Singh, Belloni, and Christian reply: We appreciate the number and quality of the responses to our article. They indicate a strong interest, which we share, in the teaching of upper-level courses such as quantum mechanics. Our article focused on the concept of time evolution to illustrate a variety of difficulties students face; we barely scratched the surface of the breadth and depth of teaching and learning issues in a standard quantum mechanics course.

We value highly the perspectives on fundamental issues from Robert Griffiths and Travis Norsen, who raised similar concerns from different viewpoints. Foundational issues in quantum mechanics are not emphasized in most undergraduate or graduate quantum mechanics curricula. Griffiths has argued that the lack of proper grounding in foundational issues is the source of many student misconceptions in quantum mechanics. The consistent histories approach 1 or Bohm's interpretation 2 may be conceptually “cleaner,” but our research has shown that many of the difficulties—for example, the confusion between the time-independent and time-dependent Schrödinger equation—are not foundational but conceptual.

As a practical matter, non-Copenhagen interpretations are not widely incorporated in quantum mechanics textbooks. We have argued that there are ways to improve student understanding within the current framework—surely, these general methods will work if and when the physics community has collectively adopted new ways of thinking about quantum mechanics.

Physics education research is well-established now, and a controlled study involving two quantum mechanics classes taught by the same instructor might be worthwhile. One class could use the standard Copenhagen interpretation while the other uses the consistent histories approach. An important question, then, is this: If both classes cover approximately the same amount of material and students in both classes are given the surveys we have developed, do students in one class significantly outperform those in the other? In addition to the written surveys, a subset of students from both classes could be interviewed to further ascertain their level of understanding. If students using the consistent histories approach significantly outperform those learning the standard Copenhagen interpretation, it may be worthwhile to develop interactive tutorials similar to those discussed in the article but using the consistent histories approach.

In response to Travis Norsen, we note that we agree with Alan Van Heuvelen, whom Norsen cites, and our approach is consistent with his advice. 3 However, intuition and foundational issues are not exactly the same things. Although a deep understanding of foundational issues may improve intuition, we can help our students develop qualitative, conceptual understanding of many aspects of quantum theory without first having to clarify every foundational issue. Our research suggests that the nature of physical intuition is not well understood, though intuition is important. 4  

As Philip Shemella has suggested, we have used other wordings for the question of interest, including the wording he recommends. Our findings are unchanged. During interviews, the interviewer has often rephrased the question when a student was unable to answer correctly. The responses were qualitatively unchanged.

As Griffiths, Norsen, and Walter Harrison imply, the use of simulations and results from physics education research to address functional issues is just a single prong in what should be a multi-pronged approach to the teaching of quantum mechanics. We agree that addressing foundational issues is just as important.

In addition to the approach taken in textbooks by Griffiths and Harrison, Richard Robinett's quantum text 5 relates pedagogical quantum models to modern experimental realizations of these systems and emphasizes connections to classical mechanics.

We agree with Norman Chonacky that a discussion of the broader role of computation in the physics curriculum is needed. We encourage interested readers to attend the American Association of Physics Teachers topical conference Computational Physics for Upper Level Courses, to be held in July 2007 (see http://www.opensourcephysics.org/CPC/index.html). Its purpose is to identify problems in which computation helps students understand key physics concepts.

1.
R. B.
Griffiths
,
Consistent Quantum Theory
,
Cambridge U. Press
,
New York
(
2002
). Some chapters and a few exercises are available at http://quantum.phys.cmu.edu.
2.
J. S.
Bell
,
Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy
,
Cambridge U. Press
,
New York
(
2004
). Also see http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics.
3.
A.
Van Heuvelen
,
Am. J. Phys.
59
,
891
(
1991
).
4.
C.
Singh
,
Am. J. Phys.
70
,
1103
(
2002
).
5.
R. W.
Robinett
,
Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples
, 2nd ed.,
Oxford U. Press
,
New York
(
2006
).