advertisement

High school students face QM basic concepts Alberto Stefanel Physics Education Research Unit, Physics Department, University of Udine, Italy stefanel@fisica.uniud.it 2. The first study 2004-2005 22 high school pupils of 18-19 years old (21 females and 1 male) The sample: the 18 students always present to the several activities. -Any previous experience in lab -Traditional approach to physics -Any knowledge on optics -9 hours - laboratory and group activities conducted by a researcher -using worksheets for students stimulating personal involvement and challenge by open inquiry (Martongelli et al. 2001; Michelini et al. 2004, Stefanel 2006b) 2 hours – tests (pre test – post test) A Stefanel, High school students face QM STE - MODENA 2009 2 04/05 Day Hours Aims (Activities) 05/06 Day Hours I/1 Pre Test(I) I/1 II / 2-3 Exploration of the polarization of light transmitted through several polaroids II / 2-3 differently crossed: with the naked eye (on the OHP) (T-C); by means of sensors (Malus and transmittivity of polaroids) (G) III / 4 Data elaboration and discussion in computer science laboratory (T and C) IV / 5 Summary of: the experimental results; the phenomenological laws. Discussion about III /4 the meaning of the different factors involved (T). Single photon simulated experiments and probabilistic interpretation (C). V/6 Operative association of a property to photons and its iconographic representation. IV Mutually exclusive properties. Hypothesis exploration: superposition of states – 5-6 statistical mixture (T – C) V 7-8 VI / 7 Phenomenology exploration of birefringent crystals (T-C) VI /9 VII / 8 Impossibility to associate a trajectory to photons (T-C) VII-10 IX 9-10 Superposition principles for the polarization and generalization for any observable VIII (T-C); Linear operators and their physical meaning (T-C) 11-1213 X / 11 Post Test,(I) VIII / 14 Table 1. Synthetic intervention. Activity: A Stefanel, Highprospect school of the in-classSTE - MODENA 2009 (I) individual work; (C) work in groups 3 of 2-3 students; students face (G) QM work in groups of 5-6 students; (T) the whole class. 3. Research methodology 3.1 Monitoring tools Monitoring tools (Aiello 1997) for the in class intervention: a) Pre/Post qualitative evaluation b) Pre/Post test (the same) 15 items - 2 open answer questions, - 13 multiple choice questions and motivation (only one exhaustive answer according to QM - Stefanel 2006a); c) worksheets for students (URDF 2002; Stefanel 2006b); d) notes written by the researcher and the class teacher during and after activity. e) Adiorecording of class discussion f) Only for the 2005/06 intervention (after the posttes): discussion on the web; final examination A Stefanel, High school students face QM STE - MODENA 2009 4 3.2 Data analysis methodology Qualitative analysis Classification of answers and comments to - the test questions, - the worksheets questions, - the stimula offered by in-class activity and by the peer interaction Quantitative Frequency, distribution, chi-test phenomenographycal approach (Marton 1986) A Stefanel, High school students face QM STE - MODENA 2009 5 Profiles Clas – Classic profile. Microscopic systems have an analogous nature to classic macroscopic systems. All their observables always own well defined values. In order to describe their evolution, the concept of trajectory can be used, even if it is necessary to use a statistical approach for lack of information about the initial state of the system under observation. Hid – Hidden variables profile (local). Microscopic systems preserve some properties of the classic macroscopic systems, in particular the trajectory even if it is not knowable/detectable. Their non-classical behaviour is due to uncontrollable disturbances, or rather to some their properties that are not directly accessible/measurable. Quant – Quantum profile. Classic and quantum systems have different nature. It is possible to associate dynamic properties to quantum systems only by means of measurements. These properties are in general imcompatible with those that characterize the state before the measurement itself. Position and trajectory loose their meaning. The process of measure can be described only as a transition between an initial and a final state. A Stefanel, High school students face QM STE - MODENA 2009 6 Q1 - Indicate three topics characterizing QM w.r.t. CM (open answers) Q2 - Measuring a physical observable, which aspect among the following ones characterizes in a peculiar way QM w.r.t. CM? (5 options+ explanation) Q3, Consider the following probabilistic previsions: K) the heads outcome in tossing coin has probability ½ to be realized; J) a photon with vertical polarization has probability ½ to pass through a polaroid at 45º. (3 options, explanations) Q4. The physical meaning of Heisenberg uncertainty relations? (5 options, explanation) Q5. Quantum indeterminism and state concept (2 options, explanation) Q6. In classic mechanics it is always possible to attribute a trajectory to a particle. Which statement can be made as far as a quantum particle is concerned (5 options, explanation) Q7. The physical meaning of the superposition principle? (5 options, explanation) Q8. Physical meaning of Ψ(x) (5 options, explanation) A Stefanel, High school students face QM STE - MODENA 2009 7 Q4. Which of the following statements better outlines the meaning of Heisenberg uncertainty relations? Answer Options In Out a) there are physical quantities pairs of the same system, which can not be simultaneously determined with arbitrary precision 2 6 b) it is not possible to measure with arbitrary precision the value of a physical observable 2 3 c) it is not possible to make the indetermination of an observable measurement arbitrary small 2 3 d) It is never possible, even not in principle, to predict precisely the outcome of a measurement 3 1 e) it is not possible to measure with arbitrary precision the position and the momentum of a particle. 5 1 NR 4 4 A Stefanel, High school students face QM STE - MODENA 2009 8 4. First study profiles summary Crossing of the data collected with the three monitoring tools coherence check between answers profiles in the pre-post test and of the other two monitoring tools (Michelini, Stefanel 2007, Girep Conf.). Classification of categorized answer On the basis of the coherent prevailing of one profile among the others, the total profile for each student was defined making use of an index (Michelini, Stefanel 2006a) build according to the C-index proposed by Müller, Wiesner (2002). A Stefanel, High school students face QM STE - MODENA 2009 9 Among all examined questions, 6 cases have shown a prevailing of the Clas and Quant profiles at the same time. Those have been classified in the Confl Profile (conflictual profile). The profiles distribution after the activities (Post), indicating changes from pre to post profiles: in 14 cases among 18. The conceptions initialy classified as Hid-profile have been mostly confirmed (4 cases) or they have evolved to the Quant profile. Only in two cases they have evolved to conflictual ideas A Stefanel, High school students face QM STE - MODENA 2009 10 X = Clas answers* pc Y = Hid answers * ph Z = Quant aswers * pq p =1 – 1,2 – 1,5 1 – no coherent answers 1,2 – 2 coherent answers 1,5 – 4 coherent answers or more A Stefanel, High school students face QM STE - MODENA 2009 11 Hid Quant Confl A Stefanel, High school students face QM STE - MODENA 2009 12 5. The second study 2005-06 17 High School pupils of 18-19 years old (12 females and 5 male) -Good experience in lab -Good practice in connection of data and models -Knowledge of light polarization phenomenology The sample : the 16 students always present to the several activities. -12 hours - laboratory and group activities conducted by a researcher (using Work-sheets) -2 hours – tests (pre test – post test) A Stefanel, High school students face QM STE - MODENA 2009 13 04/05 Day Hours Aims (Activities) 05/06 Day Hours I/1 Pre Test(I) I/1 II / 2-3 Exploration of the polarization of light transmitted through several polaroids II / 2-3 differently crossed: with the naked eye (on the OHP) (T-C); by means of sensors (Malus and transmittivity of polaroids) (G) III / 4 Data elaboration and discussion in computer science laboratory (T and C) IV / 5 Summary of: the experimental results; the phenomenological laws. Discussion about III /4 the meaning of the different factors involved (T). Single photon simulated experiments and probabilistic interpretation (C). V/6 Operative association of a property to photons and its iconographic representation. IV Mutually exclusive properties. Hypothesis exploration: superposition of states – 5-6 statistical mixture (T – C) V 7-8 VI / 7 Phenomenology exploration of birefringent crystals (T-C) VI /9 VII / 8 Impossibility to associate a trajectory to photons (T-C) VII-10 IX 9-10 Superposition principles for the polarization and generalization for any observable VIII (T-C); Linear operators and their physical meaning (T-C) 11-1213 X / 11 Post Test,(I) VIII / 14 Table 1. Synthetic intervention. Activity: A Stefanel, Highprospect school of the in-classSTE - MODENA 2009 (I) individual work; (C) work in groups 14 of 2-3 students; students face (G) QM work in groups of 5-6 students; (T) the whole class. Q2 - Measuring a physical observable, which aspect among the following ones characterizes in a peculiar way quantum mechanics w.r.t. classic mechanics? Answer options Pre Post A)Under some conditions, dicrete (not continuous) values of the measured observable are obtained 4 0 5 6 C) In general, systems initially prepared in the same state evolve in a different way when subjected to a process of measure 2 5 D) The interaction with the measurement apparatus produces a perturbation on the system 0 4 3 1 4 0 B) Results of measurements are predictable only in probabilistic terms E) The result of a measurement is affected by an ineliminable indetermination NA A Stefanel, High school students face QM STE - MODENA 2009 15 Q3, Consider the following probabilistic previsions: K) the heads outcome in tossing coin has probability ½ to be realized; J) a photon with vertical polarization has probability ½ to pass through a polaroid at 45º. Answer options Pre Post A) In the K case we do not know initial conditions precisely enough, in the J case initial conditions are known, but the phenomenon itself has a probabilist nature. 6 10 B) In both cases we do not know initial conditions precisely enough. 1 0 C) In the K case we do not know initial conditions precisely; in the J case, even knowing the initial conditions, we do not know with enough precision how the interaction photon-polaroid happens. 8 5 NR 0 0 A Stefanel, High school students face QM STE - MODENA 2009 16 Q6. In classic mechanics it is always possible to attribute a trajectory to a particle. Which statement can be made as far as a quantistic particle is concerned (chooce only one option): Answer options Pre Post A) it is possible to attribute a trajectory, but it is not possible to experimentally determine with arbitrary precision all the information needed to determine it with arbitrary precision 1 1 B) it is possible to attribute a trajectory, but all the information needed to determine it is not experimentally accessible 5 0 C) it is possible to attribute a trajectory only when a position measurement is performed 1 1 D) it is impossible to attribute a trajectory to a particle because it is always subjected to casual perturbations 3 2 E) it is not possible, even not in principle, to associate a trajectory to a particle 3 11 2 0 NR A Stefanel, High school students face QM STE - MODENA 2009 17 Q6. In classic mechanics it is always possible to attribute a trajectory to a particle. Which statement can be made as far as a quantistic particle is concerned (chooce only one option): A Stefanel, High school students face QM STE - MODENA 2009 18 Photons-polaroid interaction Hypotesis A: N 45° polarization photons =N/2 vert. pol. ph.+N/2 horiz. pol. ph ? Iconographic representation of photons property: The photon posses either A photon can not posses one properties or one simultaneously two other (just one property) different properties A Stefanel, High school students face QM = ** U The properties are incompatible STE - MODENA 2009 The interaction with a polaroid changes the property of the photon (active role of the polaroid) 19 According to the comparizon of experimental results and the hypotesis A results May we attribute a trajectory to a photon? No: 13 Yes: 2 NA : 1 Antino: NO - “if I hypotize a trajectory I do not obtain the experimental results” Marta: No – “We can not attribute for certain to each photon the followed trajectory” Stefania: NO “ We can not establish a priori the trajectory followed, because the quantum uncertainty” Dafne: NO “for each photon, I can not establish which direction it followed” A Stefanel, High school students face QM STE - MODENA 2009 20 Discussione sulla traiettoria e i sistemi quantistici in ambiente WEB Giacomo: In un sistema quantistico (almeno da quello che credo aver capito) non si può stabilire né a priori né a posteriori la traiettoria di una particella, ma si può solo sapere da dove parte e dove viene ri-(leva o vela?)-to. questo perché se si cerca di attribuire una traiettoria alla particella automaticamente si interferisce con essa e si altera la misura. Sean (replica a Giacomo): Io neppure credo che si possa stabilire la traiettoria di una particella in meccanica quantistica. Penso ciò perché abbiamo capito che lo stato di un corpo è inconoscibile fino a quando lo andiamo a rivelare/rilevare, e facendolo lo alteriamo, e ne modifichiamo le proprietà... tra una sorgente ed un rilevatore la nostra particella (fotone) penso si possa considerare essere ovunque, non potendo sapere quale sia il suo percorso ne come si sposta... A Stefanel, High school students face QM STE - MODENA 2009 21 Marta: ….é lec ito supporre che tra la sorgente e il rilevatore la particella si trovi una situazione di sovrapposizione di tutti gli stati possibili e che essa (la particella) precipiti in uno stato ben definito solo in seguito ad un’interazione con un altro corpo, ad esempio il rilevatore stesso Dafne: In meccanica quantistica non è possibile attribuire una traiettoria alla particella. Di una particella si può solo conoscere lo stato iniziale e quello finale mentre non è possibile stabilire ciò che avviene in mezzo, e più in particolare non si può stabilire con certezza quale percorso abbia seguito la particella. Con la misurazione pratica, sperimentale si può stabilire la posizione di una particella; ma nel momento in cui si effettua tale rilevazione si altera il sistema, e la particella modifica le sue proprietà. Di conseguenza se ne deduce che una particella, tra la sorgente ed il rilevatore, può seguire qualsiasi percorso, quindi non è possibile attribuirle una traiettoria precisa, e nel tentativo di determinarne la posizione (nell'atto sperimentale) vengono modificate le sue proprietà. A causa di questo indeterminismo è logico supporre che la particella, tra sorgente e rilevatore, si trovi in una sovrapposizione degli stati possibili e che sia l'interazione stessa con il rilevatore a determinare lo stato finale con cui la particella viene rilevata. A Stefanel, High school students face QM STE - MODENA 2009 22 CQ index (Michelini, Stefanel 2006a) build according to the C-index proposed by Müller, Wiesner (2002). A Stefanel, High school students face QM STE - MODENA 2009 23 A Stefanel, High school students face QM STE - MODENA 2009 24 7. Conclusions • Personal learning paths and the construction of independent lines of thinking are activated when students are actively involved in hands-on/minds-on activities, in a specific explorative contex, producing a solid understanding of phenomenology and the capability of rigorous argumentation (R4). (see also Stefanel 2001; Michelini et al 2001), The evolution of conceptual schemes of reference (R2, R3) happens not only towards typically quantum ideas, but also, for a non negligible fraction of students, towards conceptions in which the evolution of a system is described on the basis of hidden variables. A Stefanel, High school students face QM STE - MODENA 2009 25 • Conceptual nuclei (R1) about an alternative interpretation of QM, constitute the initial ideas with which some students start studying QM. - In several cases these nuclei can become stronger and lead logically to develop conceptions coherent with alternative approaches to QM or to play the role as a bridge toward a quantum orthodox vision. • Tendency in the evolution of students ideas, from a completly classic conception of phenomena, to a hidden variables ideas, to a quantum vision (R3). This process is not linear and regards part of the students of the current study (R2). Thank you A Stefanel, High school students face QM STE - MODENA 2009 26 Quantum ideas - Michele: Q2 he chooses B (Results of measurements are predictable only in probabilistic terms) in both cases, without motivations; Q3 he chooses A (intrinsic indeterminism in QM measure processes) in both cases motivating in the Post-test: "the probability of transmitting a photon with vertical polarization which passes through a polaroid at 45º is 1/2"; Q4 his answer changes from C to E, without motivations; Q6 his answer changes from A to D motivating his choice with the "uncertainty principle" According to Michele: the central knot for the quantum phenomena understanding is the uncertainty principle which takes the role of cognitive organizer. It is important to be noted that here this principle has been introduced underlying the role of incompatible observables. A Stefanel, High school students face QM STE - MODENA 2009 27 Hidden variables. Federica: Q2 (measurement ) her answer changes from A to B motivating her new choice with the fact that "measurements are predictable only in probabilistic terms because the trajectory of quantum particles is not predictable"; Q3 her answer changes from C to A (intrinsic stocatic nature of measure process) "the phenomenon of the photon transmission has a probabilistic nature, because it is not possible to predict the trajectory (we have experimentally demonstrated this fact: the number of photons passing through a polaroid is not always the same, but it varies constantly, and half of them pass on average)"; Q4 the Pre-answer is E, the Post-answer is C (it is possible to attribute a trajectory only when a position measurement is performed) motivating her choice by saying that "this is due to the fact that the measurement of a physics quantity is predictable only in terms of probability"; Q6 her answer changes from A to B including the comment: "That is why we can describe the physics observable in terms of probability" A Stefanel, High school students face QM STE - MODENA 2009 28 According to Federica: the central knot is the impossibility to predict the trajectory of a quantum system. It is implicit that this trajectory exists, but it is not accessible to us. That is way it is necessary to make probabilist predictions. This is the typical vision of a hidden variables theory, in which the hidden variable is precisely the position. A Stefanel, High school students face QM STE - MODENA 2009 29