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Playing with Quantum Toys: Julian Schwinger’s Measurement Algebra and the Material Culture of Quantum Mechanics Pedagogy at Harvard in the 1960s

Physics in Perspective volume 20, pages 8–42 (2018)Cite this article

Abstract

In the early 1960s, a PhD student in physics, Costas Papaliolios, designed a simple—and playful—system of Polaroid polarizer filters with a specific goal in mind: explaining the core principles behind Julian Schwinger’s quantum mechanical measurement algebra, developed at Harvard in the late 1940s and based on the Stern-Gerlach experiment confirming the quantization of electron spin. Papaliolios dubbed his invention “quantum toys.” This article looks at the origins and function of this amusing pedagogical device, which landed half a century later in the Collection of Historical Scientific Instruments at Harvard University. Rendering the abstract tangible was one of Papaliolios’s demonstration tactics in reforming basic teaching of quantum mechanics. This article contends that Papaliolios’s motivation in creating the quantum toys came from a renowned endeavor aimed, inter alia, at reforming high-school physics training in the United States: Harvard Project Physics. The pedagogical study of these quantum toys, finally, compels us to revisit the central role playful discovery performs in pedagogy, at all levels of training and in all fields of knowledge.

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Fig. 1

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 2

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 3

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 4

Credit: Wikimedia Commons

Fig. 5

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 6

Credit: Collection of Historical Scientific Instruments, Harvard University.

Fig. 7

Credit: Illustration by Maureen Ton, CHSI

Fig. 8

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 9

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 10

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 11

Credit: Collection of Historical Scientific Instruments, Harvard University

Fig. 12

Credit: Collection of Historical Scientific Instruments, Harvard University

Notes

  1. To know on which side to inscribe the signs, make sure that the following operation is always respected: ∣↑〉〈↑∣∣↓〉〈↓∣ (no light goes through); ∣↑〉〈↑∣∣↑〉〈↑∣ (light goes through). This is true for all quantum toys described below.

References

  1. Sherry Turkle, ed., Falling for Science: Objects in Mind (Cambridge, MA: MIT Press, 2008), 5, 37, 279. See also her other books on the topic of affect and material culture: Evocative Objects: Things We Think With (Cambridge, MA: MIT Press, 2007) and The Inner History of Devices (Cambridge, MA: MIT Press, 2008). On how regular people deal with stuff, from an anthropological point of view, see Daniel Miller, The Comfort of Things (Cambridge: Polity Press, 2008). On a theoretical approach to “stuff” see Miller, Stuff (Cambridge: Polity Press, 2010).

  2. Johan Huizinga, Homo ludens: A Study of the Play-Element in Culture (London: Routledge & Kegan Paul, 1949), 4–9.

  3. Gerard L’E. Turner, “Presidential Address: Scientific Toys,” British Journal for the History of Science 20 (1987), 377–98; Paolo Brenni, “The Evolution of Teaching Instruments and Their Use Between 1800 and 1930,” Science & Education 21 (2012), 191–226; Peter Heering and Roland Wittje, eds., Learning by Doing: Experiments and Instruments in the History of Teaching (Stuttgart: Franz Steiner Verlag, 2011); Melanie Keene, Science in Wonderland: The Scientific Fairy Tales of Victorian Britain (Oxford: Oxford University Press, 2015). Simon Schaffer, “A Science Whose Business is Bursting: Soap Bubbles as Commodities in Classical Physics,” in Things that Talk: Objects Lessons from Art and Science, ed. Lorraine Daston (New York: Zone Books, 2004), 147–92; Sofie Lachapelle, Conjuring Science. A History of Scientific Entertainment and Stage Magic in Modern France (New York: Palgrave Macmillan, 2015). A fascinating case study is Anke Te Heesen, The World in a Box: The Story of an Eighteenth-Century Picture Encyclopedia (Chicago: University of Chicago Press, 2002).

  4. Olival Freire Junior, The Quantum Dissidents: Rebuilding the Foundations of Quantum Mechanics (1950–1990) (New York: Springer, 2015), 262–63. See, for instance, Costas Papaliolios, “Experimental Test of a Hidden-Variable Quantum Theory,” Physical Review Letters 18 (1967), 622–25. See also the oral history of John Clauser, a theoretical physicist who worked on Bell’s theorem, who mentions Papaliolios’s early contribution to the field and describes him as “a charming fellow.” John Clauser, interview with Joan Bromberg, May 20, 21, and 23, 2002, Niels Bohr Library and Archives, American Institute of Physics, accessed February 6, 2016, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/25096.

  5. “Memorial Minute, Costas Papaliolios,” Harvard Gazette, May 13, 2004.

  6. This information was provided by Eric Heller in a private correspondence, email to author June 16, 2015. He believes, as do some other people he talked to, that Schwinger would never have used these in a classroom. Other undergraduate and graduate students of Schwinger’s, both at Harvard and UCLA, never saw these toys in the hands of the theoretical physicist. Kimball A. Milton, correspondence with the author, June 13, 2015; Walter Wilcox, correspondence with the author, June 12, 2015.

  7. David Kaiser, How Hippies Saved Physics: Science, Counterculture, and the Quantum Revival (New York: W. W. Norton & Company, 2011), ch. 1; American Physics and the Cold War Bubble (Chicago: University of Chicago Press, forthcoming), ch. 4, “Training Quantum Mechanics.” I would like to thank the author for sharing this chapter with me.

  8. Among recent examples of authors who really use objects as primary sources are Katie Taylor, “Mogg’s Celestial Sphere (1813): The Construction of Polite Society,” Studies in History and Philosophy of Science 40 (2009), 360–71; Melanie Keene, “‘Every Boy & Girl a Scientist’: Instruments for Children in Interwar Britain,” Isis 98 (2007), 266–89; Katharine Anderson, Mélanie Frappier, Elizabeth Neswald, and Henry Trim, “Reading Instruments: Objects, Texts and Museums,” Science & Education 22 (2013), 1167–89; Sara Schechner, “The Art of Making Leyden Jars and Batteries According to Benjamin Franklin,” eRittenhouse 26 (2015), http://www.erittenhouse.org/articles/vol-26-contents-and-authors/making-leyden-jars-and-batteries/.

  9. Michael Mahoney, “Reading a Machine,” 1999, accessed February 6, 2016, https://www.princeton.edu/~hos/h398/readmach/modeltfr.html.

  10. Laurel Ulrich, Ivan Gaskell, Sara J. Schechner, Sarah Anne Carter, and Samantha van Gerbig, Tangible Things: Making History through Objects (Oxford: Oxford University Press, 2015), 2. A review of the exhibition is found on the CHSI website: http://chsi.harvard.edu/chsi-tangible_things.html.

  11. Ken Alder, ed., “Focus: Thick Things,” Isis 98 (2007), 80–142. He first introduced this notion in his Engineering the Revolution: Arms & Enlightenment in France, 1763–1815 (Chicago: University of Chicago Press, 1997). See also: Davis Baird, Thing Knowledge: A Philosophy of Scientific Instruments (Berkeley: University of California Press, 2004). Liba Taub, ed., “Focus: The History of Scientific Instruments,” Isis 102 (2011), 689–729. Another useful concept to think with is Bill Brown, “Thing Theory,” Critical Inquiry 28 (2001), 1–22.

  12. Harvard University Archives, Costas Papaliolios Papers (unprocessed accessions 14811), Box 1, Folder Phys 251a Part I, II, III, IV., Part I, pages 1 and 3 (emphasis original). [Hereafter cited as CPP]. I would like to thank Dimitri Papaliolios for granting me permission to look at his father’s papers. Harvard College, Courses of Instruction, 1954–1966, Box 7. HUC 8500.16, Harvard University Archives.

  13. Ibid., 3 (emphasis original).

  14. Costas Papaliolios, letter to Dr. John Fowler, January 5, 1968, CPP, Box 21, Folder Quantum Toys.

  15. Quoted in Jagdish Mehra and Kimball A. Milton, Climbing the Mountain: The Scientific Biography of Julian Schwinger (Oxford: Oxford University Press, 2000), 156.

  16. Cécile Morette was responsible for establishing the school in 1951. See Toni Feder, “Path Integrals, Les Houches, and Other Adventures of Cécile DeWitt-Morette,” Physics Today 61, no. 8 (2008), 28.

  17. In a paper presented at the international Hermann Weyl conference (1985) in Kiel, Schwinger argued that his formalism was methodologically similar to an insight Weyl developed in a 1927 paper, that is, quantum theory could be logically deduced from an irreducible Abelian group of unitary ray representations. Schwinger starts with the Stern-Gerlach experiment and through his measurement algebra deduces Heisenberg’s commutation relation and Schrödinger’s equation. Schwinger, “Hermann Weyl and Quantum Mechanics,” in Exact Sciences and their Philosophical Foundations, ed. Wolfgang Deppert et al. (New York: Peter Lang, 1988), 107–29.

  18. Julian Schwinger, “Quantum Dynamics, Part 1,” typewritten manuscript (Université de Grenoble: Les Houches, 1955), 1.

  19. One example would be David Bohm’s popular Quantum Theory (New York: Prentice Hall, 1951), which does exactly that. On the changing nature of quantum mechanics’ textbook publishing see Kaiser, Hippies (ref. 7).

  20. Julian Schwinger, “Lecture Notes in Physics 251a Quantum Mechanics,” notes taken by Russell K. Hobbie (Harvard University, 1956–57).

  21. Julian Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, ed. Berthold-Georg Englert (Berlin: Springer, 2001), 29. An earlier exposition of Schwinger’s formulation of quantum mechanics—closer in spirit to the present article—is found in Julian Schwinger, Quantum Kinematics and Dynamics (New York: W. A. Benjamin, 1970).

  22. Jeremy Bernstein, “The Stern-Gerlach Experiment—Was Sind und was Sollen” (2010), arXiv, accessed January 29, 2016, http://arxiv.org/abs/1007.2435. For a complete historical treatment of the experiment, see Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, vol. 1, pt. 2 (Berlin: Springer-Verlag, 1982), 422–45. For a modern analysis of the experiment, see Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum Mechanics, vol. 1 (New York: John Wiley & Sons, 1977), 387–405.

  23. Paul Adrien Maurice Dirac, The Principles of Quantum Mechanics, 4th ed. (Oxford: The Clarendon Press, 1958).

  24. This measurement symbol was later changed in the mid-1980s to |a′a′|. See Schwinger, Quantum Mechanics (ref. 21), where he uses the latter symbolic notation. It is not the purpose of this article to explain the various iterations Schwinger’s formalism underwent from Les Houches in 1955 to the latter 1980s, when he was teaching at UCLA. See Mehra and Milton, Climbing the Mountain (ref. 15), 345–52 for a summary of Schwinger’s measurement algebra.

  25. The text describing the three principles was taken almost verbatim from Schwinger, Quantum Dynamics (ref. 18), 2. I generalized the measurement symbols, that is, M(a′) = M(a′,a′).

  26. Quoted from Mehra and Milton, Climbing the Mountain (ref. 15), 343.

  27. Harvard College, Courses of Instruction, 1944–1953, Box 6. HUC 8500.16, Harvard University Archives. The course was open to students who had taken Physics 251a and Physics 251b, Quantum Mechanics.

  28. Jeremy Bernstein, “The Charms of a Physicists,” The New York Review of Books, April 11, 1991, accessed April 25, 2015, http://www.nybooks.com/articles/archives/1991/apr/11/the-charms-of-a-physicist/. Bernstein reviews a book written by Victor Weisskopf. Bernstein completed his education in quantum mechanics using a more conventional textbook: Bohm’s Quantum Theory. See Bernstein, “The Stern-Gerlach Experiment” (ref. 22), 6. Bryce DeWitt, a Schwinger PhD student (1949), remembers that he “was merely being guided by a kind of intuition, led by the formalism itself. That is, the formalism would take a life of its own and just lead you even though it might not be completely legitimate to do so.” Quoted in Silvan Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga (Princeton: Princeton University Press, 1994), 368.

  29. Paul C. Martin, “Julian Schwinger—Personal Recollections,” in Julian Schwinger: The Physicist, the Teacher, and the Man, ed. Y. Jack Ng (Singapore: World Scientific, 1996), 85. Murray Gell-Mann did not agree with this assertion, saying he “never got anything out of that course” he took with Schwinger. Murray Gell-Mann, interview with Sara Lippincott, July 17 and 18, 1997, Archives of the California Institute of Technology, Pasadena, California, accessed April 25, 2014, http://oralhistories.library.caltech.edu/228/1/Gell-Mann_OHO.pdf. You can also hear about Gell-Mann’s experience at Harvard at Web of Stories, accessed April 25, 2015, http://www.webofstories.com/play/murray.gell-mann/26.

  30. Quoted in Walter Kohn, “Tribute to Julian Schwinger,” in Ng, ed., Julian Schwinger (ref. 29), 62.

  31. Paul C. Martin and Sheldon L. Glashow, “Biographical Memoir of Julian Schwinger, 1918–1994,” Biographical Memoirs of the National Academy of Sciences (Washington, DC: National Academy of Sciences, 2008), 10. 

  32. Quoted in David Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics (Chicago: University of Chicago Press, 2005), 46.

  33. He is most likely the same Walter H. Balcke from Winchester who was granted a patent on March 16, 1915, for a “Dancing toy figure” designed to work with a phonograph. Patent number 1,131,895. It can be explored in Google Patents.

  34. Garrett Birkhoff, letter to Walter Balcke, March 3, 1954, Collection of Historical Scientific Instruments, Folder W. H. Balcke, Lib 4927.

  35. J. L. Walsh, letter to Walter to Balcke, November 15, 1957, Collection of Historical Scientific Instruments, Folder W. H. Balcke, Lib 4927. It is indeed inscribed on the instrument: “Kindergarten / EQUILATERAL DRILL.” CHSI inventory number 1997-1-1608.

  36. J. L. Walsh, letter to Walter to Balcke, December 5, 1961, Collection of Historical Scientific Instruments, Folder W. H. Balcke, Lib 4927.

  37. Dirac, Principles of Quantum Mechanics (ref. 23), sec. 1, 4–7.

  38. Ibid., 6.

  39. The larger cubes were apparently designed for lecture demonstrations—most likely with the help of an overhead projector. Papaliolios mentions that most of them were dispersed between 1962 and 1968, hence the incomplete set at the Collection of Historical Scientific Instruments. Costas Papaliolios, letter to John Fowler, January 5, 1968, CPP, Box 21, Folder Quantum Toys.

  40. Costas Papaliolios, “Experimental Test of a Hidden-Variable Quantum Theory,” Physical Review Letters 18, no. 15 (1967), 622–25. The Polaroid sheet explaining “transmission curves” is found in CPP, Box 22, Folder Polarized Light.

  41. Papaliolios, letter to Fowler (ref. 39).

  42. Papaliolios, in a manuscript titled “Notes on Measurement Symbols ’98,” which accompanies several sets of small quantum toys, wrote down the algebra required to exhibit all the relations between the base states. For example, |+〉 = \( \frac{1}{\sqrt 2 } \)|1〉 − |0. 〉 and | −〉 = \( \frac{1}{\sqrt 2 } \)|1〉 + i|. 0〉. I would like to thank Dimitri Papaliolios for sharing this information with me, which is not included in the Harvard University Archives.

  43. “Comments regarding operator cubes,” n.d., ca. 1968, CPP, Box 21, Folder Quantum Toys.

  44. Papaliolios, letter to Fowler (ref. 39). The multiplication principles, generalized in quantum mechanics as σ i σ j = i σ k, is verified in all base states using the Pauli cubes.

  45. Papaliolios, letter to Fowler (ref. 39).

  46. Richard Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, definitive ed., 3 vols. (San Francisco: Pearson / Addison Wesley, 2006 [1963–1965]), vol. 3, sec. 5-1. I used the online version, accessed January 4, 2017, http://www.feynmanlectures.caltech.edu/, which has the same organization.

  47. Feynman, Lectures on Physics (ref. 46), sec. 11-5, n. 2. The best historical and philosophical analysis of the K-meson “strangeness” is Allan Franklin, The Neglect of Experiment (Cambridge, UK: Cambridge University Press, 1986), esp. 73–107.

  48. Untitled document, n.d., ca. 1968, CPP, Box 21, Folder Quantum Toys. Schwinger’s papers inserted in the folder are “The Algebra of Microscopic Measurement” (1959), “The Geometry of Quantum States” (1959), and “Unitary Transformations and the Action Principle” (1960).

  49. Papaliolios, “Notes on Measurement Symbols” (ref. 42). Emphasis original.

  50. John W. Robson, letter to Costas Papaliolios, July 25, 1968, CPP, Box 21, Folder Quantum Toys.

  51. “Bra-Ket Symbols for Polarizers, Retarders, etc.,” W. A. Shurcliff, February 14, 1967, CPP, Box 22, Folder Polarized Light.

  52. As a comparison, see Robert W. Spekkens, “Evidence for the Epistemic View of Quantum States: A Toy Theory,” Physical Review A 75 (2007), 032110. That is a thirty-page, double-column paper, which was commented upon and extended by several other scholars in the field. One of the concluding assertions is striking: “The toy theory contains almost no physics.” Something similar is lacking for Papaliolios’s own toys.

  53. Gerald Holton does not recall having seen these quantum toys. Conversation with the author, March 10, 2016.

  54. The best analysis of the Harvard Project Physics is David Meshoulam, “Teaching Physics as One of the Humanities: The History of (Harvard) Project Physics, 1961–1970” (PhD dissertation, University of Wisconsin–Madison, 2014).

  55. F. James Rutherford, Gerald Holton, and Fletcher G. Watson, The Project Physics Course: Unit 1—Concepts of Motion (New York: Holt, Rinehart and Winston, 1970), n.p. More on the philosophy of this program is found in Gerald Holton, “On the Educational Philosophy of the Project Physics Course,” in The Scientific Imagination (Cambridge, MA: Harvard University Press, 1998), 284–98.

  56. Gerald Holton, “The Project Physics Course, Then and Now,” Science & Education 12 (2003), 779–86, on 781. Other funding agencies were the US Office of Education, the Alfred P. Sloan Foundation, and the Ford Foundation. On the difficulty (and the politics) of getting funding from the NSF and other agencies, Meshoulam, The History of (Harvard) Project Physics (ref. 54), ch. 3. Primary sources on this topic are found in Gerald James Holton personal archive, 1919–2015 and undated. HUM 132, Harvard University Archives, “Endless Vexations with NSF, OE [National Science Foundation, Office of Education] with PPC [Project Physics Course], 1963–1970, Box 37, Folders 1 & 2. Hereafter cited as HPA.

  57. F. James Rutherford, Gerald Holton, and Fletcher G. Watson, About the Project Physics Course: An Introductory to the Teacher Resource Book (New York: Holt, Rinehart and Winston, 1971), 1. On the “impromptu remarks,” Meshoulam, The History of (Harvard) Project Physics (ref. 54), 94. A similar claim was made by British physicists in the nineteenth-century regarding the role of precision measurement in training a new generation of physicists, engineers, physics teachers, and liberal arts students. Graeme Gooday, “Precision Measurement and the Genesis of Physics Teaching Laboratories in Victorian Britain,” The British Journal for the History of Science 23 (1990), 25–51.

  58. Gerald Holton, “Project Physics: A Report on Its Aims and Current Status,” The Physics Teacher 5 (1967), 198–211. On the “New Math” during the same period Christopher Phillips, The New Math: A Political History (Chicago: University of Chicago Press, 2014). The history of the MIT PSSC program and its creator is told in Jack S. Goldstein, A Different Sort of Time: The Life of Jerrold R. Zacharias, Scientist, Engineer, Educator (Cambridge, MA: MIT Press, 1992).

  59. See, for instance, Gerald Holton, letter to Eugene Kone [American Institute of Physics], December 23, 1964, where the name of Papaliolios is listed as one of the “key people associated with Harvard Project Physics.” HPA, Box 34, Folder 4.

  60. For the influence of Harvard President James B. Conant on this movement, see Christopher Hamlin, “The Pedagogical Roots of the History of Science: Revisiting the Vision of James Bryant Conant,” Isis 107 (2016), 282–308. On science education reform in the United States, see John L. Rudolph, Scientists in the Classroom: The Cold War Reconstruction of American Science Education (New York: Palgrave, 2002).

  61. Meshoulam, The History of (Harvard) Project Physics (ref. 54), 30–36.

  62. CPP, Box 13, Folder Nat Sci 2 Fall ’62 Assignments & HW Sol’ns (Holton). The contrast could not be sharper with the instructions provided for the spring 1963 semester of Natural Sciences 2, offered this time by an associate professor named Pipkin. Harvard College, Courses of Instruction, 1954–1966, Box 7. HUC 8500.16, Harvard University Archives. The term paper suggestions had virtually no social or cultural connotation. Students had to write about “J. J. Thomson and the Electron,” “Radiocarbon Dating,” “Bubble Chambers,” “Radio Astronomy,” “Superconductivity,” and other such general topics in physics. CPP, Box 13, Folder Nat Sci 2 Spring ‘63 Complete Set (Pipkin) [Lecturer]. It is not clear whether Papaliolios was able to test his cubes during the spring 1963 semester—since quantum mechanics did not fit within the fall curriculum.

  63. He was the chief instructor in the fall of 1968 (with two TAs) and was again a TA for Holton in the Spring of 1969. In 1969–1970, 1970–1971, and 1971–1972 (now listed as full professor) Papaliolios was the chief instructor in both semesters. The course was not offered in 1972–1973, 1973–1974, or 1974–1975. In 1975–1976, he co-taught Nat. Sci. 2 in the Spring with Sheldon Glashow and in 1976–1977, with Norman Ramsey (fall). Harvard College, Courses of Instruction, 1967–1972, Box 8 and 1973–1978, Box 9. HUC 8500.16, Harvard University Archives.

  64. CPP, Box 9, Folder Nat Sci 2, Fall 1968 and Folder Nat Sci 2, 1969–70. In Box 13, Folder NS 7, 1966, a similar book list spanning several pages is requested to be put on reserve by Stephen Brush, on the official Harvard Project Physics letterhead. It is interesting to note here that Holton published in 1969 an important paper on Einstein and the Michelson interferometer experiment: Gerald Holton, “Einstein, Michelson, and the ‘Crucial’ Experiment,” Isis 60 (1969), 132–97. Reprinted in Thematic Origins of Scientific Thought: Kepler to Einstein, rev. ed. (Cambridge, MA: Harvard University Press, 1988), 279–370.

  65. On the production of all the Harvard Project Physics films by the Office national du film du Canada between 1966 and 1970, see the various documents deposited at the Service des archives de l’ONF, Fonds de l’Office national du Film du Canada and identified under the following headings: Procès-verbaux 1966–1970; Échanges_1966–1969; Production_1967–1968; and Presse_1966–1967. I would like to thank André d’Ulisse at the ONF for providing me with these files.

  66. Gerald Holton, letter to Costas Papaliolios, November 5, 1968, CPP, Box 9, Folder NS 2 Lectures ’68. Gerald Holton, “The Project Physics Course” (ref. 56), 782.

  67. Holton was clearly aware of these potential issues. He noted that a high-school teacher from Colorado complained in the May 1970 issue of The Physics Teacher that the PSSC instruments provided by Macalaster, a subsidiary of Raytheon, were getting expensive (the price of a simple ripple tank increased by 40% between 1967 and 1970) and were not always promptly delivered. At the top of the Xerox copy Holton handwrote: “Shape of complaints to come? …” HPA, Box 37, Folder 8. At the same time, Holton and others were fighting the Macalaster Scientific Corporation against misuses of their “Project Physics” trademark. In March of 1969 they were granted an injunction against the Raytheon Education Company, the parent company of Macalaster, for “appropriating the name Project Physics and giving it a grossly incomplete and inadequate package, offered for sale during the interim period, before Project Physics itself can come out with its tested, complete, final materials.” Holton caught Macalaster in the very same misdeed two years later at their exhibit booth during an American Physical Society Meeting in New York City. See the Memorandum and letters by Holton in HPA, Box 37, Folder 8.

  68. CPP, Box 15, Folder Q.M. for Nat Sci and Folder Q.M. for Nat Sci 2 (QM) 1975–76. Stephen Fulling, who is currently a professor of mathematics and physics at Texas A&M University, received an undergraduate degree in physics from Harvard between 1963 and 1967. He does not remember seeing these toys in the hands of anyone. Stephen Fulling, correspondence with the author, June 15, 2015.

  69. Wolfgang Rueckner, correspondence with the author, June 14, 2015.

  70. Linda J. Greenhouse, “Gerald Holton: The Discovery that Scientists are also Philosophers Should Not Depend on Accidents,” The Harvard Crimson, 12 December 1966. Holton “thought there ought to be two kinds of general education: one essentially civilizing the heathens at the very beginning [Nat Sci 2], and then at the end to show the physics seniors why it was all physics when they took it in their various courses, like thermodynamics here and electromagnetism there.” Gerald Holton, interview with Katherine Sopka, January 11, 1977, Niels Bohr Library and Archives, American Institute of Physics, accessed December 12, 2016, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/31279.

  71. A complete list is found in Mehra and Milton, Climbing the Mountain (ref. 15), 639–43.

  72. Quoted in Schweber, QED (ref. 28), 372.

  73. On Wheatland, see William J. H. Andrewes, “The Life and Work of David Pingree Wheatland (1898–1993),” Journal of the History of Collections 7 (1995), 261–68; “The Legacy of David Wheatland,” Nuncius 16 (2001), 687–701.

  74. Gerald Holton, letter to David Wheatland, June 19, 1980, HPA Box 16, Folder 6. See also Gerald Holton, letters to Paul Forman, June 19, 1980; October 22, 1980; November 24, 1980, HPA, Box 16, Folder 6. In other correspondence, we learn that Forman wanted to make an exhibition on the occasion of Bridgman’s centenary. It was never mounted. The presses and other equipment have inventory numbers 1980.0595.01 through 05 at the Smithsonian. I would like to thank Roger Sherman for providing me with this information.

  75. Holton to Forman, June 19, 1980 (ref. 74). Smithsonian inventory number: 1980.0594. It was exhibited in 1982 in an exhibition called “Atomic Clocks.” Again, my thanks to Roger Sherman for this information.

  76. David Wheatland, draft of letter to Gerald Holton, June 23, 1980. Collection of Historical Scientific Instruments, Correspondence folder 1979–1980.

  77. Handwritten copy of an acquisition list, Collection of Historical Scientific Instruments, Percy Williams Bridgman folder. We have in the same folder an inventory made by Chase himself dated May 29, 1956, which mentions ten such high-pressure presses in four different rooms in the Jefferson Physical Laboratory. The list was given to the CHSI by Holton.

  78. On the discovery of the muon, Peter Galison, “The Discovery of the Muon and the Failed Revolution against Quantum Electrodynamics,” Centaurus 26 (1983), 262–316.

  79. Other important material-based pedagogical analyses are what David Kaiser, Andrew Warwick, and Christopher Phillips have done respectively for Feynman’s diagrams in quantum electrodynamical research, the mundane devices used in training for the Cambridge mathematical Tripos exam, and the blackboard in the context of a military academy. Kaiser, Drawing Theories Apart (ref. 32). Andrew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago: University of Chicago Press, 2003). Christopher Phillips, “An Officer and a Scholar: Nineteenth-Century West Point and the Invention of the Blackboard,” History of Education Quarterly 55 (2015), 82–108. A more general analysis is found in David Kaiser, ed., Pedagogy and the Practice of Science: Historical and Contemporary Perspectives (Cambridge, MA: MIT Press, 2005).

  80. John G. Payne, “Physics Just for Fun—An Individualized Course Using Harvard Project Physics,” The Physics Teacher 10 (1972), 138–40.

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Acknowledgments

I would like to warmly thank Aaron Wright, David Kaiser, Kimball Milton, Stephen Fulling, Walter Wilcox, Eric Heller, Joseph Martin, Robert Crease, and Peter Pesic for their careful reading and various comments on the initial manuscript. I dedicate this article to Sam Schweber, who was with me every step of the way toward publication. His extraordinary knowledge and kindness will be dearly missed.

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  1. Collection of Historical Scientific Instruments, Harvard University, 1 Oxford St, Science Center 371, Cambridge, MA, 02138, USA

    Jean-François Gauvin

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Correspondence to Jean-François Gauvin.

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Jean-François Gauvin, PhD, is the Director of Administration for the Collection of Historical Scientific Instruments at Harvard University. He has written numerous articles and books on the material culture of science, from the seventeenth to the twentieth century.

Appendix: In the Makers’ Lab

Appendix: In the Makers’ Lab

The quantum toys can be manufactured of any size. The ones Costas Papaliolios designed, now preserved in the Collection of Historical Scientific Instruments at Harvard, are one-inch and four-inch cubes. The one-inch cubes are much easier to manipulate, and thus should be favored for hands-on operations.

The cubes are made of aluminum. You can buy square aluminum tubing online or in a good hardware store. To make a complete set of quantum toys, you will need at least 13 inches of tubing (a complete set consists of 13 cubes). The exterior dimension of the aluminum tubing is 1 inch; the interior dimension is 7/8 inches.

To secure the polarizers inside the cubes, you will need another length of square tubing, this one of brass, with an exterior dimension of 7/8 inches. The sides should be as thin as possible. Cut these smaller square pieces into lengths of 1/3 inches. You will need two per cube, for a total of twenty-six. They will be pressure-fit inside the cubes to squeeze the polarizers into place.

The polarizers are the key elements of the quantum toys. They should be cut in squares slightly less than 7/8 inches to fit inside the aluminum tubing. You will need three types of polarizers, plus a piece of unpolarized clear plastic (for the Identity cube). Results may vary with the quality of the polarizers. Here is the full list, based on Papaliolios’s own drawings:

Unpolarized clear plastic

Linear polarizer (0 degree);

Linear polarizer (45 degrees);

Halfwave plate—retarder—(fast axis 0 degree); ½λ

Halfwave plate—retarder—(fast axis 45 degrees); ½λ

Quarterwave plate—retarder—(fast axis 0 degree); ¼λ

Quarterwave plate—retarder—(fast axis 90 degrees); ¼λ

Follow the table below to assemble each quantum toys with the appropriate polarizers. Once it is done, you simply need to mark the cubes with the proper Dirac notation.

Please note the following when inscribing the cubes:

On the |↑〉〈↑| cube: inscribe |↑〉〈↑| on two opposite sides and |↓〉〈↓| on the other two sides;

On the |↓〉〈↓| cube: inscribe |↓〉〈↓| on two opposite sides and |↑〉〈↑| on the other two sides;

On the |↑〉〈↓| cube: inscribe |↑〉〈↓| on two opposite sides and |↓〉〈↑| on the other two sides;Footnote 1

On the |1〉〈1| cube: inscribe |1〉〈1| on two opposite sides and |0〉〈0| on the other two sides;

On the |0〉〈0| cube: inscribe |0〉〈0| on two opposite sides and |1〉〈1| on the other two sides;

On the |1〉〈0| cube: inscribe |1〉〈0| on two opposite sides and |0〉〈1| on the other two sides;

On the |+〉〈+| cube: inscribe |+〉〈+| on all four sides; same for |+〉〈−| and |−〉〈−| cubes;

On the σx cube: inscribe σx on all four sides; same for σy and σz cubes.

Measurement Symbols

The identity symbol, I, is simply a piece of unpolarized clear plastic (Table 3).

Table 3 Dirac notation drawn on the quantum toys in relation to the inserted polarizers. Transcribed from the Harvard University Archives, Costas Papaliolios Papers (unprocessed accessions 14811), Box 21, Folder Quantum Toys.
Full size table

The juxtaposition of polarizing filters illustrates well, for instance, the annihilation and creation process produced by the operator |↑〉〈↓|. It is also easy to understand why, from Julian Schwinger’s measurement symbolism, M(↑,↑) = M(↑) and σx σy = σz.

Take this concrete example, juxtaposing the following two quantum toys:

|+〉〈−| |+〉〈+| = 0

We know (and can show by experiment) that light will not go through. It is one of the three key algebraic principles determined by Schwinger, that is, the exclusiveness principle. Looking at the polarizers inside the cubes we find (from left to right):

¼λ linear polarizer ¼λ and

¼λ linear polarizer ¼λ

Now imagine that (unpolarized, or ambient) light comes from the left in the above schematic. It first passes through a ¼λ retarder, which does not affect its polarization. Unpolarized light continues and passes through a linear polarizer at 45° and finally through another ¼λ retarder, fast axis at 90°. It will be circularly polarized.

Light leaves the first cubes and goes into the second cube, where it falls on a ¼λ retarder, fast axis at 0°. This retarder, juxtaposed to the first ¼λ retarder, will operate a polarization shift of 90° to the original linearly polarized light. Light is now linearly polarized at 270° when it hits the next linear polarizer at 45°. Light is blocked.

Similar analyses can be performed for any combination of quantum toys.

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Gauvin, JF. Playing with Quantum Toys: Julian Schwinger’s Measurement Algebra and the Material Culture of Quantum Mechanics Pedagogy at Harvard in the 1960s. Phys. Perspect. 20, 8–42 (2018). https://doi.org/10.1007/s00016-018-0213-3

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