Student Difficulties with Energy in Quantum Mechanics

Edward F. Redish and Bao Lei

Department of Physics, University of Maryland
College Park, MD 20742-4111


Pratibha Jolly

University of Delhi

Supported in part by NSF grant DUE-9455561.

Presented 6 January, 1997 at the Phoenix meeting of the AAPT. Posted on the Web on 22 January, 1997.


The Physics Education Research Group at University of Maryland has been studying student learning of quantum mechanics. Our previously reported research shows that student difficulties exist with classical concepts that are prerequisite for learning quantum mechanics. In this talk, we report detailed studies of student difficulties in quantum mechanics arising from confusions with the classical concept of energy. Students are confused on some detailed issues concerning energy and energy diagrams such as the possible value of total classical energy, the meaning of discontinuous potential energy diagrams, and the quantization of energy levels in the quantum case. We have also found new induced confusions on classical issues that are created by students misinterpreting quantum concepts. Proposals for instruction to deal with these issues will be discussed.


A number of factors make it both desirable and possible to teach quantum mechanics to more students and at an earlier stage.

If we are to reach this new audience with these new tools, we must take into account the nature of the audience. (speculation)

To reach these students we need to understand

The UMd PERG has begun preliminary investigations of student response to instruction in QM.

Context: the third semester of a 3 semester calculus-based introductory physics course for engineers

Instruction: approximately 1/3 of the course's lectures (5 weeks) were de-voted to QM. In addition, we developed 3 tutorials.

building potential wells
MBL tutorial with cart moving along a track. Force probe + sonic ranger give F vs. x plot. Students build PE graph and con-sider classical probability.
understanding eigenvalues
Computer simulation (MUPPET). Students solve Schr. eqn. at various energies and find only a few give allowable wave functions.
the shape of the wave function
Computer simulation (MUPPET). Students explore relation between local KE and w. fn. curvature.

Research was based upon

Most students can relate the energy diagram with local velocity and many can make the connections to classical probability density.

Students scored much better than ex-pected on a final exam problem (multiple-choice multiple-response) that asked them to choose classical and quantum probabilities for a particle moving in a complex well.

Exam Problem

In the figure below is shown a plot of a one-dimensional potential energy function U[x] (lower plot) and the wavefunction (upper plot) of an eigenstate for an electron in the influence of that potential. The energy, E, associated with that state is also shown on the energy plot. Distances are measured in nanometers and energies in eV.

1.1 If the particle were moving classically (i.e., its motion were described by Newton's laws) in the potential U[x], and it had an energy E, which of the following statements would be true? List all that apply.

1.2 If the particle were moving quantum mechanically (i.e., its motion were described by the Schrödinger equation) in the potential U[x], and it had an energy E, which of the fol-lowing statements would be true? List all that apply.

Although students scored well on the multiple-choice multiple-response exam question (81%), pre-tests, homeworks, and interviews indicated that even the top students had a number of serious difficulties.

Interview protocol:

Students were (multiple region) formula specifying a potential well similar to the one on the final. They had to


The group of 6 students averaged 92% on the MCMR question. There was much that was reasonable in the interviews, but...

  1. 2/6 students had serious difficulty with the discontinuously defined function.
  2. Only one correctly stated the classically allowed energies (E> -30 eV).
  3. Some thought that the quantum energies could only have negative energies and only 2/6 stated that the energy should be discrete.

Some students had difficulty with the discontinuously defined function.

S: [The vertical lines?] Well, I was really connecting the different energy states. Because I wanted this guy to be continuous....I think they belong there. Yeah...the transition. The transitions from one state to another.

S': The...umm...the borders of the those...classically, an electron can't be there.

S'': Transitions between different states, I believe.

Some students had difficulty interpreting negative energies.

I: For a classical system...are there any values of the total energy that the particle in the system could not have?

S: In a classical system, we can not have negative energies, so you can have any-thing greater than or equal to zero but not anything less. Classically.

S': energy is going to be ... has to be positive [later accepts request to draw a negative energy and interprets motion and prob-ability approximately correctly]

S'':...the whole thing with negative energy was funny to me.

Some students had difficulty under-standing what energies were allowed in a potential energy diagram, both in classical and quantum mechanics. When asked if there were limitations on the allowed energies in QM, only one student gave the correct answer.

S': No, there is no limitation. The limitation again has to be the floor. See you can’t go below the floor. Yeah, which would mean you can’t go below -30. So the same limitation as in the classical case.

S'': It can have ..… negative or positive ….. well it can, any where between zero and negative thirty. But it couldn’t have a positive value.


Stay tuned!

This page prepared by

Edward F. Redish

University of Maryland
Physics Department
College Park, MD 20742-4111
(301) 405-6120