Implementation of a proposal to teach quantum mechanics concepts from Feynman s Multiple Paths applied to the light

Transcription

1 Implementation of a proposal to teach quantum mechanics concepts from Feynman s Multiple Paths applied to the light María De Los Ángeles Fanaro 1,2, María Rita Otero 1,2 Mariana Elgue 1 1 Núcleo de Investigación en Enseñanza de las Ciencias y la Tecnología (NIECyT) Facultad de Ciencias Exactas- Universidad Nacional del Centro de la Provincia de Buenos Aires (UNCPBA) 2 Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Argentina Abstract The questions in a didactic sequence for teaching different aspects of the light in a unified framework and in an unconventional way are presented in this work. The objectives to be attained by each question are also described hereafter. Then, the focus is set on the analysis of the double slit experience with light by using concepts of the Feynman method of Multiple Paths in quantum mechanics. The advantages of this formulation for the teaching of basic aspects of quantum mechanics in secondary school education are analyzed. Keywords Reflexion, refraction, Double slit experience, light, Feynman s Multiple Paths Introduction The behaviour of matter and light cannot be well described exclusively by the classical notions of particle and wave, but from the model provided by quantum physics. For the case of electrons and subatomic particles, Feynman s Multiple Paths offer an alternative approach that allow a meaningful study of the concepts of probability, superposition principle and correspondence principle, showing the fundamental and universal character of quantum mechanics laws (Arlego, 2008; Fanaro, Otero & Arlego, 2009; Fanaro, Arlego & Otero, 2014). In previous research a sequence of situations adapting Feynman's approach was constructed and implemented in secondary school for teaching electron behaviour. (Fanaro, Otero & Arlego, 2009, 2012). In the case of the light, an alternative approach to the usual presentation of light quantization in textbooks for teaching is proposed as applying the Feynman model (Arlego, Fanaro & Otero, 2013; Elgue, Fanaro, Arlego & Otero, 2011). The double slit experience is usually used to explain and demonstrate the wave behaviour of light, using the concepts of wavelength, optical path difference, interference and diffraction. This way of presenting and analyzing the experience aims at establishing the wave nature of electromagnetic radiation. Identifying light with a wave or with a particle is physically inappropriate because neither the corpuscular nor the wave model, or a combination of both, offer a complete description of the behaviour of light in all energy and size scales. This does not mean that the classical concepts of wave and particle are completely discarded in the description of quantum phenomena. The behaviour of a quantum objects, such as the electron or the photon, depends on the experimental setup with which they interact. For instance, in the double-slit

2 experiment, light exhibits wave behaviour when interacting with the double slit, though it is detected as a particle as localized on the screen. This duality is a manifestation of the principle of complementarity introduced by Bohr, in the framework of standard interpretation of quantum mechanics and having its root in the principle of uncertainty. In quantum theory, corpuscular and wave-like points of view are complementary rather than contradictory (Fanaro, Arlego & Otero, 2014) In the present work, the proposal is to perform the study of the behaviour of light from a perspective that shows the general character of quantum mechanics and describes the results of the double slit experience in terms of the "Feynman s Multiple Paths" approach. This method is adapted to the students mathematical level by using concepts such as the sum of vectors and trigonometric functions. In that context, the application of the technique is showed in detail by using simulations done with the Geogebra software. In particular, the probability of simple event detections of light and the distribution of light on the screen in the double slit experience is evaluated. Unlike other approaches which are based on the Feynman method (Ogborn, Hanc and Taylor, 2006; Hanc and Tuleja, 2005; Taylor, Vokos, O Meara and Thornber, 1998), the word photon is not mentioned in the present case. Moreover, questions of the type What is light? are avoided, as the focus of the study is set on the predictive character of the laws and not on its epistemological aspects. Method In the first place, the concepts were analysed and structured to teaching as based on Feynman's technique, and then the didactic sequence was elaborated. The material was used in four Physics courses at senior-level high school. The classes were composed of students aged years old. There were two one-hour-long classes per week. The instructional sequence consisted of a set of lessons divided into four stages as described in the following section. In the implementations, the general conversations on each situation were recorded in audio and then transcribed, including all the paper resolutions performed by the students (problem solving, drawings, personal synthesis, written test, etc.). From these registers, the obstacles and the way in which the sequence worked out could be analyzed. The didactic sequence The key aspect in this didactic viewpoint is the starting by presenting questions to the students. The situations elaborated at this point required students to imagine, perform experiments, develop conjectures, and answer questions. The didactic sequence was carried out starting from the experiences of reflection, refraction and the double slit experiment. These experiences were performed in the classroom using mirrors and a graduated circle specially designed to analyze the reflection and refraction, a container and different liquids for the refraction, and a metallized sheet designed with two thin slits. In all of them a common laser light was used. The three experiences were carried out by the students and the experimental results were successfully described by them afterwards. Following this, the double-slit-experience results with very low light intensity showing individual detections were presented through a sequence of real images of the double slit experiment. These images -taken from the web- support the idea of the individual detection events and the light hitting the screen in a granular way. At the beginning, individual events seem to be distributed randomly over the screen. But as time elapses, a pattern of maxima and minima is formed on the screen. In this case the light exhibits a different behaviour from previous experiences: showing a particle-like

3 aspect in the detection of individual events on the screen, and also a wave-like aspect in the concentration fringes formed on the screen, like in the experience performed in the classroom. The concept of discrete detection was highlighted in this part. An explanatory model of all these experiences was proposed: the laws of quantum mechanics for light using Feynman s path integrals technique were adapted to the mathematical level of the students. Graphic representations and basic operations with vectors, which captured the essential aspects of the theory, were used. A simulation made with the GeoGebra (R) software helped students to visualize the results of the Feynman s Multiple Paths technique applied to a simple case of emission and detection of light. The concepts of probability and minimum time were highlighted in this part. Then, these results were applied to the double slit experience to explain the results presented in the previous situation in relation to the granular detection and successive fringes of concentration. Finally, the technique to consider alternative paths was applied to the reflection and refraction of light by using GeoGebra (R) simulations. The students noticed that the results approximated to those observed in previous experiences and therefore reached the conclusion that the laws of quantum mechanics can describe the observed phenomena. As follows, the questions posed to the students are presented hereafter together with the feasible conclusions for each question. Stage1- Four experiences to study light reflection, refraction and the double slit experience Situation 1.1 Studying the reflection The students were presented with the following question: Considering that we want to look at a flat mirror which stands vertically and perpendicularly to the floor, trying to see all our body in it. What should the minimum size of the mirror be and how should it be placed? Then, the students were asked to perform the experience in the classroom as Fig. 1 shows. Fig. 1: Reflection experience performed in class Fig. 2: Refraction experience performed in class The aim of this activity was that students concluded the following: When light falls upon a flat mirror it is fully reflected causing that the angle of reflection is equal to the angle of incidence Situation 1.2- Studying the refraction The students were asked: How do we perceive a pencil or a spoon that is partially submerged in a glass of water or oil? Afterwards, the classroom experience was performed similarly to Fig. 2. The aim was to reach the following conclusion: The direction of the incident light changes when the light goes through a surface. Situation 1.3- Studying the Double Slit Experience (DSE) with laser light

4 A scheme of the Double Slit Experience was presented (Fig. 3 Left) and the students were asked: Can you imagine the light distribution in the screen? Fig. 3. Left: The scheme of the Double Slit Experience presented to the students Right: Picture of the resulting pattern obtained in a typical double slit experiment with a red laser. Then, the group performed the double slit experience in the classroom in a simple way by using a metallized sheet with two thin slits. The source of light was a red laser pointer. On the detection screen, a characteristic pattern of alternated light and dark fringes was formed, as it is showed in Fig. 3 (right). The objective was to establish the following idea when arrangement of fringes is produced; it is called interference pattern. Even though the resulting pattern is interesting in itself, its formation, i.e., the way in which the distribution evolves in time until the stationary pattern is obtained results intriguing. Then the temporal evolution of the pattern is analysed in the next situation referred to the double slit experience (DSE) done with a special detection system. This denomination has been chosen because this experience must be differentiated from the experience made in class, but avoiding any reference to the concepts of photon and laser intensity. So we presented to the students the results of the double slit experience made in a real laboratory (not in classroom) and we also noted that in this case a special laser and a large number of light sensitive detectors, which are able to record in real time the detections, were used. Due is technologically impossible to make this experience in the classroom, the results (available on the internet) provided to the students are an accessible alternative to visualize the formation of the interference pattern. Situation 1.4 Analyzing images of the double slit experience (DSE) done with a special detection system The results of the double slit experience showing individual detections were presented to the students as shown in Fig. 4 1 which contains a sequence of real images of the double slit experiment with very low light intensity: 1 Image obtained from

5 Fig. 4: Snapshots at increasing times (left to right) of detection screen in a double slit experiment with very low intensity of light. Using these snapshots is possible to discuss with the students the individual detection events, and the light hitting the screen in a granular way. First, the individual events seem to be distributed randomly over the screen. However, as time elapses, a pattern of maxima and minima is formed on the screen. Then the question given to the students was: How can these results be described? In consequence, students are able to state that in this case the light exhibits a different behaviour from previous experiences: showing a particle-like aspect in the detection of individual events on the screen, and also a wave-like aspect in the concentration fringes formed on the screen, like in the experience performed in the classroom. The concept of discrete detection can be highlighted in this part of the work. The aim of the activity was to analyze the way in which quantum mechanics can predict the alternated pattern observed in the double slit experiment. From a quantum point of view, the question is: What is the probability of detecting the source-emitted light at a given point of the detection screen? In the following stage this question is presented and discussed. Stage 2- The quantum model of light: the Sum over All Paths (SAP) technique 2 probability calculus for After analyzing the four previous experiences the students were asked the following: How can the results of all the previous experiences be explained and confirmed from a unique model? An explanatory model of all these experiences was proposed: the laws of quantum mechanics for light using Feynman s path integrals technique and adapted to the mathematical level of the students. Graphic representations and basic operations with vectors which capture the essential aspects of the theory were used. A simulation using GeoGebra( R ) software was constructed. It provided the students with an aid to visualize the results of the Sum over All Paths technique applied to a simple case of emission and detection of light, as follows: Situation 2.1- Presentation and analysis of Sum over All Paths (SAP) technique for probability calculus The technique was presented to the students in order to determine the probability of detecting light at F that was emitted at I, as follows: 1) Consider different paths connecting I with F. Figure 5 shows some of them (A, B, C, D, E, F and G) 2 The Feynman s Multiple Paths approach is called Sum over All Paths (SAP) technique with the students

6 Fig. 5: (a) Some paths connecting I with F, and their associated vectors ; (b) Representation of the path length for different "paths" ; (c) Sum of the associated vectors 2) Associate a unitary vector to each path in the plane, whose direction is proportional to the length of the light path (L). The proportionality constant depends on the "type" of light (red, green, infrared, etc.): angle ~ k. L 3) Add the vectors corresponding to all the paths in order to obtain the resultant vector, as shown schematically in the Fig. 5c. 4) The squared length of the resultant vector is proportional to the probability of detecting at point F the light emitted at I. Situation 2.2 Study contributions to the sum of the vectors with the simulation performed with GeoGebra In order to visualize the general procedure and the contributions of different paths, a simulation performed with the Geogebra software was presented: Fig. 6: Screen shot of a simulation performed with Geogebra, showing simultaneously a path and its associated vector (lengh= 1 ; angle= ωd / c). The middle point between I and F can be moved vertically to generate different "paths" The concepts of probability and minimum time were highlighted in this part. Then, these results were applied to the double slit experience as a means to explain the results noticed in the previous situation in connection with granular detection and successive fringes of concentration. Stage 3: Light reflection and refraction from Feynman s model (SAP) The technique to consider alternative paths was applied to the reflection and refraction of light, using simulations made with GeoGebra. The students noticed that the results approximated to

7 those observed in previous experiences. Thus, it is possible to conclude that the laws of quantum mechanics can describe the observed phenomena. The questions given to the students for both reflection and refraction were: How is vector contribution interpreted in terms of probabilities? How does the result obtained by adding the vectors relate with the law of reflection studied in the first experiment? For each experience, the students can be aided by the simulations made with GeoGebra in order to arrive at the following conclusions: The most probable path for the light is the shortest, i.e. that the incidence angle is equal to reflection angle (reflection law); The most probable path for the light is that for which the time is minimum, in the reflection and refraction too. Finally, the students were asked to reconsider the double slit experience presented in the image in Fig.4 and to analyze the following: What is the probability of light detection at x? This question can be responded by means of the procedure established in Stage 2. To this end, the scheme of Fig. 7 is presented to the students. In this graphic, the source (not shown) is expected to be placed on the left side of the double slit screen and, the detection screen is placed at a long distance from the slits. Also, the detection point is at a distance x on the right screen (measured from its centre). Fig. 7: Scheme of the double slit experience. As it can be observed in Fig. 7, two direct paths connecting the source with a given point x on the screen are being considered. One of the paths passes through the lower slit, while the other goes across the upper slit. But there are many alternatives, in principle infinite, of connecting the source with the detection point. One option could be, for instance, a completely arbitrary path connecting the source to one of the slits, and from there to the detection screen. However, according to previous discussion, the shortest path and its environment are the most important alternatives, and ultimately only these paths will be considered. Hence, the vectors that contribute to the probability are those identified with the direct paths, that is to say, one for each slit, and a finite set (n) of vectors associated with neighbouring paths, which contribute essentially with the same angle (in phase). Therefore, for the first slit: uuuuuuur V1( r1 x) n 1; ω. d =, c

8 where ω is proportionality constant 3 and R 1 is the distance from the slit 1 to x (see Fig. 7). In Cartesian coordinates, which are more familiar to students the expression is Analogously for the other slit uuuuuuur ω ω v1( r1 x) = n cos R1 ;sin R1. c c uuuuuuuur ω ω v2( r2 x) = n cos R2 ;sin R2. c c Performing the sum of these two vectors and squaring the result, the following expression for the probability of detection of light at distance x from the centre of the screen results in: ω c where k =. In the experimental setup D>>d and therefore x R 2 R 1 = d is a good D approximation, which allows writing Eq. (1) in terms of x as follows: This expression gives the probability of detecting light as emitted from the source at a distance x from the centre of the screen. It is a result derived purely from quantum mechanics, and indicates that the probability function have maxima and minima, in agreement with the alternation pattern observed in the experiment. In this way, students can compare the theoretical prediction with the results of the double slit experience carried out in the classroom. The expression (2) allows analyzing the dependence of the probability on the colour of the light, which was initially set in red. For instance, when using blue light in the experience, students may notice that the maxima are closer to each other and this is indeed what the probability function predicts, where now the value of k is larger, and therefore the argument of the function is also larger. Another possibility would be to fix the colour and analyze the probability dependence on slits separation. It is clear that working with the parameters of the expression can be a very enriching experience for the conceptualization of students. Stage 4: The Double Slit Experience from the quantum viewpoint The following questions were provided to the students: What characteristics have the P(x) graph? How does probability with x distance to the center of the screen change? Considering the previous Ec (2) D (distance from the slits to the wall) and d (distance between slits) and the 3 ω is the angular frequency of the corresponding classical electromagnetic wave. However, the use of this terminology is avoided with students since the aim is to conceptualize the light from a quantum point of view, bypassing classical Maxwell electromagnetism.

9 value of the proportionality constant k, which corresponds to the red laser which is k= (s -1 ) the students can draw a graphic P(x), that is a positive and periodical function. The experimental results can be only represented in an approximate way, given that the true curve is modulated as shown in Fig. 8. Fig. 8: P(x) graph from according to the experimental requirements established Therefore, the graphic displays peaks that do not have the same height, so the fringes are not equally concentrated, as shown by experimental observation. The discussion with students about the approximation obtained can be very helpful for conceptualization. The aim of this discussion is that students reached the conclusion that It is possible to observe the harmonic variation of the probability as a function of the distance to the centre of the screen, and it explains the maxima and minima of the detections observed. Conclusions This sequence of situations was implemented in four courses of secondary school. The technique adapted from Feynman s method to consider alternative paths was applied to the reflection and refraction of light, and to the double slit experience using simulations made with GeoGebra(R). The purpose is that the students can notice that the results derived from Feynman method approximated to those observed in the analyzed and performed experiences. Thus, they can reach to the conclusion that the laws of quantum mechanics are general and allow the description of the observed phenomena successfully. It is possible to anticipate that the sequence was viable because the students were able to establish the principles and the basic ideas of quantum mechanics. The analysis of the conceptualization of each situation is in progress, and it will give true knowledge of the viability of the sequence. References Arlego, M. (2008).Revista Electrónica de Investigación en Educación en Ciencias 3, 59 Arlego, M. Fanaro M. and Otero, M.R. (2013) Poceedings of World Conference on Physics Education, Istambul, 2012, edited by Mehmet Fath Taşar (Pegem Academi, Istambul, 2013), p Elgue, M. Fanaro M., Arlego M. and Otero, M.R. (2011) Enseñar el comportamiento de la luz en la escuela secundaria desde una visión actual utilizando el método de caminos múltiples de Feynman. Actas del I Congreso Internacional en Enseñanza de las Ciencias y la Matemática - II Encuentro Nacional en Enseñanza de la Matemática, Tandil, p Fanaro, M. Arlego, M y Otero M. R. (2014) The double slit experience with light from the point of view of Feynman s sum of multiple paths Revista Brasileira de Ensino de Física 3 (2)

10 Fanaro, M. Otero M. and Arlego, A. (2012) A proposal to teach the light at secondary school from the Feynman method Problems of Education in the 21 st Century 472, 27 Fanaro, M. Otero M.R and Arlego, M. (2009). Teaching the foundations of quantum mechanics in secondary school: a proposed conceptual structure Investigações em Ensino de Ciências 14, 37 Fanaro, M. Otero M.R and Arlego, M. (2012) Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman s Path Integrals Method. The Physics Teacher 50, 156 Hanc J. and Tuleja, S. (2005) The Feynman Quantum Mechanics with the help of Java applets and physlets in Slovakia. Proceedings 10 th Workshop on Multimedia in Physics (Freie Universität Berlin, 2005), p. 10. Ogborn, J. Hanc, J. and Taylor, E. (2006) Action on Stage: Historical Introduction Proceedings The Girep conference 2006, Modeling in Physics and Physics Education (AMSTEL Institute, Amsterdam, 2006), p Taylor, E.F. Vokos, S. O'Meara J.M. and Thornber, N.S. (1998). Teaching Feynman s sumover-paths quantum theory Computers in Physics 12, 190 Affiliation and address information María de los Ángeles Fanaro Núcleo de Investigación en Enseñanza de las Ciencias y la Tecnología (NIECyT) Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Argentina Universidad Nacional del Centro de la Provincia de Buenos Aires Facultad de Ciencias Exactas Pinto 399. Tandil, Provincia de Buenos Aires Argentina mfanaro@exa.unicen.edu.ar María Rita Otero Núcleo de Investigación en Enseñanza de las Ciencias y la Tecnología (NIECyT) Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Argentina Universidad Nacional del Centro de la Provincia de Buenos Aires Facultad de Ciencias Exactas Pinto 399. Tandil, Provincia de Buenos Aires Argentina rotero@exa.unicen.edu.ar Mariana Elgue Núcleo de Investigación en Enseñanza de las Ciencias y la Tecnología (NIECyT) Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Argentina Universidad Nacional del Centro de la Provincia de Buenos Aires Facultad de Ciencias Exactas Pinto 399. Tandil, Provincia de Buenos Aires Argentina marianaelgue@hotmail.com