Student difficulties with the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect
aa r X i v : . [ phy s i c s . e d - ph ] J un Student difficulties with the corrections to the energy spectrum of the hydrogen atom for theintermediate field Zeeman effect
Emily Marshman, Christof Keebaugh, and Chandralekha Singh
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260
We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogenatom for the intermediate field Zeeman effect using the degenerate perturbation theory. The investigation wascarried out in advanced quantum mechanics courses by administering free-response and multiple-choice ques-tions and conducting individual interviews with students. We find that students share many common difficultiesrelated to relevant physics concepts. In particular, students often struggled with mathematical sense-making inthis context of quantum mechanics which requires interpretation of the implications of degeneracy in the un-perturbed energy spectrum and how the Zeeman perturbation will impact the splitting of the energy levels. Wediscuss how the common difficulties often arise from the fact that applying linear algebra concepts correctly inthis context with degeneracy in the energy spectrum is challenging for students.
I. INTRODUCTION AND BACKGROUND
Quantum mechanics (QM) is challenging even for upper-level undergraduate and graduate students and students oftenstruggle with the non-intuitive subject matter and in makingconnections between mathematics and QM concepts (e.g., seeRefs. [1–10]). Prior research studies have found that studentshave difficulty connecting and applying mathematics correctlyin introductory physics contexts (e.g., see Refs. [11–13]).Mathematical sense-making in the context of solving physicsproblems can often be more difficult than when solving equiv-alent mathematics problems without the physics context [11–13]. Since working memory is constrained to a limited num-ber of chunks and students’ knowledge chunks pertaining tophysics concepts are small when they are developing exper-tise, use of mathematics in physics can increase the cognitiveload during problem solving especially if students are not pro-ficient in the mathematics involved [14]. Therefore, studentsmay struggle to integrate mathematics and physics concepts.Since mathematical sense-making while focusing on solvinga physics problem is often more challenging, students some-times make mathematical mistakes that they otherwise wouldnot make if the physics context was absent [11–13].One QM concept that involves connecting mathematics to aphysical situation is degenerate perturbation theory (DPT) inthe context of the Zeeman effect. We investigated student dif-ficulties with finding the first-order corrections to the energiesof the hydrogen atom for the Zeeman effect using DPT, whichincluded probing of difficulties in mathematical sense makingin this QM context so that the research can be used as a guideto develop learning tools to improve student understanding.While the solution for the Time-Independent Schr¨odingerEquation (TISE) for the hydrogen atom with Coulomb po-tential energy can be obtained exactly, the TISE involvingthe Zeeman effect must also include the fine structure correc-tion and cannot be solved exactly. However, since the fine-structure and, in general, the Zeeman corrections to the en-ergies are significantly smaller than the unperturbed energies,perturbation theory is an excellent method for computing thecorrections to the energies. The high degree of symmetry ofthe unperturbed Hamiltonian leads to degeneracy in the en-ergy spectrum and DPT must be used to find the perturbative corrections for the Zeeman effect.The Hamiltonian ˆ H for the system can be expressed as thesum of two terms, the unperturbed Hamiltonian ˆ H and theperturbation ˆ H ′ , i.e., ˆ H = ˆ H + ˆ H ′ . The TISE for theunperturbed Hamiltonian, ˆ H ψ n = E n ψ n , is exactly solv-able, where ψ n is the n th unperturbed energy eigenstate and E n is the unperturbed energy. The n th energy can be ap-proximated as E n = E n + E n + E n + . . . where E in for i = 1 , , .. are the i t h order corrections to the n t h energyof the system. In perturbation theory, the first-order correc-tions to the energies are E n = h ψ n | ˆ H ′ | ψ n i and the first-order corrections to the unperturbed energy eigenstates are | ψ n i = P m = n h ψ m | ˆ H ′ | ψ n i ( E n − E m ) | ψ m i , in which (cid:8) | ψ n i (cid:9) is a com-plete set of eigenstates of the unperturbed Hamiltonian ˆ H . Ifthe eigenvalue spectrum of ˆ H has degeneracy, the perturba-tive corrections are only valid provided one uses a good basis[15]. For a given ˆ H and ˆ H ′ , a good basis consists of a com-plete set of eigenstates of ˆ H that diagonalizes ˆ H ′ in eachdegenerate subspace of ˆ H .Using standard notations, the unperturbed Hamiltonian ˆ H of a hydrogen atom is ˆ H = ˆ p m − e πǫ r , which accountsonly for the interaction of the electron with the nucleus viaCoulomb attraction. The solution for the TISE for the hy-drogen atom with Coulomb potential energy gives the unper-turbed energies E n = − . eV n , where n is the principal quan-tum number. The perturbation is ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z , in which ˆ H ′ Z is the Zeeman term and ˆ H ′ fs is the fine structure term.The Zeeman term is given by ˆ H ′ Z = µ B B ext ¯ h ( ˆ L z + 2 ˆ S z ) inwhich ~B ext = B ext ˆ z is a uniform, time independent externalmagnetic field along the ˆ z -direction, µ B is the Bohr magnetonand ˆ L z and ˆ S z are the operators corresponding to the z com-ponent of the orbital and spin angular momenta, respectively.The fine structure term includes the spin-orbit coupling and arelativistic correction for the kinetic energy, and is expressedas ˆ H ′ fs = ˆ H ′ r + ˆ H ′ SO . Here, ˆ H ′ r = − ˆ p m c is the relativisticcorrection term and ˆ H ′ SO = e πǫ m c r ~S · ~L is the spin-orbitinteraction term (all notations are standard).We note that the unperturbed Hamiltonian is sphericallyymmetric since [ ˆ H , ˆ ~L ] = 0 . Therefore, for a fixed n , ˆ H for the hydrogen atom is diagonal when any complete set oforthogonal states is chosen for the angular part of the basis(consisting of the product states of orbital and spin angularmomenta). Thus, so long as the radial part of the basis is al-ways chosen to be a stationary state wavefunction R nl ( r ) forthe unperturbed hydrogen atom (for given principal and az-imuthal quantum numbers n and l ), which we will assumethroughout, the choice of a good basis amounts to choosingthe angular part of the basis (angular basis) appropriately, i.e.,ensuring that the perturbation is diagonal in each degeneratesubspace of ˆ H . Therefore, we focus on the angular basis tofind a good basis and the corrections to the energies for theZeeman effect. For the angular basis for each n , states in thecoupled representation | l, j, m j i are labeled by the quantumnumbers l, j , and m j (they are eigenstates of ˆ J and ˆ J z ) andthe total angular momentum is defined as ~J = ~L + ~S (all no-tations are standard and s has been suppressed from the states | l, j, m j i since s = 1 / is a fixed value for a hydrogen atom).On the other hand, states | l, m l , m s i in the uncoupled repre-sentation are labeled by the quantum numbers l, m l , and m s (notations are standard) and are eigenstates of ˆ L z and ˆ S z .An angular basis consisting of states in the coupledrepresentation forms a good basis for the fine structure term ˆ H ′ fs since with this choice of the angular basis, ˆ H ′ fs isdiagonal in each degenerate subspace of ˆ H . But a basisconsisting of states in the uncoupled representation forms a good basis for the Zeeman perturbation ˆ H ′ Z . (In this casewith ˆ H ′ Z only, first order PT yields the exact result since [ ˆ H , ˆ H ′ Z ] = 0 .) Therefore, for the intermediate field Zeemaneffect, in which ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z and ˆ H ′ fs and ˆ H ′ Z are treatedon equal footing (i.e., energy corrections due to the two termsare comparable E ′ fs ≈ E ′ Z ), neither a basis consisting ofstates in the coupled representation nor a basis consisting ofstates in the uncoupled representation forms a good angularbasis to find perturbative corrections for the hydrogen atomplaced in an external magnetic field. For example, in a basisconsisting of states in the coupled representation ( | l, j, m j i ),the perturbation matrix ˆ H ′ = ˆ H ′ Z + ˆ H ′ fs corresponding tothe n = 2 subspace is given below (in which γ = (cid:0) α (cid:1) α = e πǫ ¯ hc , β = µ B B ext and the basis states are chosenin the order | , , i , | , , − i , | , , i , | , , − i , | , , i , | , , i , | , , − i , and | , , − i ): ˆ H ′ = γ − β γ + β γ − β γ + 2 β γ − β √ β √ β γ − β γ + β √ β √ β γ + β . The following procedure describes what students should beable to do to determine a good basis and find the first ordercorrections to the energy spectrum for the Zeeman effect: (1)choose an initial basis consisting of a complete set of eigen-states of ˆ H , (2) write the ˆ H and ˆ H ′ matrices in the initiallychosen basis, (3) recognize ˆ H ′ in each degenerate subspace of ˆ H , (4) diagonalize the ˆ H ′ matrix in each degenerate sub-space of ˆ H to determine a good basis, and (5) identify andbe able to explain why the first-order corrections to the energyspectrum are the diagonal matrix elements of the ˆ H ′ matrixgiven by E n = h ψ n | ˆ H ′ | ψ n i in the good basis. II. METHODOLOGY
Student difficulties with the corrections to the energies ofthe hydrogen atom for the Zeeman effect using DPT were in-vestigated using five years of data involving responses from64 upper-level undergraduate students and 42 first-year grad-uate students to open-ended and multiple-choice questions ad-ministered in-class after traditional instruction in relevant con-cepts. 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 13individual think-aloud interviews (a total of 45 hours) withundergraduate and graduate students following the completionof their quantum mechanics courses. Students were providedwith all relevant information discussed in the introduction sec-tion and had lecture-based instruction in relevant concepts.Similar percentages of undergraduate and graduate studentsdisplayed difficulties with DPT.We first analyzed responses of 32 undergraduates on ques-tions related to DPT in the context of the Zeeman effect for thehydrogen atom administered in two previous years. Then, weexamined the difficulties that 32 undergraduate and 42 gradu-ate students had with identifying a good basis for the Zeemaneffect in the following three years as part of an in-class quiz af-ter traditional lecture-based instruction. The following ques-tion is representative of a series of questions that were posedafter traditional lecture-based instruction on relevant concepts(the operator ˆ H ′ , in Q1, is a proxy for the operators ˆ H ′ fs , ˆ H ′ Z ,and ˆ H ′ fs + ˆ H ′ Z that were listed individually in three separatequestions): Q1.
A perturbation ˆ H ′ acts on a hydrogen atom with the un-perturbed Hamiltonian ˆ H = − ¯ h m ∇ − e πǫ (cid:0) r (cid:1) . For theperturbation Hamiltonian ˆ H ′ , circle ALL of the representa-tions that form the angular part of a good basis and explainyour reasoning. Assume that for all cases the principal quan-tum number is fixed to n = 2 .i. Coupled representation,ii. Uncoupled representation,iii. Any arbitrary complete orthonormal basis constructedwith linear combinations of states in the coupled represen-tation with the same l (i.e., states with different l values arenot mixed),iv. Any arbitrary complete orthonormal basis constructedwith linear combinations of states in the uncoupled represen-tation with the same l (i.e., states with different l values arenot mixed),v. Neither coupled nor uncoupled representation. In order to find the perturbative corrections, one must firstchoose a good basis. Q1 focuses on the bases that form a good angular basis for the perturbation ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z , as wells the perturbations ˆ H ′ fs and ˆ H ′ Z individually. Knowledgeof the bases that form a good angular basis for the individualperturbations ˆ H ′ fs and ˆ H ′ Z can be helpful when determining a good basis for ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z .The unperturbed Hamiltonian ˆ H is spherically symmetricwith unperturbed energies only dependent on n and thereforeoptions i, ii, iii, and iv in Q1 all form a complete set of angu-lar eigenstates of ˆ H . Therefore, one must consider which setof basis states in Q1 also diagonalize the given perturbation ˆ H ′ in each degenerate subspace of the unperturbed Hamilto-nian ˆ H . In each degenerate subspace of ˆ H , the fine struc-ture term ˆ H ′ fs is diagonal if the basis is chosen to consist ofstates in the coupled representation (option i in Q1) and theZeeman term is diagonal if the basis is chosen to consist ofstates in the uncoupled representation (option ii in Q1), butnot vice versa. Therefore, for the intermediate field Zeemaneffect with ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z , neither a basis consisting ofstates in the coupled representation nor a basis consisting ofstates in the uncoupled representation forms a good basis andoption v in Q1 is correct. In order to determine a good basis,one may first choose a basis, e.g., consisting of states in eitherthe coupled or uncoupled representation and then diagonalizethe perturbation ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z in the n = 2 degeneratesubspace of the unperturbed Hamiltonian ˆ H . III. STUDENT DIFFICULTIES
Students had some difficulties with DPT in general (not re-stricted to the context of the Zeeman effect only). For exam-ple, when students were asked to determine a good basis forfinding 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 cor-rections to the wavefunction, but they did not use DPT to findthe first-order corrections to the energies. These students oftenincorrectly claimed that they did not need to use DPT since noterms in E n = h ψ n | ˆ H ′ | ψ n i “blow up”. Other students onlyfocused on the Zeeman term ˆ H ′ Z when asked to determine a good basis for finding the corrections to the energies. In par-ticular, they did not take into account the fine structure term ˆ H ′ fs . However, the fine structure term must be consideredwhen determining the corrections to the unperturbed energyspectrum for the Zeeman effect.In response to Q1, students struggled to realize that nei-ther a basis consisting of states in the coupled representationnor a basis consisting of states in the uncoupled representationforms a good basis for the perturbative corrections to the hy-drogen atom placed in an external magnetic field. The resultsare summarized in Table I. Table I shows that only 44% ofundergraduates and 33% of graduate students correctly iden-tified that option v in Q1 is the correct answer for the Zeemaneffect. Additionally, 16% of undergraduate and 17% of grad-uate students did not provide any answer to question Q1 aftertraditional instruction in relevant concepts.Below, we discuss student difficulties in selecting the repre-sentation 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 perturbation ˆ H ′ =ˆ H ′ fs + ˆ H ′ Z after traditional instruction for undergraduates (U) (num-ber of students N = 32 ) and graduate students (G) ( N = 42 ).i ii iii iv v BlankU (%) 28 22 16 13 44 16G (%) 29 17 12 12 33 17 tions to the energy spectrum, based primarily upon responsesduring the think aloud interviews. A. Difficulty understanding why diagonalizing the en-tire ˆ H ′ matrix is problematic: Many students did not realizethat when the initially chosen basis is not a good basis and theunperturbed Hamiltonian ˆ H and the perturbing Hamiltonian ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z do not commute, they must diagonalize the ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z matrix only in each degenerate subspace of ˆ H . When presented with a similar system and asked to de-termine the perturbative corrections, one interviewed studentwho attempted to diagonalize the entire ˆ H ′ matrix justified hisreasoning by incorrectly stating, “We must find the simultane-ous eigenstates of ˆ H and ˆ H ′ .” Discussions suggest that thisstudent, and others with similar difficulties often did not real-ize that when ˆ H and ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z do not commute, wecannot simultaneously diagonalize ˆ H and ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z since they do not share a complete set of eigenstates. Studentsstruggled with the fact that if ˆ H and ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z do notcommute, diagonalizing ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z produces a basis inwhich ˆ H is not diagonal. However, since ˆ H is the dominantterm and ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z provides only small corrections,we must ensure that the basis states used to determine the per-turbative corrections remain eigenstates of ˆ H . B. Incorrectly claiming that BOTH a basis consisting ofstates in the coupled representation and a basis consistingof states in the uncoupled representation are good bases:
In Q1, many students correctly identified that the good basisfor the fine structure term ˆ H ′ fs is a basis consisting of states inthe coupled representation (option i) and also correctly iden-tified that the good basis for the Zeeman term ˆ H ′ Z is a basisconsisting of states in the uncoupled representation (option iiin Q1). However, after correctly identifying the good basis forthe two perturbations individually, some students did not real-ize that neither the coupled nor the uncoupled representation(option v in Q1) forms a good basis for the Zeeman effect inwhich the perturbation is ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z . One interviewedstudent incorrectly claimed that “the coupled are a good ba-sis for ˆ H ′ fs and uncoupled are a good basis for ˆ H ′ Z , so bothcoupled and uncoupled form a good basis for ˆ H ′ fs + ˆ H ′ Z .”This student and others with this type of response thought thatsince a basis consisting of states in the coupled representation(option i in Q1) forms a good basis for the fine structure term ˆ H ′ fs and a basis consisting of states in the uncoupled repre-sentation (option ii in Q1) forms a good basis for the Zeemanterm ˆ H ′ Z , a good basis for the perturbation consisting of thesum of these two perturbations is either a basis consisting oftates in the coupled or uncoupled representation. C. Incorrectly claiming that a good basis does not existfor the Zeeman effect:
Some students argued that good ba-sis does not exist for the intermediate field Zeeman effect andstruggled to realize that the coupled representation or the un-coupled representation are not the only two possibilities forthe angular basis. One interviewed student with this type ofreasoning had difficulty understanding options iii and iv in Q1,stating: “I don’t know what a linear combination of coupledor uncoupled states is. I thought there were just coupled statesor uncoupled states.” This student and others with this typeof reasoning did not realize that a good basis could be con-structed from a linear combination of states in the coupled oruncoupled representation.Some students had difficulty realizing that any linear com-bination of states from the same degenerate subspace of ˆ H are eigenstates of ˆ H . For example, one student who correctlyidentified that neither the coupled nor the uncoupled represen-tation forms a good basis for the Zeeman effect argued that“no good basis exists since we cannot diagonalize a part of the ˆ H ′ matrix ( ˆ H ′ in the degenerate subspace of ˆ H ) without af-fecting the ˆ H matrix.” This student and others who providedsimilar incorrect reasoning claimed that by diagonalizing ˆ H ′ in the degenerate subspace of ˆ H , the ˆ H matrix would nolonger be diagonal. However, due to the degeneracy, any lin-ear combination of states from the same degenerate subspaceof ˆ H are eigenstates of ˆ H . Therefore, diagonalizing ˆ H ′ inthe degenerate subspace of ˆ H determines the special linearcombination that forms a good basis. D. Incorrectly claiming that the choice of the initial ba-sis affects corrections to the energy spectrum:
Of the stu-dents who correctly identified that a good basis for the Zee-man effect consists of special linear combinations of states inthe coupled or uncoupled representation, some did not realizethat the first order corrections to the energy spectrum would bethe same regardless of the initial choice of the basis. A good basis cannot easily be identified at the onset. In order to deter-mine a good basis and the first order corrections to the energyspectrum due to the Zeeman effect, one can initially choosea basis consisting of states in the coupled representation andthen diagonalize ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z in each degenerate subspaceof ˆ H . However, one could also initially choose a basis con-sisting of states in the uncoupled representation and then diag-onalize ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z in each degenerate subspace of ˆ H to determine a good basis and the first order corrections to theenergy spectrum due to the Zeeman effect. Regardless of the choice of the initial basis, after diagonalizing ˆ H ′ = ˆ H ′ fs + ˆ H ′ Z in each degenerate subspace of ˆ H , the first order correctionsto the energy spectrum due to the Zeeman effect will be thesame in any good basis. Many students thought that the firstorder corrections to the energies depend on the initial choiceof basis. Therefore, if one chooses a basis consisting of statesin the coupled representation, then the first order correctionsin this case would be different than those obtained had a basisconsisting of states in the uncoupled representation been cho-sen as the initial basis. However, it does not make sense ex-perimentally that the observed perturbative corrections woulddepend upon the choice of basis. Lack of appropriate connec-tion between physics and mathematics in the context of DPTfor the Zeeman effect sheds light on student epistemology andthe difficulty in mathematical sense-making in QM [11]. IV. SUMMARY AND FUTURE PLAN
Both upper-level undergraduate and graduate studentsstruggled with finding perturbative corrections to the hydro-gen atom energy spectrum for the intermediate field Zeemaneffect using DPT. Interviewed students’ responses suggestedthat some of them held epistemological beliefs inconsistentwith the framework of QM and struggled with mathematicalsense-making in the context of QM in which the paradigm isnovel [4]. After traditional instruction, some students claimedthat different initial choices of the basis before a good ba-sis has been found will yield different corrections to the en-ergy spectrum of the hydrogen atom for the Zeeman effect.These students had difficulty in connecting experimental ob-servations with quantum theory and in correctly reasoning thatsince the corrections to the energy spectrum can be measuredexperimentally, different choices of the initial basis cannotyield different physically observable corrections to the energyspectrum. Since students are still developing expertise in QMand the DPT requires appropriate integration of mathematicaland physical concepts, cognitive overload can be high whilereasoning about these problems [14]. Many advanced studentsfound it challenging to do metacognition [14] in this contextof QM and provided responses that were not consistent witheach other. We are using the difficulties as a guide in develop-ing a Quantum Interactive Learning Tutorial (QuILT) to helpstudents develop a good grasp of these concepts.
ACKNOWLEDGMENTS