Developing and evaluating an interactive tutorial on degenerate perturbation theory

We discuss an investigation of student difficulties with degenerate perturbation theory (DPT) carried out in advanced quantum mechanics courses by administering free-response and multiple-choice questions and conducting individual interviews with students. We find that students share many common difficulties related to this topic. We used the difficulties found via research as resources to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) which strives to help students develop a functional understanding of DPT. We discuss the development of the DPT QuILT and its preliminary evaluation in the undergraduate and graduate courses.


I. INTRODUCTION
Quantum mechanics (QM) is a particularly challenging subject for upper-level undergraduate and graduate students in physics [1][2][3][4]. Guided by research studies conducted to identify student difficulties with QM and findings of cognitive research, we have been developing a set of researchbased learning tools including the Quantum Interactive Learning Tutorials (QuILTs) [5][6][7]. Here, we discuss an investigation of student difficulties with degenerate perturbation theory (DPT) and the development and evaluation of a researchbased QuILT that makes use of student difficulties as resources to help them develop a solid grasp of DPT.
Perturbation theory (PT) is a powerful approximate method for finding the energies and the energy eigenstates for a system for which the Time-Independent Schrödinger Equation (TISE) is not exactly solvable. The HamiltonianĤ for the system can be expressed as the sum of two terms, the unperturbed HamiltonianĤ 0 and the perturbationĤ ′ , i.e.,Ĥ =Ĥ 0 + ǫĤ ′ with ǫ ≪ 1. The TISE for the unperturbed Hamiltonian, H 0 ψ 0 n = E 0 n ψ 0 n , is exactly solvable where ψ 0 n is the n th unperturbed energy eigenstate and E 0 n the unperturbed energy. PT builds on the solutions of the TISE for the unperturbed case. Using PT, the energies can be approximated as E n = E 0 n +E 1 n +E 2 n +· · · where E i n for i = 1, 2, 3.. are the i th order corrections to the n th energy of the system. The energy eigenstate can be approximated as ψ n = ψ 0 n + ψ 1 n + ψ 2 n + · · · where ψ i n are the i th order corrections to the n th energy eigenstate. The tutorial focuses on the following first order perturbative corrections to the energies and energy eigenstates, which are usually the dominant corrections: is a complete set of eigenstates ofĤ 0 . When the eigenvalue spectrum ofĤ 0 has degeneracy (two or more eigenstates ofĤ 0 have the same energy, i.e., two or more diagonal elements ofĤ 0 are equal), Eq. 1 from nondegenerate perturbation theory (NDPT) is still valid provided one uses a good basis. For a givenĤ 0 andĤ ′ , we define a good basis as consisting of a complete set of eigenstates ofĤ 0 that diago-nalizesĤ ′ in each degenerate subspace ofĤ 0 and keepsĤ 0 diagonal. In a good basis,Ĥ ′ is diagonal in each degenerate subspace ofĤ 0 . Therefore, the terms ψ 0 m |Ĥ ′ |ψ 0 n in Eq. 1 for the wavefunction are zero when m = n so that the expression for the corrections to the wavefunction in Eq. 1 do not have terms that diverge. In a good basis, Eq. 1 is also valid for finding the first order corrections to the energies (which are the diagonal elements of theĤ ′ matrix as given by Eq. 1).
Degenerate perturbation theory is challenging for students since not only does it require an understanding of QM but also requires a strong background in linear algebra. Students should understand that when the originally chosen basis is not a good basis for a givenĤ 0 andĤ ′ , but consists of a complete set of eigenstates ofĤ 0 , a good basis can be constructed from the originally chosen basis states by diagonalizing theĤ ′ matrix in each degenerate subspace ofĤ 0 . Students must also be able to identify degenerate subspaces ofĤ 0 , understand why bothĤ 0 andĤ ′ should be examined carefully to determine if the original basis is good and that they should only diagonalize theĤ ′ matrix in each degenerate subspace ofĤ 0 (instead of diagonalizing the entireĤ ′ matrix) to find a good basis if the original basis is not good. They should also understand why such a basis transformation does not change the diagonal nature ofĤ 0 (it is essential that the basis states are eigenstates ofĤ 0 in PT because the corrections to the unperturbed energies are small). After all of these considerations, students can use Eq. 1 to determine the perturbative corrections.

II. STUDENT DIFFICULTIES
Student difficulties with DPT were investigated using responses from 32 upper-level undergraduate students' to openended and multiple-choice questions administered after traditional instruction in relevant concepts and responses from 10 students' during individual think-aloud interviews. Below, we discuss three of the common student difficulties with DPT found via research in the context of a three dimensional Hilbert space with a two-fold degeneracy. In particular, we find that when students are given the Hamiltonian for a system, they have difficulty correctly (1) identifyingĤ ′ in the degenerate subspace ofĤ 0 , (2) identifying whether the originally chosen basis is a good basis for finding the perturbative corrections, and (3) then finding a good basis if the originally chosen basis is not already a good basis.
A. Difficulty identifyingĤ ′ in the degenerate subspace ofĤ 0 given the HamiltonianĤ: Many students had difficulty identifying theĤ ′ matrix in the degenerate subspace of H 0 , when the HamiltonianĤ for the system was provided in a matrix form. In particular, many students did not understand that in order to determineĤ ′ in the degenerate subspace of Ĥ 0 , they should start by identifying whether there is degeneracy in the energy spectrum ofĤ 0 . In fact, we find that many students incorrectly focused on the diagonal elements of the perturbationĤ ′ to determine whether there was degeneracy in the system and whether they should use DPT. However, degeneracy inĤ ′ has nothing to do with whether one should use DPT and whether one should examine that the original basis, in whichĤ 0 andĤ ′ are provided, is good.
B. Difficulty identifying whether the originally chosen basis is a good basis for finding perturbative corrections: A good basis is one that keeps the unperturbed Hamiltonian H 0 diagonal while diagonalizing the perturbationĤ ′ in the degenerate subspace ofĤ 0 . However, several students incorrectly stated that the originally chosen basis is a good basis because it consists of a complete set of eigenstates ofĤ 0 (Ĥ 0 was diagonal in the original basis) without any consideration for whetherĤ 0 had any degeneracy and the implications of the degeneracy inĤ 0 for what should be examined in theĤ ′ matrix before using Eq. 1. Other students only examined the basis in a general manner and did not focus on eitherĤ 0 or H ′ . For example, one student incorrectly stated that the basis is a good basis if "it forms a complete Hilbert space." Another student stated that "the basis vectors should be orthogonal" is the only condition to have a good basis regardless of the fact that the unperturbed HamiltonianĤ 0 had degeneracy in the situation provided.
Furthermore, when students were given a HamiltonianĤ = H 0 +Ĥ ′ in a basis (which consisted of a complete set of eigenstates ofĤ 0 ) and asked if that basis is a good basis, some students had a tendency to focus on eitherĤ 0 orĤ ′ but not both as is necessary to correctly answer the question. For example, during the interview, one student said, "Ĥ ′ must be diagonal in the good basis". Equivalently, another student claimed the basis was not a good basis "sinceĤ ′ has off-diagonal terms in this basis." These types of incorrect responses suggest that students have difficulty with the fact thatĤ ′ should only be diagonal in the degenerate subspace ofĤ 0 . Students with these types of responses focused on diagonalizing the entireĤ ′ matrix (rather than diagonalizingĤ ′ in the degenerate subspace ofĤ 0 ). They did not realize or consider the fact that ifĤ 0 andĤ ′ do not commute,Ĥ 0 may become non-diagonal if the entireĤ ′ matrix is diagonalized .
Moreover, some students had difficulty with the fact that even when the originally chosen basis is not a good basis, it may include some states that are good states (the sub-basis in the degenerate subspace is not good but the sub-basis in the non-generate subspace is good) and can be used to find the perturbative corrections using Eq. 1. When students were asked to identify whether the originally chosen states were good states, roughly one-fourth of students were unable to correctly identify whether each state in the originally chosen basis is a good state or not. For example, during the interview, one students said, "We cannot trust nondegenerate basis states for finding corrections to the energy. We must adjust all the basis states since we can't guarantee any will be the same." The students with this type of response assumed that if the unperturbed Hamiltonian has degeneracy then none of the originally chosen basis states are good states. However, any state belonging to the nondegenerate subspace ofĤ 0 is a good state. Many other students had similar difficulty.
Also, other students struggled with the fact that ifĤ ′ is already diagonal in a degenerate subspace ofĤ 0 in the original basis, the originally chosen basis is a good basis, and Eq. 1 can be used to determine the perturbative corrections without additional work to diagonalizeĤ ′ in the subspace. They attempted to diagonalize a matrix that was already diagonal.
C. Difficulty finding a good basis if the originally chosen basis is not already a good basis for finding the perturbative corrections: Many students struggled to find a good basis if the originally chosen basis was not already a good basis. They did not realize that when the originally chosen basis is not already a good basis and the unperturbed Hamiltonian H 0 and the perturbing HamiltonianĤ ′ do not commute, they must diagonalizeĤ ′ matrix only in the degenerate subspace ofĤ 0 . The most common mistake was diagonalizing the en-tireĤ ′ matrix instead of diagonalizing theĤ ′ matrix only in the degenerate subspace ofĤ 0 . For example, roughly half of students diagonalized the entireĤ ′ matrix. When asked to determine a good basis for a Hamiltonian in whichĤ 0 andĤ ′ do not commute, one interviewed student incorrectly stated, "We must find the simultaneous eigenstates ofĤ 0 andĤ ′ ." This student and many others did not realize that whenĤ 0 andĤ ′ do not commute, we cannot simultaneously diago-nalizeĤ 0 andĤ ′ since they do not share a complete set of eigenstates. Students struggled with the fact that ifĤ 0 and H ′ do not commute, then diagonalizingĤ ′ produces a basis in whichĤ 0 is no longer diagonal. SinceĤ 0 is the dominant term andĤ ′ provides only small corrections, we must ensure that the basis states used to determine the perturbative corrections in Eq. 1 remain eigenstates ofĤ 0 . IfĤ 0 andĤ ′ do commute, it is possible to diagonalizeĤ 0 andĤ ′ simultaneously to find a complete set of shared eigenstates. However, diagonalizingĤ ′ only in the degenerate subspace ofĤ 0 still produces a good basis and in general requires less algebra and fewer opportunities for mistakes. Students had difficulty with these issues partly because they did not have a solid foundation in linear algebra in order to apply it in the context of DPT.
Some students struggled with the fact thatĤ ′ should be diagonalized in the degenerate subspace ofĤ 0 while keep-ingĤ 0 diagonal. For example, one student in the interview stated, "We cannot diagonalize a part ofĤ ′ , we must diagonalize the whole thing." Students, in general, had great difficulty with the fact that the degeneracy in the eigenvalue spectrum ofĤ 0 provides the flexibility in the choice of basis in the degenerate subspace ofĤ 0 , so thatĤ ′ can be diagonalized in that subspace (even ifĤ 0 andĤ ′ do not commute) while keepingĤ 0 diagonal. For example, if we consider the case in whichĤ 0 has a two-fold degeneracy, thenĤ 0 ψ 0 a = E 0 ψ 0 a ,Ĥ 0 ψ 0 b = E 0 ψ 0 b , and ψ 0 a |ψ 0 b = 0 where ψ 0 a and ψ 0 b are normalized degenerate eigenstates of H 0 . Any linear superposition of these two states, say ψ 0 = αψ 0 a + βψ 0 b , with |α| 2 + |β| 2 = 1 must remain an eigenstate ofĤ 0 with the same energy E 0 . Many students struggled to realize that since any linear superposition of the original basis states that correspond to the degenerate subspace ofĤ 0 remains an eigenstate ofĤ 0 , we can choose that special linear superposition that diagonalizesĤ ′ in the degenerate subspace ofĤ 0 .

III. DEVELOPMENT OF THE QUILT
The development of the DPT QuILT started with an investigation of student difficulties via open-ended and multiplechoice questions administered after traditional instruction to advanced undergraduate and graduate students and conducting a cognitive task analysis from an expert perspective of the requisite knowledge [8]. The QuILT strives to help students build on their prior knowledge and addresses common difficulties. After a preliminary version was developed based upon the task analysis [8] and knowledge of common student difficulties, it underwent many iterations among the three researchers and then was iterated several times with three physics faculty members to ensure that they agreed with the content and wording. It was also administered to advanced undergraduate students in individual think-aloud interviews to ensure that the guided approach was effective, the questions were unambiguously interpreted, and to better understand the rationale for student responses. During these semi-structured interviews, students were asked to think aloud while answering the questions. Students first read the questions on their own and answered them without interruptions except that they were prompted to think aloud if they were quiet for a long time. After students had finished answering a particular question to the best of their ability, they were asked to further clarify and elaborate on issues that they had not clearly addressed earlier. Modifications and improvements were made based upon the student and faculty feedback.
The QuILT uses an inquiry-based approach to learning and actively engages students in the learning process. It includes a pretest to be administered in class after instruction in DPT. Then students engage with the tutorial in small groups in class or alone when using it as a self-paced learning tool in homework, and then they are administered a posttest in class. As students work through the tutorial, they are asked to predict what should happen in a given situation. Then, the tutorial strives to provide scaffolding and feedback as needed to bridge the gap between their initial knowledge and the level of understanding that is desired. Students are also provided checkpoints to reflect upon what they have learned and to make explicit the connections between what they are learning and their prior knowledge. They are given an opportunity to reconcile differences between their predictions and the guidance provided in the checkpoints before proceeding further.
In the QuILT, students actively engage with examples involving DPT that are restricted to a three dimensional Hilbert space (with two-fold degeneracy) to allow them to focus on the fundamental concepts without requiring cumbersome calculations that may detract from the focus on why it is important to determine if the original basis is a good basis, and if it is not good, what type of basis transformation must be performed before Eq. 1 can be used to find the corrections. In particular, for a givenĤ 0 andĤ ′ , when there is degeneracy in the eigenvalue spectrum ofĤ 0 , students learn about why all bases are not good even though they may consist of a complete set of eigenstates ofĤ 0 , how to determine if the basis is good and how to change the basis to one which is good (if the original basis is not good) so that Eq. 1 can be used to find the first order corrections.
In the QuILT, students work through different examples in which the same unperturbed HamiltonianĤ 0 is provided and they are asked to identify whether the originally chosen basis is a good basis for a givenĤ ′ . In one example,Ĥ ′ is diagonal in the degenerate subspace ofĤ 0 in the original basis provided and therefore the original basis is a good basis. In another example, the perturbationĤ ′ is not diagonal in the degenerate subspace ofĤ 0 and therefore the original basis is not a good basis. These examples help students learn that consideration of bothĤ 0 andĤ ′ is necessary to determine if the basis is good in DPT. Students are then asked to summarize in their own words why the original basis is a good basis or not.

IV. PRELIMINARY EVALUATION
Once the researchers determined that the QuILT was successful in one-on-one implementation using a think-aloud protocol, it was given to 11 upper-level undergraduates in a second-semester junior/senior level QM course and 19 firstyear physics graduate students in the second-semester of the graduate core QM course. Both undergraduate and graduate students were given a pretest after traditional instruction in relevant concepts in DPT but before working through the tutorial. The undergraduates worked through the tutorial in class for two days and were asked to work on the remainder of the tutorial as homework. The graduate students were given the tutorial as their weekly homework assignment. After working through and submitting the completed tutorial, both groups were given the posttest with questions similar to the pretest but with the degenerate subspace ofĤ 0 being different. The following were the pretest questions: 1. Consider the unperturbed Hamiltonian (a) Write an example of a perturbing HamiltonianĤ ′ in the same basis asĤ 0 such that for thatĤ 0 andĤ ′ , this basis forms a good basis (so that one can use the same expressions that one uses in non-DPT for perturbative corrections). Use ǫ as a small parameter. (b) Write an example of a perturbing HamiltonianĤ ′ in the same basis asĤ 0 such that for thatĤ 0 andĤ ′ , this basis does NOT form a good basis (so that we cannot use the basis for perturbative corrections using Eq. 1).
Use ǫ as a small parameter. with ǫ ≪ 1, determine the first order corrections to the energies. You must show your work.

GivenĤ
ǫ ≪ 1, determine the first order corrections to the energies. You must show your work. Question 1 focuses on student difficulties A and B. In Question 1, students must be able to correctly identify the degenerate subspace ofĤ 0 . They must then determine how to construct anĤ ′ matrix in order to make sure that the basis used to representĤ 0 andĤ ′ in matrix form is a good basis in part (a) and not a good basis in part (b). For question 1(a), in order for the basis to be a good basis, the constructedĤ ′ matrix must be diagonal in the degenerate subspace ofĤ 0 . For question 1(b), in order for the basis not to be a good basis, theĤ ′ matrix must be non-diagonal in the degenerate subspace ofĤ 0 . Question 2 also focuses on student difficulties A and B. Students must first identifyĤ ′ andĤ 0 in the degenerate subspace ofĤ 0 . Once they identifyĤ ′ in the degenerate subspace of H 0 , they must determine whether the originally chosen basis is a good basis. In particular, they must realize that in question 2,Ĥ ′ is diagonal in the degenerate subspace ofĤ 0 and therefore the original basis is a good basis.
Question 3 focuses on student difficulties A, B, and C. Students must first identifyĤ ′ andĤ 0 in the degenerate subspace ofĤ 0 . Once they identifyĤ ′ in the degenerate subspace of H 0 , they must determine whether the originally chosen basis is a good basis. In question 3,Ĥ ′ is not diagonal in the degenerate subspace ofĤ 0 . Thus, the original basis is not a good basis and they must find a good basis for perturbative corrections.
The open-ended questions were graded using rubrics which were developed by the researchers together. A subset of questions was graded separately by them. After comparing the grading, they discussed any disagreements and resolved them with a final inter-rater reliability of better than 90%. Table I shows the performance of undergraduates and graduate students on the pretest and posttest. The pretest was scored for completeness for both groups but the posttest counted differently towards the course grade for the two groups. One reason for why the undergraduates' performance on the posttest is better than that of graduate students may be that the course grade for the posttest was based on correctness for the undergraduates but on completeness for the graduate students. Table I also includes the average gain, G, and normalized gain [9], g. The normalized gain is defined as the posttest percent minus the pretest percent divided by (100-pretest percent). The posttest scores are significantly better than the pretest scores on all of these questions for both groups.
To investigate retention of learning, the undergraduates were given questions 1(a) and 1(b) again as part of their final exam. The final exam was six weeks after students engaged with the tutorial. The average score on question 1(a) was 97.8% and on question 1(b) was 91.0%. In question 1(a), all 11 students provided anĤ ′ matrix that was diagonal in the degenerate subspace ofĤ 0 . In question 1(b), 10 out of 11 students provided anĤ ′ matrix that was not diagonal in the degenerate subspace ofĤ 0 . These results are encouraging.

V. SUMMARY
Using the common difficulties of advanced students with DPT as resources, we developed and evaluated a researchbased QuILT which focuses on helping students reason about and find the perturbative corrections using DPT. It strives to provide appropriate scaffolding and feedback using a guided inquiry-based approach to help students develop a functional understanding of DPT. The preliminary evaluation shows that the QuILT was effective in improving undergraduate and graduate students' understanding of the fundamentals of DPT.