We discuss an investigation of students' difficulties with systems of non-interacting identical particles. The investigation was carried out in advanced quantum mechanics courses at both the undergraduate and graduate levels by administering free-response and multiple-choice questions and conducting individual interviews with students. This research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) that helps students develop a functional understanding of the concepts necessary for determining the many-particle Hamiltonian and stationary state wavefunctions for a system of non-interacting identical particles.
REFERENCES
1.
E.
Galvez
and
C.
Holbrow
, “
Interference with correlated photons: Five quantum mechanics experiments for undergraduates
,” Am. J. Phys.
73
, 127
–140
(2005
).2.
M.
Belloni
and
W.
Christian
, “
Time development in quantum mechanics using a reduced Hilbert space approach
,” Am. J. Phys.
76
, 385
–392
(2008
).3.
D.
McIntyre
,
J.
Tate
, and
C.
Manogue
, “
Integrating computational activities into the upper-level Paradigms in Physics curriculum at Oregon State University
,” Am. J. Phys.
76
, 340–346
(2008
).4.
M. N.
Beck
and
M.
Beck
, “
Witnessing entanglement in an undergraduate laboratory
,” Am. J. Phys.
84
, 87–94
(2016
).5.
R.
Aspden
,
M.
Padgetta
, and
G.
Spalding
, “
Video recording true single-photon double-slit interference
,” Am. J. Phys.
84
, 671–677
(2016
).6.
A.
Kohnle
et al, “
New introductory quantum mechanics curriculum
,” Eur. J. Phys.
35
, 015001
(2014
).7.
A.
Kohnle
et al, “
Developing and evaluating animations for teaching quantum mechanics concepts
,” Eur. J. Phys.
31
, 1441
–1455
(2010
).8.
R.
Muller
and
H.
Wiesner
, “
Teaching quantum mechanics on an introductory level
,” Am. J. Phys.
70
, 200–209
(2002
).9.
D.
Zollman
et al, “
Quantum mechanics for everyone: Hands-on activities integrated with technology
,” Am. J. Phys.
70
, 252–259
(2002
).10.
M.
Wittmann
et al, “
Investigating student understanding of quantum physics: Spontaneous models of conductivity
,” Am. J. Phys.
70
, 218–226
(2002
).11.
E.
Gire
and
E.
Price
, “
The structural features of algebraic quantum notations
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
11
, 020109
(2015
).12.
A.
Maries
and
C.
Singh
, “
Can students apply the concept of ‘which-path’ information learned in the context of Mach Zehnder Interferometer to the double slit experiment?
,” Am. J. Phys.
88
(7
), 542
–550
(2020
).13.
P.
Emigh
et al, “
Student understanding of time dependence in quantum mechanics
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
11
, 020112
(2015
).14.
G.
Passante
et al, “
Examining student ideas about energy measurements on quantum states across undergraduate and graduate levels
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
11
, 020111
(2015
).15.
V.
Dini
and
D.
Hammer
, “
Case study of a successful learner's epistemological framings of quantum mechanics
,” Phys. Rev. Phys. Educ. Res.
13
, 010124
(2017
).16.
C.
Keebaugh
et al, “
Improving student understanding of corrections to the energy spectrum of the hydrogen atom for the Zeeman effect
,” Phys. Rev. Phys. Educ. Res.
15
, 010113
(2019
).17.
R.
Sayer
et al, “
Case study evaluating just-in-time teaching and peer instruction using clickers in a quantum mechanics course
,” Phys. Rev. Phys. Educ. Res.
12
, 020133
(2016
).18.
P.
Justice
et al, “
Improving student understanding of quantum mechanics underlying the Stern-Gerlach experiment using a research-validated multiple-choice question sequence
,” Eur. J. Phys.
40
, 055702
(2019
).19.
P.
Justice
et al, “
Student understanding of Fermi energy, the Fermi-Dirac distribution and total electronic energy of a free electron gas
,” Eur. J. Phys.
41
, 015704
(2020
).20.
S.
Devore
and
C.
Singh
, “
An interactive learning tutorial on quantum key distribution
,” Phys. Rev. Phys. Educ. Res.
16
, 010126
(2020
).21.
C.
Singh
, “
Student understanding of quantum mechanics
,” Am. J. Phys.
69
, 885–895
(2001
).22.
C.
Singh
, “
Student understanding of quantum mechanics at the beginning of graduate instruction
,” Am. J. Phys.
76
, 277–287
(2008
).23.
G.
Zhu
and
C.
Singh
, “
Surveying students' understanding of quantum mechanics in one spatial dimension
,” Am. J. Phys.
80
, 252–259
(2012
).24.
C.
Singh
, “
Student difficulties with quantum mechanics formalism
,” AIP Conf. Proc.
883
(1
), 185–188
(2007
).25.
E.
Marshman
and
C.
Singh
, “
Framework for understanding student difficulties in quantum mechanics
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
11
, 020119
(2015
).26.
C.
Singh
and
E.
Marshman
, “
Review of student difficulties in quantum mechanics
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
11
, 020117
(2015
).27.
A.
Mason
and
C.
Singh
, “
Do advanced students learn from their mistakes without explicit intervention?
,” Am. J. Phys.
78
(7
), 760–767
(2010
).28.
B.
Brown
,
A.
Mason
, and
C.
Singh
, “
Improving performance in quantum mechanics with explicit incentives to correct mistakes
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
12
, 010121
(2016
).29.
C.
Singh
,
M.
Belloni
, and
W.
Christian
, “
Improving students' understanding of quantum mechanics
,” Phys. Today
59
(8
), 43–49
(2006
).30.
G.
Zhu
and
C.
Singh
, “
Improving students' understanding of quantum mechanics via the Stern-Gerlach experiment
,” Am. J. Phys.
79
, 499–507
(2011
).31.
G.
Zhu
and
C.
Singh
, “
Improving students' understanding of quantum measurement: II. Development of research-based learning tools
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
8
, 010118
(2012
).32.
G.
Zhu
and
C.
Singh
, “
Improving student understanding of addition of angular momentum in quantum mechanics
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
9
, 010101
(2013
).33.
G.
Zhu
and
C.
Singh
, “
Improving students' understanding of quantum measurement: I. Investigation of difficulties
,” Phys. Rev. Spec. Top.-Phys. Educ. Res.
8
, 010117
(2012
).34.
S. Y.
Lin
and
C.
Singh
, “
Categorization of quantum mechanics problems by professors and students
,” Eur. J. Phys.
31
, 57
–68
(2010
).35.
C.
Singh
, “
Interactive learning tutorials on quantum mechanics
,” Am. J. Phys.
76
, 400–405
(2008
).36.
E.
Marshman
and
C.
Singh
, “
Interactive tutorial to improve student understanding of single photon experiments involving a Mach-Zehnder interferometer
,” Eur. J. Phys.
37
, 024001
(2016
).37.
E.
Marshman
and
C.
Singh
, “
Validation and administration of a conceptual survey on the formalism and postulates of quantum mechanics
,” Phys. Rev. Phys. Educ. Res.
15
, 020128
(2019
).38.
E.
Marshman
and
C.
Singh
, “
Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics
,” Eur. J. Phys.
38
, 025705
(2017
).39.
E.
Marshman
and
C.
Singh
, “
Investigating and improving student understanding of quantum mechanics in the context of single photon interference
,” Phys. Rev. Phys. Educ. Res.
13
, 010117
(2017
).40.
E.
Marshman
and
C.
Singh
, “
Investigating and improving student understanding of the expectation values of observables in quantum mechanics
,” Eur. J. Phys.
38
, 045701
(2017
).41.
R.
Sayer
,
A.
Maries
, and
C.
Singh
, “
A quantum interactive learning tutorial on the double-slit experiment to improve student understanding of quantum mechanics
,” Phys. Rev. Phys. Educ. Res.
13
, 010123
(2017
).42.
A.
Maries
et al, “
Effectiveness of interactive tutorials in promoting ‘which-path’ information reasoning in advanced quantum mechanics
,” Phys. Rev. Phys. Educ. Res.
13
, 020115
(2017
).43.
C.
Keebaugh
et al, “
Investigating and addressing student difficulties with the corrections to the energies of the hydrogen atom for the strong and weak field Zeeman effect
,” Eur. J. Phys.
39
, 045701
(2018
).44.
C.
Keebaugh
et al, “
Investigating and addressing student difficulties with a good basis for finding perturbative corrections in the context of degenerate perturbation theory
,” Eur. J. Phys.
39
, 055701
(2018
).45.
C.
Keebaugh
et al, “
Improving student understanding of a system of identical particles with a fixed total energy
,” Am. J. Phys.
87
, 583–593
(2019
).46.
C.
Keebaugh
et al, “
Improving student understanding of fine structure corrections to the energy spectrum of the hydrogen atom
,” Am. J. Phys.
87
, 594–605
(2019
).47.
D.
Griffiths
, Introduction to Quantum Mechanics
, 2nd ed. (
Prentice Hall
,
Englewood Cliffs, NJ
, 2005
).48.
J.
Townsend
, A Modern Approach to Quantum Mechanics
, 2nd ed. (
University Science Books
,
Sausalito, CA
, 2012
).49.
J. J.
Sakurai
and
J.
Napolitano
, Modern Quantum Mechanics
, 2nd ed. (
Addison-Wesley
,
Reading, MA
, 2011
).50.
51.
52.
L.
Landau
and
E.
Lifshitz
, Quantum Mechanics: Non-Relativistic Theory
, 3rd ed. (
Butterworth-Heinemann
,
New York
, 1981
).53.
N.
Zettilli
, Quantum Mechanics: Concepts and Applications
, 2nd ed. (
Wiley
,
New York
, 2009
).54.
R.
Shankar
, Principles of Quantum Mechanics
, 2nd ed. (
Springer
,
New York
, 1994
).55.
See <https://www.physport.org/methods/method.cfm?G=QUILTS> for the QuILTs.
56.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0006910 for additional details about the investigation and findings.
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2022
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