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Student difficulties with the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect

We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect using the degenerate perturbation theory. The investigation was 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 relevant physics concepts. In particular, students often struggled with mathematical sense-making in this context of quantum mechanics which requires interpretation of the implications of degeneracy in the unperturbed energy spectrum and how the Zeeman perturbation will impact the splitting of the energy levels. We discuss how the common difficulties often arise from the fact that applying linear algebra concepts correctly in this context with degeneracy in the energy spectrum is challenging for students.


I. INTRODUCTION AND BACKGROUND
Quantum mechanics (QM) is challenging even for upperlevel undergraduate and graduate students and students often struggle with the non-intuitive subject matter and in making connections between mathematics and QM concepts (e.g., see Refs. [1][2][3][4][5][6][7][8][9][10]). Prior research studies have found that students have difficulty connecting and applying mathematics correctly in introductory physics contexts (e.g., see Refs. [11][12][13]). Mathematical sense-making in the context of solving physics problems can often be more difficult than when solving equivalent mathematics problems without the physics context [11][12][13]. Since working memory is constrained to a limited number of chunks and students' knowledge chunks pertaining to physics concepts are small when they are developing expertise, use of mathematics in physics can increase the cognitive load during problem solving especially if students are not proficient in the mathematics involved [14]. Therefore, students may struggle to integrate mathematics and physics concepts. Since mathematical sense-making while focusing on solving a physics problem is often more challenging, students sometimes make mathematical mistakes that they otherwise would not make if the physics context was absent [11][12][13].
One QM concept that involves connecting mathematics to a physical situation is degenerate perturbation theory (DPT) in the context of the Zeeman effect. We investigated student difficulties with finding the first-order corrections to the energies of the hydrogen atom for the Zeeman effect using DPT, which included probing of difficulties in mathematical sense making in this QM context so that the research can be used as a guide to develop learning tools to improve student understanding.
While the solution for the Time-Independent Schrödinger Equation (TISE) for the hydrogen atom with Coulomb potential energy can be obtained exactly, the TISE involving the Zeeman effect must also include the fine structure correction and cannot be solved exactly. However, since the finestructure and, in general, the Zeeman corrections to the energies are significantly smaller than the unperturbed energies, perturbation theory is an excellent method for computing the corrections to the energies. The high degree of symmetry of the unperturbed Hamiltonian leads to degeneracy in the energy spectrum and DPT must be used to find the perturbative corrections for the Zeeman effect.
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 +Ĥ ′ . The TISE for the unperturbed Hamiltonian,Ĥ 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 is the unperturbed energy. The n th energy 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. In perturbation theory, the first-order corrections to the energies are E 1 n = ψ 0 n |Ĥ ′ |ψ 0 n and the firstorder corrections to the unperturbed energy eigenstates are n is a complete set of eigenstates of the unperturbed HamiltonianĤ 0 . If the eigenvalue spectrum ofĤ 0 has degeneracy, the perturbative corrections are only valid provided one uses a good basis [15]. For a givenĤ 0 andĤ ′ , a good basis consists of a complete set of eigenstates ofĤ 0 that diagonalizesĤ ′ in each degenerate subspace ofĤ 0 . Using standard notations, the unperturbed HamiltonianĤ 0 of a hydrogen atom isĤ 0 =p 2 2m − e 2 4πǫ0 1 r , which accounts only for the interaction of the electron with the nucleus via Coulomb attraction. The solution for the TISE for the hydrogen atom with Coulomb potential energy gives the unperturbed energies E 0 is a uniform, time independent external magnetic field along theẑ-direction, µ B is the Bohr magneton andL z andŜ z are the operators corresponding to the z component of the orbital and spin angular momenta, respectively. The fine structure term includes the spin-orbit coupling and a relativistic correction for the kinetic energy, and is expressed 1 m 2 c 2 r 3 S · L is the spin-orbit interaction term (all notations are standard).
We note that the unperturbed Hamiltonian is spherically symmetric since [Ĥ 0 ,ˆ L] = 0. Therefore, for a fixed n,Ĥ 0 for the hydrogen atom is diagonal when any complete set of orthogonal states is chosen for the angular part of the basis (consisting of the product states of orbital and spin angular momenta). Thus, so long as the radial part of the basis is always chosen to be a stationary state wavefunction R nl (r) for the unperturbed hydrogen atom (for given principal and azimuthal quantum numbers n and l), which we will assume throughout, the choice of a good basis amounts to choosing the angular part of the basis (angular basis) appropriately, i.e., ensuring that the perturbation is diagonal in each degenerate subspace ofĤ 0 . Therefore, we focus on the angular basis to find a good basis and the corrections to the energies for the Zeeman effect. For the angular basis for each n, states in the coupled representation |l, j, m j are labeled by the quantum numbers l, j, and m j (they are eigenstates ofĴ 2 andĴ z ) and the total angular momentum is defined as J = L + S (all notations are standard and s has been suppressed from the states |l, j, m j since s = 1/2 is a fixed value for a hydrogen atom).
On the other hand, states |l, m l , ms in the uncoupled representation are labeled by the quantum numbers l, m l , and m s (notations are standard) and are eigenstates ofL z andŜ z . An angular basis consisting of states in the coupled representation forms a good basis for the fine structure term H ′ f s since with this choice of the angular basis,Ĥ ′ f s is diagonal in each degenerate subspace ofĤ 0 . But a basis consisting of states in the uncoupled representation forms a good basis for the Zeeman perturbationĤ ′ Z . (In this case withĤ ′ Z only, first order PT yields the exact result since [Ĥ 0 ,Ĥ ′ Z ] = 0.) Therefore, for the intermediate field Zeeman effect, in whichĤ ′ =Ĥ ′ f s +Ĥ ′ Z andĤ ′ f s andĤ ′ Z are treated on equal footing (i.e., energy corrections due to the two terms are comparable E ′ f s ≈ E ′ Z ), neither a basis consisting of states in the coupled representation nor a basis consisting of states in the uncoupled representation forms a good angular basis to find perturbative corrections for the hydrogen atom placed in an external magnetic field. For example, in a basis consisting of states in the coupled representation (|l, j, m j ), the perturbation matrixĤ ′ =Ĥ ′ Z +Ĥ ′ f s corresponding to the n = 2 subspace is given below (in which γ = α 8 2 13.6 eV, α = e 2 4πǫ0hc , β = µ B B ext and the basis states are chosen in the order |0, 1 2 The following procedure describes what students should be able to do to determine a good basis and find the first order corrections to the energy spectrum for the Zeeman effect: (1) choose an initial basis consisting of a complete set of eigenstates ofĤ 0 , (2) write theĤ 0 andĤ ′ matrices in the initially chosen basis, (3) recognizeĤ ′ in each degenerate subspace ofĤ 0 , (4) diagonalize theĤ ′ matrix in each degenerate subspace ofĤ 0 to determine a good basis, and (5) identify and be able to explain why the first-order corrections to the energy spectrum are the diagonal matrix elements of theĤ ′ matrix given by E 1 n = ψ 0 n |Ĥ ′ |ψ 0 n in the good basis.

II. METHODOLOGY
Student difficulties with the corrections to the energies of the hydrogen atom for the Zeeman effect using DPT were investigated using five years of data involving responses from 64 upper-level undergraduate students and 42 first-year graduate students to open-ended and multiple-choice questions administered in-class after traditional instruction in relevant concepts. The undergraduates were in an upper-level QM course, and graduate students were in a graduate-level QM course. Additional insight about the difficulties was gained from 13 individual think-aloud interviews (a total of 45 hours) with undergraduate and graduate students following the completion of their quantum mechanics courses. Students were provided with all relevant information discussed in the introduction section and had lecture-based instruction in relevant concepts. Similar percentages of undergraduate and graduate students displayed difficulties with DPT.
We first analyzed responses of 32 undergraduates on questions related to DPT in the context of the Zeeman effect for the hydrogen atom administered in two previous years. Then, we examined the difficulties that 32 undergraduate and 42 graduate students had with identifying a good basis for the Zeeman effect in the following three years as part of an in-class quiz after traditional lecture-based instruction. The following question is representative of a series of questions that were posed after traditional lecture-based instruction on relevant concepts (the operatorĤ ′ , in Q1, is a proxy for the operatorsĤ ′ f s ,Ĥ ′ Z , andĤ ′ f s +Ĥ ′ Z that were listed individually in three separate questions): Q1. A perturbationĤ ′ acts on a hydrogen atom with the unperturbed HamiltonianĤ 0 = −h 2 2m ∇ 2 − e 2 4πǫ0 1 r . For the perturbation HamiltonianĤ ′ , circle ALL of the representations that form the angular part of a good basis and explain your reasoning. Assume that for all cases the principal quantum number is fixed to n = 2. i. Coupled representation, ii. Uncoupled representation, iii. Any arbitrary complete orthonormal basis constructed with linear combinations of states in the coupled representation with the same l (i.e., states with different l values are not mixed), iv. Any arbitrary complete orthonormal basis constructed with linear combinations of states in the uncoupled representation with the same l (i.e., states with different l values are not mixed), v. Neither coupled nor uncoupled representation.
In order to find the perturbative corrections, one must first choose a good basis. Q1 focuses on the bases that form a good angular basis for the perturbationĤ ′ =Ĥ ′ f s +Ĥ ′ Z , as well as the perturbationsĤ ′ f s andĤ ′ Z individually. Knowledge of the bases that form a good angular basis for the individual perturbationsĤ ′ f s andĤ ′ Z can be helpful when determining a good basis forĤ ′ =Ĥ ′ f s +Ĥ ′ Z . The unperturbed HamiltonianĤ 0 is spherically symmetric with unperturbed energies only dependent on n and therefore options i, ii, iii, and iv in Q1 all form a complete set of angular eigenstates ofĤ 0 . Therefore, one must consider which set of basis states in Q1 also diagonalize the given perturbation H ′ in each degenerate subspace of the unperturbed Hamilto-nianĤ 0 . In each degenerate subspace ofĤ 0 , the fine structure termĤ ′ f s is diagonal if the basis is chosen to consist of states in the coupled representation (option i in Q1) and the Zeeman term is diagonal if the basis is chosen to consist of states in the uncoupled representation (option ii in Q1), but not vice versa. Therefore, for the intermediate field Zeeman effect withĤ ′ =Ĥ ′ f s +Ĥ ′ Z , neither a basis consisting of states in the coupled representation nor a basis consisting of states in the uncoupled representation forms a good basis and option v in Q1 is correct. In order to determine a good basis, one may first choose a basis, e.g., consisting of states in either the coupled or uncoupled representation and then diagonalize the perturbationĤ ′ =Ĥ ′ f s +Ĥ ′ Z in the n = 2 degenerate subspace of the unperturbed HamiltonianĤ 0 .

III. STUDENT DIFFICULTIES
Students had some difficulties with DPT in general (not restricted to the context of the Zeeman effect only). For example, when students were asked to determine a good basis for finding the corrections to the energies of the hydrogen atom, some students did not even realize that DPT should be used. Other students knew that they had to use DPT to find the corrections to the wavefunction, but they did not use DPT to find the first-order corrections to the energies. These students often incorrectly claimed that they did not need to use DPT since no terms in E 1 n = ψ 0 n |Ĥ ′ |ψ 0 n "blow up". Other students only focused on the Zeeman termĤ ′ Z when asked to determine a good basis for finding the corrections to the energies. In particular, they did not take into account the fine structure term H ′ f s . However, the fine structure term must be considered when determining the corrections to the unperturbed energy spectrum for the Zeeman effect.
In response to Q1, students struggled to realize that neither a basis consisting of states in the coupled representation nor a basis consisting of states in the uncoupled representation forms a good basis for the perturbative corrections to the hydrogen atom placed in an external magnetic field. The results are summarized in Table I. Table I shows that only 44% of undergraduates and 33% of graduate students correctly identified that option v in Q1 is the correct answer for the Zeeman effect. Additionally, 16% of undergraduate and 17% of graduate students did not provide any answer to question Q1 after traditional instruction in relevant concepts.
Below, we discuss student difficulties in selecting the representation that forms a good basis in Q1 and finding the correc-TABLE I. The percentages of undergraduate (U) and graduate (G) students who chose the options i-v in Q1 for the perturbationĤ ′ = H ′ f s +Ĥ ′ Z after traditional instruction for undergraduates (U) (number of students N = 32) and graduate students (G) (N = 42).