With current computer hardware and software, the solution and visualization of quantum-mechanical bound-state problems is relatively easy.  In one dimension, the usual numerical technique, called the shooting method,[i] is quite reliable for determining most energy eigenstates and their associated energy eigenfunctions and energy eigenvalues (as long as the artificial boundaries do not get in the way).  This technique also makes it easy to extend the realm of quantum-mechanical bound states to those not often considered in introductory physics courses, where the infinite square well and harmonic oscillator reign supreme, by exploring the qualitative form of energy eigenfunctions for a wide variety of potential energy wells. 

“It is nice to know that the computer understands the problem. 
But I would like to understand it, too.”

Over 35 years ago, French and Taylor[iii] outlined an approach to teach students and teachers alike how to understand “qualitative plots of bound-state wave functions.”   They described five fundamental statements based on the quantum-mechanical concepts of probability and energy (total and potential) which could be used to deduce the shape of energy eigenfunctions.  Despite these important and easy-to-follow statements, this approach and set of techniques has not been universally adopted in the teaching of quantum mechanics.[iv]  For example, recent studies have shown that students’ conceptual understanding of quantum mechanics on all levels (from modern physics courses to graduate courses) is surprisingly lacking[v] and that misconceptions are universal,[vi] including that of the relationship between the potential energy function vs. position and the energy eigenfunction shape vs. position.  At the same time, the teaching of quantum mechanics in introductory physics has become increasingly important given the modern technological applications that are based on quantum theory (e.g. PET scans and MRIs).  However, most treatments of quantum theory on the introductory level are cursory at best, leaving students with the impression that quantum mechanics is little more than abstract mathematics (a belief that remains with students in their future courses). 

The focus of this paper is similar to that of Ref. 3, but we have made the process more accessible at the introductory level by the inclusion of simulations via Physlets[vii],[viii] which allow us to visualize[ix] many more quantum wells than are typically considered in an introductory physics course.  We have designed Physlet-based exercises to give students a conceptual understanding of energy eigenfunction shape and its relation to the potential energy function through the Schrödinger equation by giving them concrete examples of how this analysis works.  The energy-eigenvalue Physlet can be configured to numerically solve almost any one-dimensional bound-state problem, allowing many different scenarios to be easily considered, which aids students in their conceptual development.  This focus also allows us to make use of certain classical mechanics concepts in determining the shape of quantum-mechanical energy eigenfunctions.  Such an approach can help students explore the similarities and differences between classical and quantum mechanics.  This paper also compliments recent papers in this Journal[x] which have suggested ways to make quantum mechanics more accessible to the introductory physics audience.


[i] In the shooting method, the energy eigenfunction is forced to be zero at a boundary on the left, xleft, and the technique finds energy eigenfunctions by hunting (shooting) for numerical solutions of the time-independent Schrödinger equation where the state is also zero on the boundary on the right, xright.  This approach is equivalent to placing the problem of interest into an infinite square well with walls at xleft and xright.

[ii] H. M. Nussenzveig, Differential Effects in Semiclassical Scattering, Cambridge U. P., New York (1992).

[iii] A. P. French and E. F. Taylor, “Qualitative plots of bound state wave functions,” Am. J. Phys. 39, 961-962 (1971).

[iv] See for example, C. Singh, M. Belloni, and W. Christian, “Improving Students’ Understanding of Quantum Mechanics,” Physics Today, August, 43-49 (2006).

[v] E. Cataloglu and R. Robinett, “Testing the Development of Student Conceptual and Visualization Understanding in Quantum Mechanics through the Undergraduate Career,” Am. J. Phys. 70, 238-251 (2002).

[vi] C. Singh, “Student understanding of quantum mechanics,”   Am. J. Phys., 69 (8), 885-896, (2001).

[vii] W. Christian and M. Belloni, Physlet® Physics: Interactive Illustrations, Explorations, and Problems for Introductory Physics (Prentice Hall, Upper Saddle River, NJ, 2004).

[viii] M. Belloni, W. Christian and A. J. Cox, Physlet® Quantum Physics: An Interactive Introduction (Prentice Hall, Upper Saddle River, NJ, 2006).

[ix] D. Styer, “Quantum Mechanics: See it Now,” AAPT Kissimmee, FL January, 2000 and available online at:

[x] C. G. Hood, “Teaching about Quantum Theory,” Phys. Teach. 31, 290-293 (1993); A. Hobson, “Teaching Quantum Theory in the Introductory Course,” Phys. Teach. 34, 202-210 (1996); M. Normandeau, “Putting the Humanity Back into Quantum Physics,” Phys. Teach. 43, 524-526 (2005).