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Home , Stationary state

Investigating student understanding of the stationary state wavefunction for a system of

Christof Keebaugh Department of Physics and Astronomy, Franklin and Marshall College, Lancaster, PA 17603

Emily Marshman Department of Physics and Astronomy, Community College of Allegheny County, Pittsburgh, PA 15260

Chandralekha Singh Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260

We discuss an investigation of student difficulties with concepts related to the many-particle stationary state wavefunction for a system of non-interacting fermions or bosons in cases in which the many-particle stationary state wavefunction can be written as the product of the spatial and parts. The investigation was carried out in advanced 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 these concepts. Many students struggled to write a many-particle stationary state wavefunction consistent with the symmetrization requirements for the system, i.e., a completely antisymmetric wavefunction for a system of fermions or a completely symmetric wavefunction for a system of bosons. We discuss the common difficulties pertaining to these concepts that can be used as a guide to develop research-validated learning tools.

2020 PERC Proceedings edited by Wolf, Bennett, and Frank; Peer-reviewed, doi.org/10.1119/perc.2020.pr.Keebaugh Published by the American Association of Physics Teachers under a Creative Commons Attribution 4.0 license. Further distribution must maintain the cover page and attribution to the article's authors.

266 I. INTRODUCTION AND BACKGROUND two non-interacting identical particles has terms such as

Quantum mechanics (QM) is a particularly challenging ψna (xi)ψnb (xj), where ψna (xi) and ψnb (xj) are the single- th subject for upper-level undergraduate and graduate students particle wavefunction for the i particle with coordinate xi th in physics as evidenced by many investigations, e.g., see Refs. in the state na and for the j particle with coordinate xj in [1–20]. We have been involved in a number of research the state nb, respectively. studies aimed at investigating student reasoning in QM and If we have a system of two non-interacting identical improving student understanding of QM [21–35]. However, fermions, the two-particle stationary state wavefunction must there have been few investigations into student difficulties be completely antisymmetric. There are two ways to with fundamental concepts involving a system of identical construct a completely antisymmetric wavefunction: the particles. Through researching students’ understanding and spatial part of the wavefunction could be completely reasoning about a system of identical particles, we have symmetric and the spin part of the wavefunction could found many common student difficulties that can hinder be completely antisymmetric or the spatial part of the the development of a consistent and coherent knowledge wavefunction could be completely antisymmetric and the spin structure pertaining to these concepts. part of the wavefunction could be completely symmetric. In nature, there are two general types of particles: fermions If we have a system of two non-interacting identical with a half-integer spin (e.g., electrons and bosons, the two-particle stationary state wavefunction must protons) and bosons with an integer spin quantum number be completely symmetric. There are two ways to construct (e.g., and mesons). A system of N identical particles a completely symmetric wavefunction: the spatial and consists of N particles of the same type (e.g., electrons). For spin parts of the wavefunction could both be completely a system of identical particles in (e.g., symmetric or the spatial and spin parts of the wavefunction five identical tennis balls), each particle can be distinguished could both be completely antisymmetric. from all the other particles. In contrast, in quantum When considering the spin part of the wavefunction for a mechanics, identical particles are indistinguishable and there single-particle, we use the notation |si, msi i (in which si and is no measurement that can be performed to distinguish msi are the quantum numbers corresponding to the total spin these identical particles from one another. For example, if and z-component of the spin for the ith particle, respectively). the coordinates of two identical particles are interchanged, ˆ2 ˆ The states |s1, ms1 i are eigenstates of S1 and S1z and the there is no physical that would reflect this ˆ2 ˆ states |s2, ms2 i are eigenstates of S2 and S2z. We use the interchange. Furthermore, one property that distinguishes following abbreviated notation for a spin-1/2 particle, e.g., these two types of particles is that two or more bosons can an electron: |↑i1 = |s1, ms i = |1/2, 1/2i1 and |↓i1 = occupy the same single-particle , but two or 1 |s1, ms1 i = |1/2, −1/2i1 for electron 1 in the “spin up" more fermions can never occupy the same single-particle and “spin down" state, respectively, and |↑i2 = |s2, ms i = quantum state. The restriction for fermions is known as the 2 |1/2, 1/2i2, and |↓i2 = |s2, ms2 i = |1/2, −1/2i2 for Pauli exclusion principle and is consistent with a system of electron 2 in the “spin up" and “spin down" state, respectively. fermions having a completely antisymmetric wavefunction. When considering the spin part of the wavefunction for To reflect the indistinguishability of these identical particles the two spin-1/2 particles in the uncoupled representation in and make the statistical properties of fermions and bosons the product space, we will use the notation |↑↑i = |↑i |↑i , consistent with observations, the wavefunction for a system 1 2 |↑↓i = |↑i |↓i , |↓↑i = |↓i |↑i , and |↓↓i = |↓i |↓i of identical fermions must be completely antisymmetric and 1 2 1 2 1 2 for the basis states. In order to satisfy the symmetrization the wavefunction for a system of identical bosons must be requirement for a system of two spin-1/2 particles, the completely symmetric. Here we focus on student difficulties spin part of the wavefunction must be either completely with the many-particle stationary state wavefunction that is symmetric or antisymmetric so that the overall wavefunction a solution to the Time-Independent Schrödinger Equation is antisymmetric. In the case of two spin-1/2 particles, (TISE) for a system of non-interacting identical particles. |↑↑i, |↓↓i, √1 (|↑↓i + |↓↑i) are completely symmetric spin Unless otherwise stated, throughout, we refer to the stationary 2 state wavefunction as the wavefunction. states of the two-fermion wavefunction and often referred to as the “triplet" states. The state √1 (|↑↓i − |↓↑i) is the Even though the spatial and spin parts of 2 the wavefunction can be entangled in many completely antisymmetric normalized spin state of the two- situations, we will only consider many-particle fermion wavefunction, often referred to as the “singlet" state. The following are examples of completely antisymmetric wavefunctions Ψ(x1, x2, x3, . . . , ms1 , ms2 , ms3 ,...) = normalized many-particle stationary state wavefunctions ψ(x1, x2, x3,...)χ(ms1 , ms2 , ms3 ,...) in one spatial dimension that can be written as the product of the spatial for a system of two spin-1/2 fermions in which the spatial part of the wavefunction ψ(x1, x2, x3,...) and the spin part of the wavefunction is antisymmetric and the spin part part of the wavefunction χ(ms , ms , ms ,...), in which of the wavefunction is symmetric (or vice-versa, assume 1 2 3 1 n1 6= n2): Ψ(x1, x2, ms , ms ) = √ {ψn (x1)ψn (x2) − xi denotes the spatial coordinate and msi denotes the 1 2 2 1 2 th z-component of spin quantum number of the i particle. ψn2 (x1)ψn1 (x2)}|↑i1|↑i2 and Ψ(x1, x2, ms1 , ms2 ) = ψ (x )ψ (x ) √1 {|↑i |↓i − |↓i |↑ i}. The spatial part of the wavefunction of a system of n1 1 n1 2 2 1 2 1 2

267 For a spin-1 boson, |si, msi i = {|1, −1i, |1, 0i, |1, 1i} bosons must be completely symmetric. Since the spin part for each particle. When considering the spin part of the of the wavefunction given in Q1 is symmetric, the spatial wavefunction for two spin-1 particles in the uncoupled part of the wavefunction must also be symmetric to ensure representation in the product space (3 × 3 = 9 that the overall wavefunction is completely symmetric. Two dimensional), we will use the notation |1, 1i1|1, 1i2, possible symmetric spatial states for the two spin-1 bosons

|1, 1i1|1, 0i2, |1, 1i1|1, −1i2, |1, 0i1|1, 1i2, |1, 0i1|1, 0i2, are ψ(x1, x2) = ψn1 (x1)ψn1 (x2) and (assume n1 6= n2) |1, 0i |1, −1i , |1, −1i |1, 1i , |1, −1i |1, 0i , and ψ(x , x ) = √1 [ψ (x )ψ (x ) + ψ (x )ψ (x )]. 1 2 1 2 1 2 1 2 2 n1 1 n2 2 n2 1 n1 2 |1, −1i1|1, −1i2 for the basis states. Q2. Write one possible spin part of the wavefunction The wavefunction for a system of identical bosons for two electrons if the spatial part of the wavefunction is must be completely symmetric. The spatial and ψ(x , x ) = √1 [ψ (x )ψ (x )+ψ (x )ψ (x )]. If it is spin parts of the many-particle stationary state 1 2 2 n1 1 n2 2 n2 1 n1 2 not possible to write a spin part of the wavefunction with the wavefunction can either be both symmetric or both given spatial part of the wavefunction, write “not possible" antisymmetric. For example, the following are examples and state the reason. of completely symmetric many-particle stationary state wavefunctions for a system of two spin-1 bosons: The overall wavefunction for the two electrons must be completely antisymmetric. Since the spatial part of the Ψ(x1, x2, ms1 , ms2 ) = ψn1 (x1)ψn1 (x2)|1, 1i1|1, 1i2 √1 wavefunction given in Q2 is symmetric, the spin part of and Ψ(x1, x2, ms1 , ms2 ) = {ψn1 (x1)ψn2 (x2) − 2 the wavefunction must be antisymmetric to ensure that the ψ (x )ψ (x )} √1 {|1, 1i |1, 0i − |1, 0i |1, 1i } in n2 1 n1 2 2 1 2 1 2 overall wavefunction is completely antisymmetric. The which both the spatial and spin parts of the wavefunction are antisymmetric spin state for the two spin-1/2 fermions is symmetric or antisymmetric (assume n1 6= n2), respectively. χ(m , m ) = √1 [|↑↓i − |↓↑i]. s1 s2 2 II. METHODOLOGY Q3 below was posed as an in-class clicker question to 16 Student difficulties with determining the many-particle undergraduate students in a junior/senior level undergraduate stationary state wavefunction for a system of identical course following instruction on identical fermions or bosons were first investigated using three years particles. The students first answered the question of data involving responses to open-ended and multiple- individually and then answered it a second time after choice questions administered after traditional instruction in discussing the question with their peers in small groups. relevant concepts from 57 upper-level undergraduate students Q3. Choose all of the following statements that are correct in a junior/senior level QM course and 30 graduate students about bosons. (1) The spin of a boson is an integer. (2) in the second semester of the graduate core QM course. The overall wavefunction of identical bosons can be anti- Additional insight concerning these difficulties was gained symmetric. (3) Two bosons cannot occupy the same single- from responses of 14 students during a total of 81 hours of particle state. individual “think-aloud" interviews. Only option (1) is correct for question Q3. Option (2) We discuss student responses to several questions that were is incorrect because the overall wavefunction for a system posed either as in class clicker questions or as open-ended of identical bosons MUST be symmetric and option (3) is quiz questions. Additional insight into these difficulties incorrect because two or more bosons can occupy the same was gleaned during the individual think-aloud interviews single-particle state. in which students were asked questions pertaining to these Question Q4 was posed during the think-aloud interviews issues. To probe whether students are able to identify to investigate students’ proficiency at identifying whether the and generate a many-particle stationary state wavefunction spin part of a wavefunction is a symmetric or antisymmetric including spin, the following two questions were posed to wavefunction. The question focuses on a system of two the students. Questions Q1 and Q2 were posed on a quiz spin-1/2 particles (s = 1/2, s = 1/2). The students following traditional instruction on concepts involving a 1 2 were familiar with the shorthand notation |↑↑i = |↑i |↑i , system of identical particles to 30 graduate students and 25 1 2 |↑↓i = |↑i |↓i , |↓↑i = |↓i |↑i , and |↓↓i = |↓i |↓i . undergraduate students. Students were told that the particles 1 2 1 2 1 2 Q4. For the spin part of the wavefunction (spin state) of a are confined in one spatial dimension and that ψn1 , ψn2 , etc., are the single-particle stationary state wavefunctions. two-particle system given below, identify whether the spin Below, we discuss the questions students were asked: Q1. state is symmetric, antisymmetric, or neither symmetric nor Write one possible spatial part of the wavefunction for antisymmetric with respect to exchange of the two particles. two indistinguishable spin-1 bosons if the spin part of Explain your reasoning. (a) |↑↑i (b) |↓↓i (c) |↑↓i (d) √1 (|↑↓i + |↓↑i) (e) √1 (|↑↓i − |↓↑i) the wavefunction (expressed in terms of the uncoupled 2 2 representation) is χ(m , m ) = √1 [|1 1i |1 0i + s1 s2 2 1 2 In Q4, options (a), (b), and (d) are symmetric spin states |1 0i1|1 1i2]. If it is not possible to write a spatial part of (triplet states) since exchanging the two particles results in the wavefunction with the given spin part of the wavefunction, the same state. Option (e) in Q4 is an antisymmetric spin state write “not possible" and state the reason. (singlet state) since exchanging the two particles results in the The overall wavefunction for the two indistinguishable original state multiplied by -1. Option (c) in Q4 is neither a symmetric nor antisymmetric spin state.

268 a two-particle state consisting of product states of spatial TABLE I. The percentages of graduate (N = 30) and undergraduate states involving anti-symmetric spatial wavefunction and (N = 25) students who correctly answered questions Q1 and Q2 after traditional instruction. symmetric spin states. Question Type of Particle Graduate (%) Undergraduate (%) B. Difficulty with the symmetrization requirements for Q1 Bosons 33 48 the overall many-particle stationary state wavefunction: Nature demands that the many-particle wavefunction Q2 Electrons 30 44 for a system of indistinguishable bosons be completely symmetric and the many-particle wavefunction for a system III. STUDENT DIFFICULTIES of indistinguishable fermions be completely antisymmetric. Many students struggled to recognize and generate the Therefore, in order to identify and generate a many- completely symmetric many-particle wavefunction for a particle wavefunction for a system of indistinguishable system of indistinguishable bosons in question Q1 and particles, students must be able to determine a completely the completely antisymmetric many-particle wavefunction symmetric/antisymmetric wavefunction involving both for a system of indistinguishable fermions in question spatial and spin degrees of freedom. However, some students Q2. TableI summarizes the percentages of students who struggled to correctly identify the appropriate symmetrization answered questions Q1 and Q2 correctly for a system of requirement for a system of identical particles. Other students two indistinguishable fermions or bosons after traditional had difficulty determining whether the spatial part of the instruction. wavefunction or the spin part of the wavefunction is Below, we discuss several of difficulties students had symmetric or antisymmetric when considering each part of that interfered with their ability to write the completely the wavefunction separately. In interviews, students with this symmetric/antisymmetric many-particle stationary state type of difficulty often were not able to correctly identify wavefunction for a system of indistinguishable particles. the symmetry of the overall wavefunction that included both A. Difficulty applying Pauli’s exclusion principle the spatial and spin parts of the wavefunction. Lastly, some correctly for a system of identical fermions: Some students students struggled to construct the overall many-particle struggled to realize that Pauli’s exclusion principle states stationary state wavefunction due to the fact that it is the that no two fermions can be in the same single-particle product of the spatial and spin parts. state. Students with this type of difficulty often incorrectly C. Difficulty identifying the correct symmetrization overgeneralized the Pauli exclusion principle to state that requirement for a system of identical particles: Students no two fermions can occupy the same spatial state or the also had difficulty identifying that the many-particle same spin state. Often students had difficulty realizing that wavefunction for a system of identical bosons must be two fermions can be in the same spatial state if they are completely symmetric. For example, in response to Q3, in different spin states or vice-versa, so that the overall 43% of the undergraduate students incorrectly answered that wavefunction is antisymmetric. During the interview, several the overall wavefunction of identical bosons can be anti- students incorrectly applied the Pauli exclusion principle symmetric (option (2)). Even after peer discussion, 31% when considering a separable many-particle stationary state again incorrectly selected option (2) as correct. Written wavefunction that can be expressed as the direct product explanations and interviews suggest that these students of the spatial and the spin parts of the wavefunction. For knew that the many-particle wavefunction for a system example, one interviewed student correctly stated that no of identical particles (bosons or fermions) must obey a two fermions can be in the same single-particle state, but symmetrization requirement, but could not correctly identify then went on to incorrectly claim that “this means two which symmetrization requirement corresponds to which fermions could not exist in the same single-particle stationary particle. For example, one interviewed student in response to state (pointing to a case in which they were in the same questions Q1 and Q2 stated that “there is a symmetrization spatial state)". This student and others with this type of requirement for fermions and a different symmetrization difficulty often had difficulty realizing that if the two spin- requirement for bosons." But the student was unable to 1/2 fermions are in different spin states then it is possible recognize the appropriate symmetrization requirements and for the two fermions to be in the same single-particle spatial wrote the same symmetric wavefunction for both a system state, i.e. ψn1 (x1)ψn1 (x2) is a possible spatial part of the of indistinguishable fermions and indistinguishable bosons. wavefunction. In response to Q1, some students attempted to generate By a similar argument, several students incorrectly claimed a completely antisymmetric wavefunction for a system of that two fermions could not exist in the same spin state as identical bosons. Similarly, in response to Q2, some students this too would violate the Pauli exclusion principle. For generated a completely symmetric wavefunction for a system example, some interviewed students claimed that |↑↑i and of identical fermions. Additionally, 67% of the graduates |↓↓i were not possible spin states for the system of two and 28% of undergraduate students incorrectly provided a spin-1/2 fermions regardless of whether the two fermions symmetric spin part of the wavefunction in response to Q2 were in different single-particle spatial states. These students resulting in an overall symmetric many-particle wavefunction did not consider the fact that the two fermions could be in for the two electrons.

269 Identifying the correct symmetrization requirement for wavefunctions is a completely symmetric wavefunction. the overall many-particle stationary state wavefunction is During the interviews, some students incorrectly claimed challenging due in part to the fact that one must consider that in order for the overall many-particle stationary state the symmetry of both the spatial and spin parts of the wavefunction to be antisymmetric, both the spatial and spin wavefunction before determining the overall symmetry of parts of the wavefunction must be antisymmetric. the many-particle stationary state wavefunction. This F. Difficulty identifying that the spatial and spin can be confusing for students who simply memorized the parts of the wavefunction can both be antisymmetric appropriate combinations of the spatial and spin parts of the for a system of identical bosons: Some students had wavefunction rather than developing an understanding of the difficulty realizing that both the spatial and spin parts of the symmetry of a wavefunction comprised of two separate parts. wavefunction can be antisymmetric to produce an overall D. Incorrectly determining the symmetry based on symmetric wavefunction for a system of identical bosons. the appearance of a +/- sign in the many-particle A number of students correctly reasoned that the overall wavefunction: Some students incorrectly applied a heuristic many-particle wavefunction for a system of indistinguishable by which they claimed that a wavefunction is symmetric bosons must be completely symmetric. However, some if the wavefunction is written in terms of a sum. These of these students went on to incorrectly claim that both students simply looked for all “+" signs to determine that the spatial part and spin part of the wavefunction must be a wavefunction is symmetric. They claimed that any symmetric. These students did not realize that a many- wavefunction written as terms added together is a symmetric particle stationary state wavefunction in which both the wavefunction. By a similar logic, these same students looked spatial part and spin part are antisymmetric is completely for a “-" sign to determine whether a given wavefunction is symmetric. antisymmetric. Some claimed that any wavefunction that had For example, students were asked to construct the spin part at least one negative sign was antisymmetric. For example, of a two-particle stationary state wavefunction for two spin- in response to question Q4(a), one interviewed student 1 bosons whose spatial part of the wavefunction is given by √1 [ψ (x )ψ (x ) − ψ (x )ψ (x )]. One interviewed incorrectly claimed that the spin part of the wavefunction 2 n1 1 n2 2 n2 1 n1 2 given by |↑↑i is neither symmetric nor antisymmetric as “the student incorrectly claimed that “it is not possible to write wavefunction is not a sum so it can’t be symmetric and there the spin part since the spatial part is antisymmetric. There is not a minus sign, so it can’t be antisymmetric." However, is no way to make the whole wavefunction symmetric." This the spin part of the wavefunction given by |↑↑i is symmetric student and many others with this type of difficulty did not as the exchange of the two particles results in the same realize that by choosing an antisymmetric spin part of the wavefunction, thus there need not be a plus sign in order for wavefunction, the overall two-particle wavefunction would a wavefunction to be symmetric. Other students used similar be completely symmetric. reasoning when determining the symmetry of the spin part of the wavefunction. IV. SUMMARY AND FUTURE OUTLOOK Students with this type of difficulty often struggled to Our investigation suggests that upper-level undergraduate write the overall many-particle stationary state wavefunction and graduate students have many common difficulties with for a system of identical particles with the appropriate fundamental concepts involving a system of non-interacting symmetrization requirement. For example, students who identical particles. We are using the difficulties found incorrectly claimed that the spin part of the wavefunction |↑↑i in this context as a guide to develop and validate a is neither symmetric nor antisymmetric often also incorrectly Quantum Interactive Learning Tutorial (QuILT). The QuILT claimed that it is not possible to write a many-particle focuses on helping students learn that the wavefunction for stationary state wavefunction for two fermions with this spin a system of fermions must be completely antisymmetric state. and the wavefunction for a system of bosons must be completely symmetric and then construct the many-particle E. Difficulty identifying that the spatial part is stationary state wavefunction for a system of non-interacting antisymmetric and the spin part is symmetric (or vice identical particles (fermions or bosons) consistent with the versa) for the wavefunction for a system of identical symmetrization requirements. In particular, the QuILT strives fermions: Some students correctly identified that the to help students understand and operationalize completely many-particle stationary state wavefunction for a system symmetric or completely anti-symmetric functions with of identical fermions must be completely antisymmetric, respect to the exchange of any two particles’ coordinates but incorrectly claimed that both the spatial and spin parts and use this knowledge to construct a many-particle wave of the wavefunction must be completely antisymmetric. function consisting of both spatial and spin parts. For example, when asked to determine the spatial part The initial in-class evaluation of the validated QuILT is of the wavefunction for two fermions if the spin part encouraging. of the wavefunction was the singlet state, some students incorrectly claimed that the spatial part of the wavefunction ACKNOWLEDGEMENTS is √1 [ψ (x )ψ (x ) − ψ (x )ψ (x )]. They struggled 2 n1 1 n2 2 n2 1 n1 2 We thank the NSF for award PHY-1806691. to realize that the product of two completely antisymmetric

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