We present a simple technique for evaluating multiple-choice questions and their answers beyond the usual measures of difficulty and the effectiveness of distractors. The technique involves the construction and qualitative consideration of item response curves and is based on item response theory from the field of education measurement. To demonstrate the technique, we apply item response curve analysis to three questions from the Force Concept Inventory. Item response curve analysis allows us to characterize qualitatively whether these questions are efficient, where efficient is defined in terms of the construction, performance, and discrimination of a question and its answer choices. This technique can be used to develop future multiple-choice examination questions and a better understanding of results from existing diagnostic instruments.

Several years ago, we introduced the idea of item response curves (IRC), a simplistic form of item response theory (IRT), to the physics education research community as a way to examine item performance on diagnostic instruments such as the Force Concept Inventory (FCI). We noted that a full-blown analysis using IRT would be a next logical step, which several authors have since taken. In this paper, we show that our simple approach not only yields similar conclusions in the analysis of the performance of items on the FCI to the more sophisticated and complex IRT analyses but also permits additional insights by characterizing both the correct and incorrect answer choices. Our IRC approach can be applied to a variety of multiple-choice assessments but, as applied to a carefully designed instrument such as the FCI, allows us to probe student understanding as a function of ability level through an examination of each answer choice. We imagine that physics teachers could use IRC analysis to identify prominent misconceptions and tailor their instruction to combat those misconceptions, fulfilling the FCI authors’ original intentions for its use. Furthermore, the IRC analysis can assist test designers to improve their assessments by identifying nonfunctioning distractors that can be replaced with distractors attractive to students at various ability levels.

This article investigates the properties of a few interacting particles trapped in a few wells and how these properties change under adiabatic tuning of interaction strength and inter-well tunneling. While some system properties are dependent on the specific shapes of the traps and the interactions, this article applies symmetry analysis to identify generic features in the spectrum of stationary states of few-particle, few-well systems. Extended attention is given to a simple but flexible three-parameter model of two particles in two wells in one dimension. A key insight is that two limiting cases, hard-core repulsion and no inter-well tunneling, can both be treated as emergent symmetries of the few-particle Hamiltonian. These symmetries are the mathematical consequences of infinite barriers in configuration space. They are necessary to explain the pattern of degeneracies in the energy spectrum, to understand how degeneracies are broken for models away from limiting cases, and to explain separability and integrability. These symmetry methods are extendable to more complicated models and the results have practical consequences for stable state control in few-particle, few-well systems with ultracold atoms in optical traps.

We show that in complete agreement with classical mechanics, the dynamics of any quantum mechanical wave packet in a linear gravitational potential involves the gravitational and the inertial mass only as their ratio. In contrast, the spatial modulation of the corresponding energy wave function is determined by the third root of the product of the two masses. Moreover, the discrete energy spectrum of a particle constrained in its motion by a linear gravitational potential and an infinitely steep wall depends on the inertial as well as the gravitational mass with different fractional powers. This feature might open a new avenue in quantum tests of the universality of free fall.

Spectroscopic labels for a few particles with spin that are harmonically trapped in one-dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the non-interacting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multi-component fermions and bosons transform under adiabatic evolution from non-interacting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases N = 3 − 5, are provided.PACS numbers: 03.75. Mn, 05.30.Fk The quantum system of a few particles in one dimension, confined by an external field and interacting via a short-range potential, has been a touchstone model for nuclear, atomic, statistical, and mathematical physics since the 1930's. This system possesses enough symmetry and regularity to be analytically tractable in many cases, but also manifests a rich enough phenomenological structure to serve as physically significant model for realistic systems. This class of models has been a playground for studying the interplay of integrability and solvability, the emergence of universality in few-body phenomena, and the transition from few-body to many-body physics.This article focuses specifically on the quantum system of N particles of mass m confined in a one-dimensional harmonic trap with frequency ω and interacting via an effectively zero-range potential. The Hamiltonian for the model iŝwith the particle coordinates x i expressed in units of /mω. This Hamiltonian has received intense attention recently because it serves as a model for neutral cold atoms with interactions tunable through the manipulation of Feshbach and confinement-induced resonances trapped in extremely prolate, approximately harmonic optical wells [1][2][3][4]. The model has been used to investigate few-body and many-body properties of one-dimensional trapped ultracold gases of bosons [5][6][7][8][9][10][11], fermions [12][13][14][15][16][17][18][19][20][21][22][23], and mixtures of bosons and fermions [24][25][26]. The so-called unitary limit g → ∞ of hard-core repulsive interactions is known as the TonksGirardeau gas and has been realized in the laboratory with bosons [27,28], as has the metastable super-TonksGirardeau limit g → −∞ [29]. Recent experiments in * Electronic address: harshman@american.edu which the state of a few trapped, interacting fermions can be controlled and measured [30][31][32][33] have generated excitement that tunable fermi gases will allow simulation of solid-state phenomena such as the transition to ferromagnetism [18,21,23,34].The solutions of Hamiltonian (1) form a complete basis for the spatial states of N distinguishabl...

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.PACS numbers: 03.67. Mn, 03.65.Nk, The interaction of particle systems via scattering is a fundamental theoretical and experimental paradigm. The quantum information theory of particle scattering is, however, still in its infancy. Results, theoretical and computational, exist for the entanglement between the momenta [1] or the angular momenta [2] of two particles generated in scattering, but many problems remain open. The challenges are partly technical due to the greater complexity of dealing with entanglement in continuous variable systems [3] and partly conceptual as in defining a measure of entanglement that has meaningful properties under space-time symmetry transformations. See, for example, the literature on spin-entanglement of relativistic particles [4,5] where different types of entanglement (between two particles, between two particles' spins, and between a single particle's spin and momentum) have been discussed and occasionally confused.In this letter, we examine how some of these difficulties may be resolved by combining two approaches: (1) the generalized tensor product structures (TPSs) and observable-dependent entanglement developed by Zanardi and others [6], and (2) the representation theory of space-time symmetry groups, which has a long and fruitful history in quantum mechanics. Using these methods, TPSs for single particle and multi-particle systems are explored. These methods allow one to distinguish between TPSs that are symmetry invariant and/or dynamically invariant and TPSs that are not, and, in the latter case, to obtain quantitative expressions for the change of entanglement. The reason why certain TPSs have entanglement measures which are symmetry or dynamically invariant is that the space-time symmetries or the time evolution operator, respectively, act as a product of local unitaries with respect to these TPSs.As an application of these general concepts and methods, we will study non-relativistic elastic scattering of two particles. In this context, several interesting results emerge. First, there are single particle TPSs that are invariant under transformations between inertial reference frames, and these TPSs allow one to define intraparticle entanglement between momentum and spin degrees of freedom in a Galilean invariant manner. Second, there are multiple, inequivalent two particle TPSs that are symmetry invariant. In particular, these TPSs can be used to define Galilean invariant entanglement between the internal and external degrees of freedom of the two particle system. Finally, this internal-external entanglement is also dynamically invariant, i.e., i...

We show that for a finite-dimensional Hilbert space, there exist observables that induce a tensor product structure such that the entanglement properties of any pure state can be tailored. In particular, we provide an explicit, finite method for constructing observables in an unstructured d-dimensional system so that an arbitrary known pure state has any Schmidt decomposition with respect to an induced bipartite tensor product structure. In effect, this article demonstrates that in a finite-dimensional Hilbert space, entanglement properties can always be shifted from the state to the observables and all pure states are equivalent as entanglement resources in the ideal case of complete control of observables.PACS numbers: 03.65.Aa, 03.65.UdThe entanglement of a quantum state is only defined with respect to a tensor product structure within the Hilbert space that represents the quantum system. In turn, a tensor product structure of the Hilbert space is induced by the algebra of observables. Zanardi and colleagues [1,2] have provided criteria for the algebra of observables of a finite-dimensional system to induce a tensor product structure. The algebra of observables must be partitioned into subalgebras that satisfy two mathematical requirements, the subalgebras must be independent and complete (see Corollary 3 for a precise formulation of Zanardi's Theorem), and one physical requirement, the subalgebras must be locally accessible. Such observableinduced partitions of the Hilbert space have been referred to as virtual subsystems and can be thought of as a generalization from entanglement between subsystems to entanglement between degrees of freedom (see also [3,4]). This mathematical framework has found applications to studies of multi-level encoding [5], decoherence [6], operator quantum error correction [7], entanglement in fermionic systems [8], single-particle entanglement [9,10], and entanglement in scattering systems [11].In this Letter, we extend this mathematical framework and prove what we call the Tailored Observables Theorem (Theorem 6): observables can be constructed such that any pure state in a finite-dimensional Hilbert space H = C d has any amount of entanglement possible for any given factorization of the dimension d of H. This means all pure states are equivalent as entanglement resources in the ideal case of complete control of observables. To establish the framework, we provide a brief, relatively self-contained introduction to Zanardi's Theorem and obtain some necessary preliminary results about observable algebras in finite dimensions. We then prove Theorem 6, which applies to bipartite tensor product structures, and present an illustrative example. We will also provide a corollary of the theorem (Corollary 7) applied to multipartite tensor product structures. Before delving into the technical details, we present a more intuitive discussion of this result. this Hilbert space could represent states of a quantum system composed from N subsystems each represented by Hilbert spaces H i = C ki . Fo...