, such treatment can favour a seamless connection with the usual approach to confined systems using the time-independent Schrödinger equation. 1
This procedure corresponds to finding the poles of the Green function G (xi , x f , E ) = ∑n 7
ϕn* (x i ) ϕn* (x f ) E − E n + iϵ
.
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Figure 2. Sum over paths simulation of a diffraction grating. In the left-hand window the final probability distribution, and the real and imaginary parts of the amplitude, drawn lighter and in a smaller scale. In the right-hand window, the construction of the resultant arrow at the detector point, with the phasors corresponding to individual possible paths (small black arrows) and the resulting arrow (red, in a heavier brush stroke). Parameters that can be varied in this case, besides the position of the detector, include the grating pitch d, the light wavelength, the source–detector distance D.
4. Selected simulation examples Simulations traditionally employed with the sum over path approach refer to diffraction and interference at slit experiments. We tried to enlarge the panorama by exploring other experimental situations such as the Lloyd mirror, demonstrating a phase loss for ‘external’ reflection, and the MZ interferometer, for which we designed a rather complex simulation including removable arms. We tried to render simulations as interactive and visually engaging as possible and to make them as close as possible to real experimental situations.
4.1. Interference and diffraction
Figures 2 and 3 show the general form of most of our simulations: two separate graphical views are paired in adjacent windows, while the algebra view is usually left hidden, although it can be brought up if necessary. The left graphical window is used for the geometrical setup of the experiment, for interactive sliders and boxes, and visualization of the final probability distribution. The right-hand graphics window shows the sum of phasors corresponding to the paths arriving at the detector point (which can be chosen through a slider). We found this general structure to be easy to understand and manipulate for users, and to provide a good understanding of the functioning of the sum over paths algorithm. Figure 2 shows diffraction from a grating; figure 3 diffraction from a single slit of variable width. Other cases are conceptually very similar to the ones shown.
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Figure 3. Single slit diffraction. The width of the slit can be varied through the parameter d. This simulation is used to provide a qualitative understanding of the uncertainty relation.
4.2. Refraction at an interface
In this case we propose two different simulations (figure 4). The first one is a re-implementation of Taylorʼs original example [2]. The experimental situation represented is an isotropic light source near an interface between two media, with a detector placed beyond the interface. The shape of the ‘Cornu spiral’ which appears in the right-hand graphic window in figure 4(a) shows that the paths near that of minimum time give a larger contribution to the final amplitude. Also, the minimum time path becomes more and more dominant, as the wavelength decreases with respect to the source–detector distance. Paths which are very far from the ray of geometrical optics give essentially no contribution, since they go round in ‘curls’ at the ends of the spiral. We also produced a GeoGebra file representing the phenomenon of refraction at an interface as seen in a different experimental setup, more similar to those used in real experiments. In this case the situation represented is the pointing of a ‘collimated beam’ towards the interface with a certain variable incidence angle α. The resulting probability distribution found at some distance inside the material has a peak at the position given by Snellʼs law. Since nine discretization points are used for the source, and five for the interface, 45 paths are computed for each detector point (in this case, the phasor sum is not visualized). This example is meant to show relatively advanced students how somewhat realistic situations can be dealt with using the sum over paths approach. 4.3. Lloyd mirror and phase loss upon reflection
The explanation of some interferometry experiments in the sum over paths perspective needs a detailed analysis of the consequences of reflection at an interface. For example, a general property of beam splitters is the relation δ1 + δ 2 = π , where δ1 and δ2 are the phase shifts between transmitted and reflected rays at the two inputs [38]. The peculiar features of the kind of MZ interferometer which we represent in our simulation depend on the use of beam splitters with δ1 = π and δ 2 = 0 , that is, in which the reflected ray has a π phase shift with 9
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Figure 4. Refraction of a photon at the interface between two materials with different refraction indices n1 and n2. (a) Traditional way of representing the phenomenon, with paths nearer to the ray of geometrical optics shown in a different shade (shifting towards red in the online version). (b) A different experimental arrangement for the same phenomenon, showing the bending of a collimated ray directed towards an interface.
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Figure 5. The Lloyd mirror.
respect to the transmitted ray on one of the inputs, but no phase change on the second one [39]. Also, applying to phasors associated with Feynman paths the appropriate phase shift upon reflection is essential for obtaining the correct energy levels for bound systems. For these reasons we discuss the phase change associated with reflection early in our sequence, using the Lloyd mirror experiment and simulation. Other reconstructions neglect to point out the phase change at reflection because they do not consider experiments where interference between reflected and direct paths takes place. The Lloyd mirror setup (figure 5) consists in a light source emitting photons, which can reach a detector (screen) either through a direct path, or after being reflected by a mirror placed orthogonally to a the screen. The discussion of this experiment is simple: the interference pattern between the two possible photon paths has a minimum at the origin, showing that the direct and reflected paths (which, since the point at the beginning on the screen coincides with the point at the end of the mirror, have identical lengths) are in phase opposition. This proves that the phase associated to a path reflected on a mirror surface must be shifted by π. 4.4. MZ interferometer
The MZ interferometer (figure 6) is useful both as a conceptual tool [27, 40, 41], and as a training ground for performing actual calculations. In its base form, the experiment is used with the two arms having the same optical path length, so no phase shift is introduced between the two photon paths. In this case, one of the detectors (detector 2 in figure 6) has zero probability of detecting the photon, while the other detects the photon with certainty. The main point of interest in the experiment is to compare this result to what happens when either one of the two arms has been blocked: in this case, detectors A and B have the same 1 probability P = 2 of detecting the photon, since no interference happens. Thus one can conclude that results are incompatible with the hypothesis that, in the experiment with the full setup, the photon has gone through only one of the two possible paths. Although it is certainly possible to introduce the same idea using the two slit experiment, we found that reducing the outcome possibilities to only two detectors, rather than a continuous screen, has definite educational advantages. In the simulation we built, checkboxes allow one to compare the two 11
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Figure 6. The Mach–Zehnder interferometer. Parameters to be freely varied include the
width of a dielectric film which varies the optical path length at one of the arms of the interferometers (which can also be removed by setting its width to zero), and the photon wavelength. Checkboxes ‘Remove1’ and ‘Remove2’ reduce the interferometer setup to only one of the two arms, in both cases leading to equal probabilities at detectors.
situations, highlighting the impossibility of assigning a definite path to the photon. Also, the optical path of one of the arms can be varied through the insertion of a dielectric film (reflection at the interface is neglected in this case) of variable width. This offers the possibility for students to actually compute probabilities of detection, in a situation which is slightly different from the usual case of two-slit interference. Finally, we note that the MZ interferometer, when described in terms of two component state vectors and matrices, is formally identical to the Stern–Gerlach apparatus for spin 1 2 particles. Thus, the apparatus can in principle be used as a Rosetta stone [42] connecting the Feynman and Dirac formalisms and allowing one to proceed with the treatment of spin and light polarization. 4.5. The limits of geometrical optics and classical mechanics
We present simulations aimed at showing students the emergence of classical theories in the short wavelength limit. The first one (figure 7) represents the parabolic mirror. It is well known that the Fermat principle applied to the case of a parabolic mirror leads one to prove that all rays coming from a very distant source (ideally at infinite distance) perpendicularly to the directrix of the parabola are reflected towards its focus. Using the sum over paths approach with a mirror surface, whose focal distance is comparable with the wavelength of incident light, leads to a Gaussian-like probability distribution of the photon being detected along the axis of the parabola. This probability distribution has a maximum in the focus, and becomes more and more narrowly concentrated around the peak as the photon wavelength decreases and becomes negligibly short with respect to the relevant length scale (i.e. the focal distance of the parabola). The second simulation (figure 8) represents a double slit experiment with finite length slits and massive particles. In this case, the wavelength corresponds of course to the De 12
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Figure 7. Parabolic mirror. In the case shown, the detector point is near to, but not coincident with, the parabola focus. In the simulation, besides the detector position, the photon wavelength can be varied to demonstrate convergence to a Dirac delta-like distribution for small wavelengths.
Figure 8. Two-slit interference with massive particles: varying the particle mass, one obtains either quantum interference (a) for small values of the mass or classical particle heaps and (b) for high mass values. Some of the small ripple effects in the right figure are actually spurious—a numerical artifact due to the issue that when the wavelength considered becomes too small, interference effects start appearing between the points by which the slits have been divided. This issue is always present in the simulations, and prevents the possibility of reducing the wavelength too much; but of course it is an effect which would in principle disappear in the continuum limit.
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Figure 9. Simulation of the infinite square potential well. (a) If the energy corresponds to one of the potential eigenvalues, the paths differing for an integer number of back and forth trips in the hole are in phase, and the corresponding resultant amplitude has a peak. (b) For all other values, the amplitude in the limit m → ∞ will vanish. Note that for certain choices of the source or detector positions not all the eigenvalues will be visible as peaks; in particular, those corresponding to wave functions possessing a node at the source or detector position will disappear. h
Broglie wavelength λ = p , and is inversely proportional to the particle mass. In the simulation the particle mass is indeed the most important variable parameter; and the result of the two-slit interference is a well-defined interference figure for small mass values, which transforms as the mass increases into the classical expected result of two ‘heaps’ of particles, one behind each of the slits. Of course in this case also the above discussion relating the wavelength of the particle to the relevant length scale applies. 14
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4.6. The infinite square potential well
We use the ‘particle in a box’ model as the paradigmatic example for the treatment of bound systems in the sum over paths approach (figure 9). A quantum object is confined in a square potential well with infinite depth and width AB = 2a . We take one of the points inside the hole, which can be chosen through sliders, to be the source (S), and another one to be the detector (D). Four basic possible routes exist from the source to the detector: the direct S → D path, the S → A → D and S → B → D reflected paths, and the S → A → B → D comprising two reflections. Theoretically, all the paths which can be constructed by adding to any of the above an arbitrary number of full back and forth routes should be considered; in the simulation, we only add up m = 4 derived paths for each one of the four basic types. The phasor corresponding to each path is computed by the usual rules, with each reflection contributing a π phase loss. Through simple calculations one can derive that all paths corresponding to the same ‘family’ are in phase if
4ka − 2π = 2nπ
with n = 0, 1, 2, ...
which leads to the quantization conditions k = for the infinite potential well.
(n + 1) π 2a
and E =
(4) h2(n + 1)2 , 8 m(2a)2
well-known results
5. Testing the sequence with student teachers (ST) 5.1. Organization of the study
We carried out a preliminary study with a group of 12 ST preparing for service as high school level educators. All of them were either currently working as teachers, or had done so in the past, but they lacked a formal qualification. The group was inhomogeneous: most of the members were non-physicists (mathematicians or engineers), and their background in physics varied. Three out of 12 ST had never previously had any formal training in modern physics; five had followed one college course, four had taken more than one. Three of them had previously taught wave optics in high school. We proposed ST a pre-test, in which we asked mostly open questions. In particular, we asked to name differences between quantum and classical physics [27]; to interpret the phenomenon of two-slit interference in terms of photons; to explain the meaning of the uncertainty principle, and of wave particle dualism. The course lasted 8 h divided into sessions of two hours each, plus a final exam. Between the third and final session, home exercises were proposed, which included both formal aspects (i.e. calculation of probabilities of detection) and more conceptual issues. The exercises were to be solved collectively, through an online discussion. The result was a very lively and participated exchange, from which we took many positive indications. In the last session, we proposed a post-test, which contained some of the pre-test questions, some items inspired by existing conceptual repositories on the understanding of quantum mechanics [14, 43, 44] and quantitative exercises similar to those previously assigned. 5.2. Pre-test results
As expected, ST were initially extremely confused regarding the conceptual aspects of quantum theory. In particular, most ST were unable to name differences between classical and quantum physics, beyond the very vague idea that the latter concerns the microscopic world. None of them was able to provide a satisfying definition of the uncertainty principle. 11 ST out of 12 could not write anything meaningful about an interpretation of the two-slit 15
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Figure 10. Post-test item requiring ST to compute detection probabilities.
experiment in terms of photons; only one student provided a partial answer writing that the interference pattern can be interpreted as distribution of probability. Regarding wave–particle dualism, seven ST seemed to believe that it referred exclusively to a property of light, and three did not answer the question. 5.3. General understanding of computational aspects
One of the points that were tested through the final questionnaire was whether students could use the sum over paths approach and phasor method to actually compute probabilities of detection in simple cases. One of the post-test items (reported in figure 10) required to compute the ratio of the probabilities of finding the photon at two detectors W and Z after a ‘double two slit’ setup. The question was similar to a home assigned exercise, which, however, was formulated in open form and had different numerical values. Nine out of twelve ST provided the correct answer to this item, thus demonstrating that students acquired a good mastery of the sum over paths method, at least in the simplest cases. The post-test also contained a very similar question, in which the ratio of probabilities had to be computed by taking into account the acquired information that the photon had not been detected by a detector present on slit D. This item was meant to test both technical aspects and the conceptual understanding of the role of which way measurement in quantum physics. Eleven ST provided the correct answer, thus corroborating the conclusion, which will be discussed later, that students also reached a good level of appropriation of this conceptual issue. 5.4. Interpretation of the single photon interference
In one of the home assigned exercises we asked students to explore the MZ simulation (figure 6) and analyze it in terms of single photons, obtaining in general quite satisfactory explanations. In the post-test we proposed a question very similar to the pre-test item 16
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Figure 11. Post-test open question on single photon interpretation.
concerning single photon interpretation of two-slit interference (see figure 11) and this time the outcome was quite good: most ST provided accurate descriptions and showed a noticeable confidence in talking about photons. Examples of such answers include: ‘The presence of the material introduces a phase shift in the paths of the photon passing through A, so that they are out of phase with the paths passing through B and this changes the probability distribution on the screen. ‘In terms of paths, all possible paths passing through A experience a phase change of π when the film is applied. So, while in the first experiment the vectors were in phase and the probability of detection was maximum, in the second the vectors are in phase opposition and give probability P = 0.’ Indeed, during the online discussion already it had become clear that most ST were acquiring a correct attitude of imagining, and referring to, a single photon taking all possible paths and interfering with itself. 5.5. Information and the role of measurement
Data on the role of measurement in quantum physics were collected through a multiple choice item in the post-test [44], reported in figure 12, and an open question in the final exam. This item also probes the concept of ‘wave particle dualism’, as these issues are inextricably intertwined. The results were quite satisfactory: 11 ST provided the correct answer (b), while one answered (c), signaling a realist position. In the final exam, an open question asked them to discuss the role of measurement in quantum physics, in comparison with classical mechanics. Answers produced by ST were, with almost no exception, extremely satisfactory, displaying conceptual insight and a precise and secure use of language. Practically all students, including those who had never previously been trained in modern physics, stated the problem correctly, and showed good understanding of the central message of the experiment: indistinguishable alternative processes leading to the same results add up their amplitudes and produce interference, while alternatives which can be distinguished, because ‘which way’ information has been recorded, do not. A majority of students elaborated on the idea that, in the Feynman picture, acquisition of information on the system restricts its possible paths to only those 17
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Figure 12. Post-test multiple choice question on the role of measurement.
compatible with the acquired information (which is equivalent to the wave function collapse idea); and convincingly made the point that the very fact of acquiring information about the system, and not some sort of disturbance, should be deemed responsible for the restriction in the possible paths. We may say that students showed a good level of appropriation [45] and mastery of this conceptual issue. 5.6. The uncertainty principle
Both the pre- and post-test contained identical open questions requiring an explanation of the meaning of Heisenberg’s principle. In table 1 the results from the pre- and post-test are reported, grouping answers in classes of similarity. Reading the table, one sees the appearance of five essentially correct analyses in the post-test; also, answers stating that the origin of uncertainty is a perturbation due to measurement have disappeared. However, some critical points remain. At least three ST give almost identical incorrect or partial answers in the preand post-test. One ST still does not provide any answer; and two ST write in the post-test partially correct formulations, which are however somewhat vague and do not show a secure and precise use of language. We conclude that concerning this conceptual point our results were less than optimal, and in future implementations our sequence can be improved in this respect.
6. Conclusions We presented a teaching sequence in introductory quantum physics and results of its first testing with ST. We use the sum over paths method both as a computational tool and as a conceptual point of view, from which we analyze some epistemological themes of quantum physics, aiming at making them clearer and more acceptable for students. Based on the results of this study, our approach appears to be very promising, leading ST to take hold of quantum concepts and acquire an expert-like language in a very short time. Our proposal makes extensive use of interactive simulations, which we designed using the open source software GeoGebra. With respect to the possibility of using more sophisticated (and of 18
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Table 1. Answers to the pre-and post-tests concerning the uncertainty principle. Essentially correct answers in the post-test are in bold.
Pre-test
Post-test
Impossible to determine simultaneously with certainty position and velocity of a quantum object
5
Microscopic systems are disturbed by measurement
2
Impossible to know position and velocity of a particle with the same precision
1
Impossible to determine with precision the position of an electron around the nucleus Introduction of a probabilistic element
1
1
No answer
1
Limitation to the possibility of knowing with arbitrary precision the values of two coupled/incompatible quantities (either reporting the correct formula or mentioning an inverse relationship between the two uncertainties) Impossible to determine simultaneously with certainty position and velocity of a quantum object Limitation to the possibility of measuring with arbitrary precision position and velocity (vague or unclear formulation) Impossible to determine with precision the position of an electron around the nucleus Impossible to know position or velocity of a particle with arbitrary precision No answer
5
2
2
1
1 1
course more visually engaging) platforms, such as Java applets, we found two main advantage. First, GeoGebra makes the mathematical models behind the simulations completely transparent and easily accessible to the user, and avoids producing the impression that complex and exotic algorithms are at work. Second, simulations are built on an interface which many teachers already use in their educational practice. This can encourage them to modify and improve the provided examples, adapting them to their own needs. Results from the first trial suggest some possible future directions as the most natural and promising: • The final part of the sequence can be expanded, including additional simulations and exercises, towards a more extensive treatment of bound systems and scattering from potential barriers (tunnel effect) from the point of view of Feynman paths. Besides its intrinsic value, such an extension would also allow to discuss more in depth the meaning of the uncertainty principle, and favour a smooth connection with the formulation of quantum theory based on the Schrödinger equation. • Similarly, the description of the MZ interferometer can be translated from the language of the sum over paths to that of vectors and matrices. The experiment can then be used to build a bridge from the Feynman approach towards the Dirac notation and the matrix formalism for discrete state systems. • A further effort of simplification, and the production of more extensive teaching materials, will in future be necessary in the perspective of making our sequence into a complete, self-contained approach to the teaching of quantum physics in high school.
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