Investigating student understanding of the stationary state wavefunction for a system of identical particles 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 spin parts. 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 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 na (xi) nb (xj), where na (xi) and nb (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 quantum number (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., photons 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 classical mechanics (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 jsi; 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, ^2 ^ The states js1; ms1 i are eigenstates of S1 and S1z and the there is no physical observable that would reflect this ^2 ^ states js2; 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: j"i1 = js1; ms i = j1=2; 1=2i1 and j#i1 = occupy the same single-particle quantum state, but two or 1 js1; ms1 i = j1=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 j"i2 = js2; ms i = quantum state. The restriction for fermions is known as the 2 j1=2; 1=2i2; and j#i2 = js2; ms2 i = j1=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 j""i = j"i j"i , consistent with observations, the wavefunction for a system 1 2 j"#i = j"i j#i , j#"i = j#i j"i , and j##i = j#i j#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. j""i, j##i, p1 (j"#i + j#"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 p1 (j"#i − j#"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 (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 (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 ) = p f n (x1) n (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. n2 (x1) n1 (x2)gj"i1j"i2 and Ψ(x1; x2; ms1 ; ms2 ) = (x ) (x ) p1 fj"i j#i − j#i j" ig. 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, jsi; msi i = fj1; −1i; j1; 0i; j1; 1ig 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 j1; 1i1j1; 1i2, possible symmetric spatial states for the two spin-1 bosons j1; 1i1j1; 0i2, j1; 1i1j1; −1i2, j1; 0i1j1; 1i2, j1; 0i1j1; 0i2, are (x1; x2) = n1 (x1) n1 (x2) and (assume n1 6= n2) j1; 0i j1; −1i , j1; −1i j1; 1i , j1; −1i j1; 0i , and (x ; x ) = p1 [ (x ) (x ) + (x ) (x )]: 1 2 1 2 1 2 1 2 2 n1 1 n2 2 n2 1 n1 2 j1; −1i1j1; −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 (x ; x ) = p1 [ (x ) (x )+ (x ) (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.