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Testing the development of student conceptual and visualization understanding in quantum mechanics through the undergraduate career
1.R. W. Robinett, Web-based quantum mechanics tutorials for undergraduates, NSF Award 9950702. For more information, updated periodically, go to www.ehr.nsf.gof/pirs_prs_web/search/default.asp. See also our own web site, http://www.phys.psu.edu/faculty/RobinettR/QM/QMVI/QMVI.html for downloadable copies of much of the background material generated by this study.
2.E. Cataloglu, “Development of an achievement test in quantum mechanics: The quantum mechanics visualization instrument (QMVI),” Ph.D. thesis, Penn State University, in preparation.
3.D. Hestenes, M. Wells, and G. Swackhammer, “Force concept inventory,” Phys. Teach. 30, 141–158 (1992);
3.D. Hestenes and M. Wells, “A mechanics baseline test,” Phys. Teach. 30, 159–166 (1992);
3.R. K. Thornton and D. Sokoloff, “Assessing student learning of Newton’s laws: the force and motion conceptual evaluation and the evaluation of active learning laboratory and lecture curricula,” Am. J. Phys. 66, 338–352 (1998).
4.T. O’Kuma, C. Hieggelke, D. Maloney, and A. Van Heuvelen, “Developing conceptual surveys in electricity and magnetism,” Announcer 28, 81 (1998);
4.T. O’Kuma, C. Hieggelke, D. Maloney, and A. Van Heuvelen, “Preliminary interpretation of the CSE/CSM/CSEM student results,” Announcer 29, 82 (1999);
4.T. O’Kuma, C. Hieggelke, D. Maloney, and A. Van Heuvelen, “Some results form the conceptual survey of electricity and magnetism,” Announcer 30, 77 (2000).
5.For example, in addition to the standard use of the infinite well as a bound state model, along with the Pauli principle, to help explain aspects of the structure of solids and even nuclear matter, the infinite well can be used to discuss modern aspects of quantum mechanics ranging from supersymmetry [F. Schwabl, Quantum Mechanics (Springer-Verlag, Berlin, 1990)] to wave packet revivals (Refs. 789).
6.See, e.g., M. Nauenberg, C. Stroud, and J. Yeazell, “The classical limit of an atom,” Sci. Am. 270, 44–49 (1994).
7.R. Bluhm, V. A. Kostelecký, and J. A. Porter, “The evolution and revival structure of localized wave packets,” Am. J. Phys. 64, 944–953 (1996).
8.R. W. Robinett, “Visualizing the collapse and revival of wave packets in the infinite square well using expectation values,” Am. J. Phys. 68, 410–420 (2000).
9.D. F. Styer, “Quantum revivals versus classical periodicity in the infinite square well,” Am. J. Phys. 69, 56–62 (2001).
10.N. S. Rebello and D. Zollman, “Conceptual understanding of quantum mechanics after using hands-on and visualization instructional materials,” Research on Teaching and Learning Quantum Mechanics (National Association for Research in Science Teaching, 1999), pp. 2–6.
11.R. Steinberg, M. Wittmann, L. Bao, and E. F. Redish, “The influence of student understanding of classical physics when learning quantum mechanics,” Research on Teaching and Learning Quantum Mechanics (National Association for Research in Science Teaching, 1999), pp. 41–44.
12.E. F. Redish and B. Lei, “Student difficulties with energy in quantum mechanics,” www.physics.umd.edu/rgroups/ripe/perg/quantum/aapt97qe.htm
13.L. Bao, P. Jolly, and E. F. Redish, “Student difficulties with quantum mechanics,” www.physics.umd.edu/rgroups/ripe/ph421/bao/talks/qm9608.htm
14.L. Bao, E. F. Redish, and R. Steinberg, “Student misunderstandings of the quantum wavefunction,” Summer AAPT Announcer 28, 92 (1998).
15.D. F. Styer, “Common misconceptions regarding quantum mechanics,” Am. J. Phys. 64, 31–34 (1996);
15.D. F. Styer, Am. J. Phys. 64, 1202(E) (1996).
16.S. Vokos, P. S. Schaffer, B. S. Ambrose, and L. C. McDermott, “Student understanding of the wave nature of matter: Diffraction and interference of particles,” Phys. Ed. Res. Suppl. 68, S42–S51 (2000).
17.C. Singh, “Student understanding of quantum mechanics,” Am. J. Phys. 69, 885–895 (2001).
18.For example, in the famous textbook by L. Schiff, Quantum Mechanics, 1st ed. (McGraw-Hill, New York, 1949) of half a century ago, the figures illustrating the solutions of the simple harmonic oscillator problem are reproduced from an even earlier work, by L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935) which, in turn, were drawn, by a draftsman, on graph paper.
19.B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena (Springer-Verlag, New York, 2000).
20.S. Brandt and S. Dahmen, The Picture Book of Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 2001).
21.J. Bayfield, Quantum Evolution: An Introduction to Time-Dependent Quantum Mechanics (Wiley, New York, 1999).
22.J. R. Hiller, I. D. Johnston, and D. F. Styer, Quantum Mechanics Simulations: The Consortium for Upper-Level Physics Software (Wiley, New York, 1995).
23.S. M. McMurry, Quantum Mechanics (Addison-Wesley, Reading, PA, 1993).
24.J. Jarecki, Graphical Schrödingers Equation (Physics Academic Software, Raleigh, 1998).
25.For example, one of the first pedagogical papers describing the simplest aspects of wave packet propagation and interaction with potential barriers and wells was written by A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177–186 (1967). The authors used one of the most powerful computers then available at the Livermore Lab (now dwarfed by modern PCs in speed) and had “…the probability density projected on a cathode-ray tube. From the tube, photographs are made, and in turn are processed into the successive frames of a film.” Typical software packages now allow students to reproduce these results with arbitrary initial conditions and potential parameters, almost instantaneously. See Ref. 26 for more references on time-dependent wave packet propagation.
26.A rather comprehensive list of references to discussions of wave packet propagation in many model quantum mechanical systems, both for scattering and for bound states, in the pedagogical literature is contained in M. Doncheski and R. W. Robinett, “Anatomy of a quantum ‘bounce,’ ” Eur. J. Phys. 20, 29–37 (1999).
27.Scientific Visualization in Mathematics and Science teaching, edited by D. A. Thomas (Association for the Advancement of Computing in Education, Virginia, 1995).
28.B. H. McCormick, T. A. DeFanti, and M. D. Brown, “Visualization in scientific computing,” Comput. Graph. 21, 14–18 (1987).
29.F. M. Dwyer, “The effect of over responses in improving visually programming science instruction,” J. Res. Sci. Teach. 9, 47–55 (1972).
30.W. G. Holliday, “The effects of verbal and adjunct pictorial-verbal information in science instruction,” J. Res. Sci. Teach. 12, 77–83 (1975).
31.J. W. Rigney and K. A. Lutz, “Effect of graphics analogies of concepts in chemistry on learning and attitude,” J. Ed. Psych. 68, 305–311 (1976).
32.R. R. Gotwals, “Scientific Visualization in Chemistry: Better Living Through Chemistry, Better Chemistry Through Pictures: Scientific Visualization for Secondary Chemistry Students,” in Ref. 27, pp. 153–180.
33.GRE®: Practicing to take the Physics Test (Educational Testing Service, Princeton, 1997). This contains three complete Physics GRE examinations as well as data on student responses.
34.R. A. Serway, C. Moses, and C. Moyer, Modern Physics, 2nd ed. (Saunders College, Fort Worth, TX, 1997).
35.K. S. Krane, Modern Physics, 2nd ed. (Wiley, New York, 1996).
36.A. Beiser, Concepts of Modern Physics, 5th ed. (McGraw-Hill, New York, 1995).
37.P. Tipler and R. A. Llewellyn, Modern Physics, 3rd ed. (W. H. Freeman, New York, 1999).
38.T. A. Moore, Six Ideas That Shaped Physics, Unit Q, Particles Behave Like Waves (McGraw-Hill/WCB, Boston, 1998).
39.D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Englewood Cliffs, NJ, 1995).
40.C. Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum Mechanics, Vols. I and II (Wiley-Interscience, New York, 1977) (Translated from the French by S. Hemley, N. Ostrowsky, and D. Ostrowsky).
41.I. N. Levine, Quantum Chemistry (Prentice Hall, Upper Saddle River, NJ, 2000).
42.R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford U. P., New York, 1997).
43.A very similar question was asked in the 1996 Physics GRE (GR9677, Question No. 17) (Ref. 33). The correct response rate listed for that question is 40% from the student population (presumably mostly physics seniors intending to go to graduate school) taking the test.
44.A. P. French and E. F. Taylor, “Qualitative plots of bound state wave functions,” Am. J. Phys. 39, 961–962 (1971);
44.An Introduction to Quantum Mechanics, M.I.T. Introductory Physics Series (Norton, New York, 1978).
45.Question No. 97 of the 1996 Physics GRE (GR9677) has a listed correct response rate of 11% (Ref. 33).
46.C. M. McCallum, “Book and Media reviews,” J. Chem. Phys. 74, 343 (1997).
47.R. L. Liboff, Introductory Quantum Mechanics, 2nd ed. (Addison-Wesley, Reading, PA, 1991).
48.R. W. Robinett, “Quantum and classical probability distributions for position and momentum,” Am. J. Phys. 63, 823–832 (1995);
48.see also Ref. 42, pp. 114–115.
49.Many of the background details necessary for the module under development are discussed in depth in M. A. Doncheski and R. W. Robinett, “Comparing the classical and quantum probability densities for an asymmetric infinite well,” Eur. J. Phys. 21, 217–228 (2000).
50.M. Andrews, “Wave packets bouncing off walls,” Am. J. Phys. 66, 252–254 (1998).
51.Unlike studies of student understanding of concepts related to such topics as introductory mechanics or E&M, where sample sizes of order are often readily obtainable from large service courses, the enrollments in the kinds of courses we consider (ModPh-, UgQM1-, and GrQM1-type classes) are often 10–30 at many US institutions. It may well be that data from a number of different sites will have to be collected, compared, and integrated over in order to obtain similar sample sizes.
52.The written responses have already been used extensively to assess detailed student understanding, especially during the test of the original V0.3 version, helping to suggest improvements which were implemented in V0.4. More detailed discussions of the written responses and many other issues will be included in an upcoming paper [E. Cataloglu and R. W. Robinett (in preparation)].
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