Developing and evaluating an interactive tutorial on degenerate perturbation theory

0.08 MB·

by  Christof Keebaugh

Developingandevaluating aninteractivetutorial ondegenerate perturbation theory
Christof Keebaugh, Emily Marshman, and Chandralekha Singh
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260
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
conductingindividualinterviewswithstudents.Wefindthatstudentssharemanycommondifficultiesrelatedto
thistopic.WeusedthedifficultiesfoundviaresearchasresourcestodevelopandevaluateaQuantumInteractive
LearningTutorial(QuILT)whichstrivestohelpstudentsdevelopafunctionalunderstandingofDPT.Wediscuss
thedevelopmentoftheDPTQuILTanditspreliminaryevaluationintheundergraduateandgraduatecourses.
7
1
0 I. INTRODUCTION havetermsthatdiverge.Inagoodbasis,Eq.1isalsovalidfor
2 finding the first order corrections to the energies (which are
Quantum mechanics (QM) is a particularly challenging thediagonalelementsoftheHˆ′matrixasgivenbyEq.1).
n subject for upper-level undergraduate and graduate students
a Degenerateperturbationtheoryis challengingforstudents
in physics [1–4]. Guided by research studies conducted to
J since not only does it require an understanding of QM but
identify student difficulties with QM and findings of cogni-
5 alsorequiresastrongbackgroundinlinearalgebra. Students
tive research, we have been developing a set of research-
shouldunderstandthatwhentheoriginallychosenbasisisnot
basedlearningtoolsincludingtheQuantumInteractiveLearn-
] agoodbasisforagivenHˆ0andHˆ′,butconsistsofacomplete
h ingTutorials(QuILTs)[5–7]. Here,we discussaninvestiga-
p tion of student difficulties with degenerate perturbation the- setofeigenstatesofHˆ0,agoodbasiscanbeconstructedfrom
d- ory(DPT)andthedevelopmentandevaluationofaresearch- theoriginallychosenbasisstatesbydiagonalizingtheHˆ′ma-
e based QuILT that makes use of student difficulties as re- trixineachdegeneratesubspaceofHˆ0. Studentsmustalsobe
. sourcestohelpthemdevelopasolidgraspofDPT. abletoidentifydegeneratesubspacesofHˆ0,understandwhy
s
c Perturbationtheory(PT)isapowerfulapproximatemethod bothHˆ0andHˆ′shouldbeexaminedcarefullytodetermineif
si forfindingtheenergiesandtheenergyeigenstatesforasystem theoriginalbasisisgoodandthattheyshouldonlydiagonal-
y forwhichtheTime-IndependentSchro¨dingerEquation(TISE) izetheHˆ′matrixineachdegeneratesubspaceofHˆ0(instead
h is not exactly solvable. The Hamiltonian Hˆ for the system ofdiagonalizingtheentireHˆ′ matrix)to finda goodbasisif
p
[ can be expressed as the sum of two terms, the unperturbed the original basis is not good. They should also understand
HamiltonianHˆ0andtheperturbationHˆ′,i.e.,Hˆ =Hˆ0+ǫHˆ′ whysuchabasistransformationdoesnotchangethediagonal
1 with ǫ ≪ 1. The TISE for the unperturbed Hamiltonian, natureofHˆ0(itisessentialthatthebasisstatesareeigenstates
5v Hˆ0ψn0 = En0ψn0, is exactly solvable where ψn0 is the nth ofHˆ0 inPTbecausethecorrectionstotheunperturbedener-
0 unperturbed energy eigenstate and En0 the unperturbed en- giesaresmall). Afteralloftheseconsiderations,studentscan
2 ergy. PT builds on the solutions of the TISE for the unper- useEq. 1todeterminetheperturbativecorrections.
1 turbed case. Using PT, the energiescan be approximatedas
0 En =En0+En1+En2+··· whereEni fori=1,2,3..aretheith II. STUDENTDIFFICULTIES
1. ordercorrectionstothenth energyofthesystem. Theenergy
0 eigenstatecanbeapproximatedasψn =ψn0+ψn1+ψn2+··· Student difficulties with DPT were investigated using re-
7 whereψi aretheithordercorrectionstothenthenergyeigen- sponsesfrom32upper-levelundergraduatestudents’toopen-
n
1 state. The tutorial focuses on the following first order per- ended and multiple-choice questions administered after tra-
v: turbative corrections to the energies and energy eigenstates, ditional instruction in relevant concepts and responses from
i whichareusuallythedominantcorrections: 10 students’ during individual think-aloud interviews. Be-
X
low,wediscussthreeofthecommonstudentdifficultieswith
ar InEEqn1.=1,hψ|ψn0n0|Hiˆ′|iψsn0aic,om|ψpn1leit=esmeXt6=onfeh(iψEgm0en0n|sH−ˆta′Et|eψm0sn0o)if|Hψˆm00.i.Wh(e1n) DHfinPildTbethfrotautsnpwdahcveeinawsritetuhsdeeaanrcttwshoain-rfeothgldeivcdeoenngttehenxeetrHoafcayma.itlhItnorenepiaadnritmifcoeurnlaasri,osynwsae-l
(cid:8) (cid:9)
theeigenvaluespectrumofHˆ0 hasdegeneracy(twoormore tem, they have difficulty correctly (1) identifying Hˆ′ in the
eigenstatesofHˆ0 havethesameenergy,i.e.,twoormoredi- degeneratesubspaceofHˆ0,(2)identifyingwhethertheorigi-
agonalelementsofHˆ0 areequal),Eq. 1fromnondegenerate nallychosenbasisisagoodbasisforfindingtheperturbative
perturbationtheory(NDPT)isstillvalidprovidedoneusesa corrections,and(3)thenfindingagoodbasisiftheoriginally
goodbasis. ForagivenHˆ0andHˆ′,wedefineagoodbasisas chosenbasisisnotalreadyagoodbasis.
consistingofa completeset ofeigenstatesof Hˆ0 thatdiago- A. Difficulty identifyingHˆ′ in the degeneratesubspace
nalizesHˆ′ ineachdegeneratesubspaceofHˆ0 andkeepsHˆ0 of Hˆ0 given the Hamiltonian Hˆ: Many students had diffi-
diagonal. Ina goodbasis, Hˆ′ isdiagonalineachdegenerate cultyidentifyingtheHˆ′ matrixinthedegeneratesubspaceof
subspaceof Hˆ0. Therefore,the termshψm0 |Hˆ′|ψn0i in Eq. 1 Hˆ0,whentheHamiltonianHˆ forthesystemwasprovidedina
forthewavefunctionarezerowhenm6=nsothattheexpres- matrixform. Inparticular,manystudentsdidnotunderstand
sion for the correctionsto the wavefunctionin Eq. 1 do not that in order to determine Hˆ′ in the degenerate subspace of
Hˆ0,theyshouldstartbyidentifyingwhetherthereisdegener- if the unperturbedHamiltonian has degeneracythen none of
acyintheenergyspectrumofHˆ0. Infact,wefindthatmany the originally chosen basis states are good states. However,
students incorrectly focused on the diagonalelements of the anystatebelongingtothenondegeneratesubspaceofHˆ0isa
perturbation Hˆ′ to determine whether there was degeneracy goodstate. Manyotherstudentshadsimilardifficulty.
inthesystemandwhethertheyshoulduseDPT.However,de- Also,otherstudentsstruggledwiththefactthatifHˆ′isal-
generacyinHˆ′hasnothingtodowithwhetheroneshoulduse readydiagonalinadegeneratesubspaceofHˆ0intheoriginal
DPTandwhetheroneshouldexaminethattheoriginalbasis, basis, the originally chosen basis is a good basis, and Eq. 1
inwhichHˆ0andHˆ′areprovided,isgood. canbeusedtodeterminetheperturbativecorrectionswithout
B. Difficulty identifying whether the originally chosen additionalwork to diagonalizeHˆ′ in the subspace. They at-
basis is a goodbasis for finding perturbative corrections: temptedtodiagonalizeamatrixthatwasalreadydiagonal.
A good basis is one that keeps the unperturbedHamiltonian C.Difficultyfindingagoodbasisiftheoriginallychosen
Hˆ0 diagonal while diagonalizing the perturbation Hˆ′ in the basis is not already a good basis for finding the pertur-
degeneratesubspaceofHˆ0. However,severalstudentsincor- bative corrections: Many students struggled to find a good
rectly stated that the originally chosen basis is a good basis basisiftheoriginallychosenbasiswasnotalreadyagoodba-
becauseitconsistsofacompletesetofeigenstatesofHˆ0(Hˆ0 sis. Theydidnotrealizethatwhentheoriginallychosenbasis
wasdiagonalintheoriginalbasis)withoutanyconsideration is notalreadya goodbasis andthe unperturbedHamiltonian
for whether Hˆ0 had any degeneracyand the implications of Hˆ0andtheperturbingHamiltonianHˆ′donotcommute,they
thedegeneracyinHˆ0 forwhatshouldbeexaminedintheHˆ′ must diagonalizeHˆ′ matrix only in the degeneratesubspace
matrixbeforeusingEq. 1. Otherstudentsonlyexaminedthe ofHˆ0. Themostcommonmistakewasdiagonalizingtheen-
basis in a generalmannerand did notfocuson either Hˆ0 or tireHˆ′ matrixinsteadofdiagonalizingtheHˆ′ matrixonlyin
Hˆ′. Forexample,onestudentincorrectlystatedthatthebasis thedegeneratesubspaceofHˆ0. Forexample,roughlyhalfof
isagoodbasisif“itformsacompleteHilbertspace.”Another studentsdiagonalizedtheentireHˆ′matrix.Whenaskedtode-
studentstatedthat“thebasisvectorsshouldbeorthogonal”is terminea goodbasisforaHamiltonianinwhichHˆ0 andHˆ′
theonlyconditiontohaveagoodbasisregardlessofthefact do not commute, one interviewed student incorrectly stated,
that the unperturbed Hamiltonian Hˆ0 had degeneracy in the “We must find the simultaneous eigenstates of Hˆ0 and Hˆ′.”
situationprovided. This student and many others did not realize that when Hˆ0
Furthermore,whenstudentsweregivenaHamiltonianHˆ = and Hˆ′ do not commute, we cannot simultaneously diago-
Hˆ0+Hˆ′inabasis(whichconsistedofacompletesetofeigen- nalize Hˆ0 and Hˆ′ since they do not share a complete set of
statesofHˆ0)andaskedifthatbasisisagoodbasis,somestu- eigenstates. Students struggled with the fact that if Hˆ0 and
dentshadatendencytofocusoneitherHˆ0orHˆ′butnotboth Hˆ′ donotcommute,thendiagonalizingHˆ′ producesa basis
asisnecessarytocorrectlyanswerthequestion.Forexample, inwhichHˆ0isnolongerdiagonal.SinceHˆ0isthedominant
duringtheinterview,onestudentsaid,“Hˆ′ mustbediagonal termandHˆ′ providesonlysmallcorrections,wemustensure
inthegoodbasis”. Equivalently,anotherstudentclaimedthe
thatthebasisstatesusedtodeterminetheperturbativecorrec-
basis was nota good basis “since Hˆ′ has off-diagonalterms tions in Eq. 1 remain eigenstates of Hˆ0. If Hˆ0 and Hˆ′ do
inthisbasis.”Thesetypesofincorrectresponsessuggestthat commute,itispossibletodiagonalizeHˆ0 andHˆ′ simultane-
studentshavedifficultywiththefactthatHˆ′shouldonlybedi-
ously to find a completeset of shared eigenstates. However,
agonalinthedegeneratesubspaceofHˆ0. Studentswiththese diagonalizingHˆ′ onlyinthedegeneratesubspaceofHˆ0 still
typesofresponsesfocusedondiagonalizingtheentireHˆ′ma- producesagoodbasisandingeneralrequireslessalgebraand
trix(ratherthandiagonalizingHˆ′ inthedegeneratesubspace feweropportunitiesformistakes. Studentshaddifficultywith
of Hˆ0). They did not realize or consider the fact that if Hˆ0 these issuespartlybecausetheydid nothavea solid founda-
andHˆ′donotcommute,Hˆ0maybecomenon-diagonalifthe tioninlinearalgebrainordertoapplyitinthecontextofDPT.
entireHˆ′matrixisdiagonalized. Some students struggled with the fact that Hˆ′ should be
Moreover, some students had difficulty with the fact that diagonalized in the degenerate subspace of Hˆ0 while keep-
even when the originallychosen basis is nota good basis, it ing Hˆ0 diagonal. For example, one student in the interview
may include some states that are good states (the sub-basis stated, “We cannot diagonalize a part of Hˆ′, we must di-
in the degenerate subspace is not good but the sub-basis in agonalize the whole thing.” Students, in general, had great
the non-generate subspace is good) and can be used to find difficulty with the fact that the degeneracy in the eigen-
theperturbativecorrectionsusingEq. 1. Whenstudentswere value spectrum of Hˆ0 provides the flexibility in the choice
asked to identify whether the originally chosen states were of basis in the degenerate subspace of Hˆ0, so that Hˆ′ can
good states, roughly one-fourth of students were unable to
be diagonalized in that subspace (even if Hˆ0 and Hˆ′ do
correctlyidentifywhethereachstate intheoriginallychosen
not commute) while keeping Hˆ0 diagonal. For example, if
basis is a good state or not. For example, during the inter-
view,onestudentssaid,“Wecannottrustnondegeneratebasis we consider the case in which Hˆ0 has a two-fold degener-
states for finding corrections to the energy. We must adjust acy,thenHˆ0ψa0 =E0ψa0, Hˆ0ψb0 =E0ψb0, and hψa0|ψb0i=0
all the basis states since we can’t guarantee any will be the where ψa0 and ψb0 are normalized degenerate eigenstates of
same.” The students with this type of response assumed that Hˆ0. Any linear superpositionof these two states, say ψ0 =
αψ0+βψ0,with|α|2+|β|2 = 1mustremainaneigenstate itisnotgood,whattypeofbasistransformationmustbeper-
a b
ofHˆ0 withthesameenergyE0. Manystudentsstruggledto formed before Eq. 1 can be used to find the corrections. In
realizethatsinceanylinearsuperpositionoftheoriginalbasis particular, fora givenHˆ0 and Hˆ′, when there is degeneracy
states that correspond to the degenerate subspace of Hˆ0 re- in the eigenvalue spectrum of Hˆ0, students learn about why
mainsaneigenstateofHˆ0, wecanchoosethatspeciallinear allbasesarenotgoodeventhoughtheymayconsistofacom-
superpositionthatdiagonalizesHˆ′inthedegeneratesubspace pletesetofeigenstatesofHˆ0,howtodetermineifthebasisis
ofHˆ0. goodandhowtochangethebasistoonewhichisgood(ifthe
originalbasisisnotgood)sothatEq.1canbeusedtofindthe
firstordercorrections.
III. DEVELOPMENTOFTHEQUILT
IntheQuILT,studentsworkthroughdifferentexamplesin
ThedevelopmentoftheDPTQuILTstartedwithaninves- whichthesameunperturbedHamiltonianHˆ0isprovidedand
tigation of student difficulties via open-ended and multiple- theyareaskedtoidentifywhethertheoriginallychosenbasis
choice questions administered after traditional instruction to is a good basis for a given Hˆ′. In one example, Hˆ′ is diag-
advanced undergraduateand graduate students and conduct- onal in the degenerate subspace of Hˆ0 in the original basis
ing a cognitive task analysis from an expert perspective of
providedand therefore the original basis is a good basis. In
the requisite knowledge [8]. The QuILT strives to help stu- another example, the perturbation Hˆ′ is not diagonal in the
dents build on their prior knowledgeand addresses common degeneratesubspaceofHˆ0 andthereforetheoriginalbasisis
difficulties. Afterapreliminaryversionwasdevelopedbased
notagoodbasis.Theseexampleshelpstudentslearnthatcon-
upon the task analysis [8] and knowledge of common stu-
siderationofbothHˆ0 andHˆ′isnecessarytodetermineifthe
dentdifficulties,itunderwentmanyiterationsamongthethree
basisisgoodinDPT.Studentsarethenaskedtosummarizein
researchers and then was iterated several times with three
theirownwordswhytheoriginalbasisisagoodbasisornot.
physics faculty membersto ensure that they agreed with the
content and wording. It was also administered to advanced
IV. PRELIMINARYEVALUATION
undergraduatestudentsinindividualthink-aloudinterviewsto
ensure that the guided approach was effective, the questions Once the researchersdetermined that the QuILT was suc-
wereunambiguouslyinterpreted,andtobetterunderstandthe cessful in one-on-one implementation using a think-aloud
rationaleforstudentresponses. Duringthesesemi-structured protocol, it was given to 11 upper-level undergraduatesin a
interviews,studentswereaskedtothinkaloudwhileanswer- second-semester junior/senior level QM course and 19 first-
ing the questions. Students first read the questions on their year physicsgraduatestudentsin the second-semesterof the
ownandansweredthemwithoutinterruptionsexceptthatthey graduatecore QM course. Both undergraduateand graduate
were prompted to think aloud if they were quiet for a long students were given a pretest after traditional instruction in
time. Afterstudentshadfinishedansweringaparticularques- relevantconceptsinDPT butbeforeworkingthroughthetu-
tion to the best of their ability, they were asked to further torial.Theundergraduatesworkedthroughthetutorialinclass
clarify and elaborate on issues that they had not clearly ad- fortwodaysandwereaskedtoworkontheremainderofthe
dressed earlier. Modificationsand improvementswere made tutorialas homework. The graduatestudents were given the
baseduponthestudentandfacultyfeedback. tutorialastheirweeklyhomeworkassignment.Afterworking
TheQuILTusesaninquiry-basedapproachtolearningand through and submitting the completed tutorial, both groups
activelyengagesstudentsinthelearningprocess. Itincludes were given the posttest with questions similar to the pretest
apretesttobeadministeredinclassafterinstructioninDPT. butwiththedegeneratesubspaceofHˆ0 beingdifferent. The
Thenstudentsengagewiththetutorialinsmallgroupsinclass followingwerethepretestquestions:
oralonewhenusingitasaself-pacedlearningtoolinhome- 1. ConsidertheunperturbedHamiltonian
work, and then they are administered a posttest in class. As
3 0 0
students work through the tutorial, they are asked to predict Hˆ0 =V00 3 0.
what should happen in a given situation. Then, the tuto-
0 0 7
rial strives to providescaffolding and feedback as needed to
(a) Write an example of a perturbingHamiltonian Hˆ′ in the
bridge the gap between their initial knowledgeand the level
same basis as Hˆ0 such that for that Hˆ0 and Hˆ′, this basis
of understandingthat is desired. Students are also provided
formsagoodbasis(sothatonecanusethesameexpressions
checkpoints to reflect upon what they have learned and to
thatoneusesinnon-DPTforperturbativecorrections). Useǫ
makeexplicittheconnectionsbetweenwhattheyarelearning
asasmallparameter.
andtheir priorknowledge. Theyaregivenan opportunityto
reconciledifferencesbetweentheir predictionsandthe guid- (b) Write an example of a perturbing Hamiltonian Hˆ′ in
anceprovidedinthecheckpointsbeforeproceedingfurther. thesamebasisasHˆ0 suchthatforthatHˆ0 andHˆ′,thisbasis
In the QuILT, students actively engage with examples in- doesNOTformagoodbasis(sothatwecannotusethebasis
volvingDPTthatarerestrictedtoathreedimensionalHilbert forperturbativecorrectionsusingEq. 1).
space (with two-fold degeneracy) to allow them to focus on Useǫasasmallparameter.
thefundamentalconceptswithoutrequiringcumbersomecal- 5 0 −4ǫ
culationsthatmaydetractfromthefocusonwhyitisimpor- 2. Given Hˆ = Hˆ0 +ǫHˆ′ = V0 0 1−4ǫ 0 
tant to determine if the originalbasis is a good basis, and if −4ǫ 0 1+6ǫ
good basis and they must find a good basis for perturbative
TABLE I. Average pretest and posttest scores, gain (G) and nor-
corrections.
malized gain (g) for undergraduate students (number of students
N =11)andgraduatestudents(numberofstudentsN =19). Theopen-endedquestionsweregradedusingrubricswhich
weredevelopedbytheresearcherstogether.Asubsetofques-
UndergraduateStudents GraduateStudents
tions was graded separately by them. After comparing the
Pre Post G g Pre Post G g
grading,theydiscussedanydisagreementsandresolvedthem
1(a) 23.1 100 +76.9 1.00 67.5 88.2 +20.7 0.64
with a finalinter-raterreliability of better than90%. Table I
1(b) 15.4 100 +84.6 1.00 51.3 73.7 +22.4 0.46
2 32.3 100 +67.7 1.00 30.0 94.8 +64.8 0.93 shows the performance of undergraduates and graduate stu-
3 2.6 100 +97.4 1.00 11.7 86.0 +74.3 0.84 dentson the pretest andposttest. The pretestwas scoredfor
completenessforbothgroupsbuttheposttestcounteddiffer-
entlytowardsthecoursegradeforthetwogroups.Onereason
withǫ ≪ 1,determinethefirstordercorrectionstotheener- for why the undergraduates’ performance on the posttest is
gies. Youmustshowyourwork. better than that of graduate students may be that the course
2 ǫ ǫ grade for the posttest was based on correctness for the un-
3. GivenHˆ =Hˆ0+ǫHˆ′ =V0 ǫ 2 ǫ ,with dergraduates but on completeness for the graduate students.
 ǫ ǫ 3  Table I also includes the average gain, G, and normalized
ǫ ≪ 1, determine the first order correctionsto the energies. gain [9], g. The normalized gain is defined as the posttest
Youmustshowyourwork. percentminusthepretestpercentdividedby(100-pretestper-
Question1focusesonstudentdifficultiesAandB.InQues- cent). The posttest scores are significantly better than the
tion 1, studentsmustbe able to correctlyidentifythe degen- pretestscoresonallofthesequestionsforbothgroups.
eratesubspaceofHˆ0. Theymustthendeterminehowtocon- To investigate retention of learning, the undergraduates
structanHˆ′matrixinordertomakesurethatthebasisusedto were given questions 1(a) and 1(b) again as part of their fi-
representHˆ0andHˆ′inmatrixformisagoodbasisinpart(a) nal exam. The final exam was six weeks after students en-
gaged with the tutorial. The average score on question 1(a)
andnotagoodbasisinpart(b).Forquestion1(a),inorderfor
thebasistobeagoodbasis,theconstructedHˆ′matrixmustbe was97.8%andonquestion1(b)was91.0%.Inquestion1(a),
diagonalinthedegeneratesubspaceofHˆ0.Forquestion1(b), all11studentsprovidedanHˆ′matrixthatwasdiagonalinthe
in order for the basis not to be a good basis, the Hˆ′ matrix degenerate subspace of Hˆ0. In question 1(b), 10 out of 11
mustbenon-diagonalinthedegeneratesubspaceofHˆ0. students providedan Hˆ′ matrix that was not diagonalin the
Question2alsofocusesonstudentdifficultiesAandB.Stu- degeneratesubspaceofHˆ0. Theseresultsareencouraging.
dentsmustfirstidentifyHˆ′andHˆ0inthedegeneratesubspace
V. SUMMARY
of Hˆ0. Once they identifyHˆ′ in the degeneratesubspaceof
Hˆ0,theymustdeterminewhethertheoriginallychosenbasis Using the common difficulties of advanced students with
is a goodbasis. In particular,they mustrealize thatin ques- DPT as resources, we developed and evaluated a research-
tion 2, Hˆ′ is diagonalin the degeneratesubspace of Hˆ0 and basedQuILTwhichfocusesonhelpingstudentsreasonabout
thereforetheoriginalbasisisagoodbasis. and find the perturbativecorrectionsusing DPT. It strives to
Question3focusesonstudentdifficultiesA,B,andC.Stu- provideappropriatescaffoldingandfeedbackusing a guided
dentsmustfirstidentifyHˆ′andHˆ0inthedegeneratesubspace inquiry-basedapproachtohelpstudentsdevelopafunctional
of Hˆ0. Once they identifyHˆ′ in the degeneratesubspaceof understandingofDPT.Thepreliminaryevaluationshowsthat
Hˆ0, they must determine whether the originally chosen ba- theQuILTwaseffectiveinimprovingundergraduateandgrad-
uatestudents’understandingofthefundamentalsofDPT.
sis is a good basis. In question 3, Hˆ′ is not diagonalin the
degeneratesubspaceof Hˆ0. Thus, the originalbasisis nota ACKNOWLEDGEMENTS
WethanktheNSFforawardPHY-1505460.
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