# Light and matter diffraction from the unified viewpoint of Feynman’s Sum of all Paths

### Abstract

In this work, we present a pedagogical strategy to describe the diffraction phenomenon based on a didactic adaptation of the Feynman’s path integrals method, which uses only high school mathematics. The advantage of our approach is that it allows to describe the diffraction in a fully quantum context, where superposition and probabilistic aspects emerge naturally. Our method is based on a time-independent formulation, which allows modelling the phenomenon in geometric terms and trajectories in real space, which is an advantage from the didactic point of view. A distinctive aspect of our work is the description of the series of transformations and didactic transpositions of the fundamental equations that give rise to a common quantum framework for light and matter. This is something that is usually masked by the common use, and that to our knowledge has not been emphasized enough in a unified way. Finally, the role of the superposition of nonclassical paths and their didactic potential are briefly mentioned.### References

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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 1 37-64

Fanaro M.; Otero M. R. and Arlego M. (2012A)proposal to teach the light at secondary school from the Feynman method. Problems of Education in the 21st Century 47 47 27-39

Feynman R. (1985) QED The strange theory of light and matter (London: Penguin Books) Princeton

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Gitin A. (2013) Huygens–Feynman–Fresnel principle as the basis of applied optics Applied Optics 52 7419-34

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Halliday, D. Resnick R. and Walker J. (2011) Fundamentals of Physics (New Jersey: John Wiley & Sons)

Hanc J.; Tuleja S. and Hancova M. (2003) Simple derivation of Newtonian mechanics from the principle of least action. American Journal of Physics 71 4, 386-391

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Malgieri M.; Onorato P. and De Ambrosis A. (2017)Test on the effectiveness of the sum over paths approach in favoring the construction of an integrated knowledge of quantum physics in high school Phys. Rev. Phys. Educ. Res. 13 019901

Malgieri M.; Onorato P. and De Ambrosis A. (2015)Insegnare la fisica quantistica a scuola: un percorso basato sul metodo dei cammini di Feynman. Giornale di Fisica 56 1 45

Maurines L. (2010) Geometrical Reasoning in Wave Situations: The case of light diffraction and coherent illumination optical imaging International Journal of Science Education 32 1895–1926

Michelini M.; Stefanel A. Interpreting Diffraction Using the Quantum Model Proceedings GIREP Conference 2006: Modeling in Physics and Physics Education (Amsterdam: University of Amsterdam, 2008) p. 811-815.

Ogborn J. and Taylor F. “Quantum physics explains Newton's laws of motion” Physics Education 40 26-34 (2005)

Peskin M. and Schroeder V. (2015) An Introduction To Quantum Field Theory (Boulder: Westview Press)

Ramil A. ; López A. J. and Vincitorio F. (2007) Improvements in the analysis of diffraction phenomena by means of digital images. Am. J. Phys. 75, 999

Sawant R.; Samuel J.; Sinha A.; Sinha S. and Sinha U. (2014) Nonclassical Paths in Quantum Interference Experiments Phys. Rev. Lett. 113 120406

Shankar, R. (1980) Quantum Mechanics (New York: Plenum Press)

Styer D. (2000) The Strange World of Quantum Mechanics. (New York: Cambridge University Press)

Taylor E. F.; Vokos S.; O’Meara J.M. and Thornber N. (1998) Teaching Feynman’s Sum Over Paths Quantum Theory Computers in Physics 12 190-99

Temes J. B. (2003) Teaching Electromagnetic Waves to Electrical Engineering Students: An Abridged Approach. IEEE Transactions on Education 46 283-88

Tonomura A.; Endo J.; Matsuda T.; Kawasaki T. and Ezawa H. (1989) Demonstration of single-electron build-up of an interference pattern. Am. J. Phys. 57 117–120

Wosilait K.; Heron P.; Shaffer P. and McDermott L. (1999) Adressing students’difficulties in applying a wave model to the interference and diffraction of light. Am. J. Phys. 67 S5

Wua X.Y. ; Zhang, B.J.; Yang, J.H.; Chia L.X.; Liua X.J.; Wua Y.H.; Wanga Q.C.; Wanga, Y.; Lib J.W.; and Guoc Y.Q. (2010) Quantum theory of light diffraction Journal of Modern Optics 57, 2020 2082–2091

Beau M. (2012) Feynman path integral approach to electron diffraction for one and two slits: analytical results Eur. J. Phys. 33 1023

Born M. and Wolf E. (1999) Principles of Optics (Cambridge: Cambridge University Press)

Chevallard Y. (1985) La transposition didactique. Du savoir savant au savoir enseigne. (Grenoble: La Pense´e Sauvage Edition)

Colin P. and Viennot L. (2001) Using two models in optics: Students’ difficulties and suggestions for teaching Am. J. Phys. 69 S36

Dobson K.; Lawrence I. and Britton P. (2006) The A to B of quantum physics Physics Education 35 6

Fanaro M. ;Otero M. R. and Arlego M. (2012B) Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman’s Path Integrals Method. The Physics Teacher 50 156-158

Fanaro M. and Otero M. R. (2008) Basics Quantum Mechanics teaching in Secondary School: One Conceptual Structure based on Paths Integrals Method Lat. Am. J. Phys. Educ. 2 2 103-12

Fanaro M.; Arlego M. and Otero M. R. (2014)The double slit experience with light from the point of view of Feynman's sum of multiple paths. Rev. Bras. Ensino Fís. 36 2 1-7

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 1 37-64

Fanaro M.; Otero M. R. and Arlego M. (2012A)proposal to teach the light at secondary school from the Feynman method. Problems of Education in the 21st Century 47 47 27-39

Feynman R. (1985) QED The strange theory of light and matter (London: Penguin Books) Princeton

Feynman R. and Hibbs A. R. (1965) Quantum Mechanics and Path Integrals (First Edition)

Gitin A. (2013) Huygens–Feynman–Fresnel principle as the basis of applied optics Applied Optics 52 7419-34

Goldstein H.; Poole C.; Safko J. (2013) Classical Mechanics (Essex: Person)

Halliday, D. Resnick R. and Walker J. (2011) Fundamentals of Physics (New Jersey: John Wiley & Sons)

Hanc J.; Tuleja S. and Hancova M. (2003) Simple derivation of Newtonian mechanics from the principle of least action. American Journal of Physics 71 4, 386-391

Malgieri M.; Onorato P. and De Ambrosis A. (2014) Teaching quantum physics by the sum over paths approach and GeoGebra simulations Eur. J. Phys. 35 055024

Malgieri M.; Onorato P. and De Ambrosis A. (2017)Test on the effectiveness of the sum over paths approach in favoring the construction of an integrated knowledge of quantum physics in high school Phys. Rev. Phys. Educ. Res. 13 019901

Malgieri M.; Onorato P. and De Ambrosis A. (2015)Insegnare la fisica quantistica a scuola: un percorso basato sul metodo dei cammini di Feynman. Giornale di Fisica 56 1 45

Maurines L. (2010) Geometrical Reasoning in Wave Situations: The case of light diffraction and coherent illumination optical imaging International Journal of Science Education 32 1895–1926

Michelini M.; Stefanel A. Interpreting Diffraction Using the Quantum Model Proceedings GIREP Conference 2006: Modeling in Physics and Physics Education (Amsterdam: University of Amsterdam, 2008) p. 811-815.

Ogborn J. and Taylor F. “Quantum physics explains Newton's laws of motion” Physics Education 40 26-34 (2005)

Peskin M. and Schroeder V. (2015) An Introduction To Quantum Field Theory (Boulder: Westview Press)

Ramil A. ; López A. J. and Vincitorio F. (2007) Improvements in the analysis of diffraction phenomena by means of digital images. Am. J. Phys. 75, 999

Sawant R.; Samuel J.; Sinha A.; Sinha S. and Sinha U. (2014) Nonclassical Paths in Quantum Interference Experiments Phys. Rev. Lett. 113 120406

Shankar, R. (1980) Quantum Mechanics (New York: Plenum Press)

Styer D. (2000) The Strange World of Quantum Mechanics. (New York: Cambridge University Press)

Taylor E. F.; Vokos S.; O’Meara J.M. and Thornber N. (1998) Teaching Feynman’s Sum Over Paths Quantum Theory Computers in Physics 12 190-99

Temes J. B. (2003) Teaching Electromagnetic Waves to Electrical Engineering Students: An Abridged Approach. IEEE Transactions on Education 46 283-88

Tonomura A.; Endo J.; Matsuda T.; Kawasaki T. and Ezawa H. (1989) Demonstration of single-electron build-up of an interference pattern. Am. J. Phys. 57 117–120

Wosilait K.; Heron P.; Shaffer P. and McDermott L. (1999) Adressing students’difficulties in applying a wave model to the interference and diffraction of light. Am. J. Phys. 67 S5

Wua X.Y. ; Zhang, B.J.; Yang, J.H.; Chia L.X.; Liua X.J.; Wua Y.H.; Wanga Q.C.; Wanga, Y.; Lib J.W.; and Guoc Y.Q. (2010) Quantum theory of light diffraction Journal of Modern Optics 57, 2020 2082–2091

Published

2018-05-24

How to Cite

ARELGO, Marcelo; FANARO, Maria de los Angeles.
Light and matter diffraction from the unified viewpoint of Feynman’s Sum of all Paths.

**European Journal of Physics Education**, [S.l.], v. 8, n. 2, may 2018. ISSN 1309-7202. Available at: <http://eu-journal.org/index.php/EJPE/article/view/164>. Date accessed: 19 june 2018. doi: https://doi.org/10.20308/ejpe.v8i2.164.
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Section

Classroom Physics

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