61 research outputs found
Bound states in the one-dimensional two-particle Hubbard model with an impurity
We investigate bound states in the one-dimensional two-particle Bose-Hubbard
model with an attractive () impurity potential. This is a
one-dimensional, discrete analogy of the hydrogen negative ion H problem.
There are several different types of bound states in this system, each of which
appears in a specific region. For given , there exists a (positive) critical
value of , below which the ground state is a bound state.
Interestingly, close to the critical value (), the ground
state can be described by the Chandrasekhar-type variational wave function,
which was initially proposed for H. For , the ground state is no
longer a bound state. However, there exists a second (larger) critical value
of , above which a molecule-type bound state is established and
stabilized by the repulsion. We have also tried to solve for the eigenstates of
the model using the Bethe ansatz. The model possesses a global \Zz_2-symmetry
(parity) which allows classification of all eigenstates into even and odd ones.
It is found that all states with odd-parity have the Bethe form, but none of
the states in the even-parity sector. This allows us to identify analytically
two odd-parity bound states, which appear in the parameter regions
and , respectively. Remarkably, the latter one can be \textit{embedded}
in the continuum spectrum with appropriate parameters. Moreover, in part of
these regions, there exists an even-parity bound state accompanying the
corresponding odd-parity bound state with almost the same energy.Comment: 18 pages, 18 figure
Integrability and weak diffraction in a two-particle Bose-Hubbard model
A recently introduced one-dimensional two-particle Bose-Hubbard model with a
single impurity is studied on finite lattices. The model possesses a discrete
reflection symmetry and we demonstrate that all eigenstates odd under this
symmetry can be obtained with a generalized Bethe ansatz if periodic boundary
conditions are imposed. Furthermore, we provide numerical evidence that this
holds true for open boundary conditions as well. The model exhibits
backscattering at the impurity site -- which usually destroys integrability --
yet there exists an integrable subspace. We investigate the non-integrable even
sector numerically and find a class of states which have almost the Bethe
ansatz form. These weakly diffractive states correspond to a weak violation of
the non-local Yang-Baxter relation which is satisfied in the odd sector. We
bring up a method based on the Prony algorithm to check whether a numerically
obtained wave function is in the Bethe form or not, and if so, to extract
parameters from it. This technique is applicable to a wide variety of other
lattice models.Comment: 13.5 pages, 11 figure
Forschung zum Mathematischen Argumentieren – Ein deskriptiver Review von PME Beiträgen
Die Mathematik ist eine beweisende Wissenschaft. Mathematisches Argumentieren
und Beweisen (MA&B) sind zentrale Aktivitäten der Mathematik
und gehören zu den wichtigsten zu erlernenden Fähigkeiten im schulischen
und universitären Bereich (Brunner, 2014). Gerade in der Sekundarstufe
wurde der Fokus auf Argumentieren in den letzten Jahren weltweit
durch curriculare Änderungen verstärkt, entsprechend ist MA&B auch innerhalb
der Didaktik der Mathematik wieder zunehmend in den Forschungsmittelpunkt
gerückt. In diesem Beitrag wird Argumentieren im
Sinne Toulmins relativ offen verstanden, insbesondere werden auch nicht
deduktives Schlussfolgern und Beweisen als Spezialfälle des Argumentierens
verstanden (Reiss & Ufer, 2009)
Exact real-time dynamics of the quantum Rabi model
We use the analytical solution of the quantum Rabi model to obtain absolutely
convergent series expressions of the exact eigenstates and their scalar
products with Fock states. This enables us to calculate the numerically exact
time evolution of and for all regimes of the
coupling strength, without truncation of the Hilbert space. We find a
qualitatively different behavior of both observables which can be related to
their representations in the invariant parity subspaces.Comment: 8 pages, 7 figures, published versio
Dynamical correlation functions and the quantum Rabi model
We study the quantum Rabi model within the framework of the analytical
solution developed in Phys. Rev. Lett. 107,100401 (2011). In particular,
through time-dependent correlation functions, we give a quantitative criterion
for classifying two regions of the quantum Rabi model, involving the
Jaynes-Cummings, the ultrastrong, and deep strong coupling regimes. In
addition, we find a stationary qubit-field entangled basis that governs the
whole dynamics as the coupling strength overcomes the mode frequency.Comment: 8 pages, 8 figures. Revised version, accepted for publication in
Physical Review
Bound States in the Continuum Realized in the One-Dimensional Two-Particle Hubbard Model with an Impurity
We report a bound state of the one-dimensional two-particle (bosonic or
fermionic) Hubbard model with an impurity potential. This state has the
Bethe-ansatz form, although the model is nonintegrable. Moreover, for a wide
region in parameter space, its energy is located in the continuum band. A
remarkable advantage of this state with respect to similar states in other
systems is the simple analytical form of the wave function and eigenvalue. This
state can be tuned in and out of the continuum continuously.Comment: A semi-exactly solvable model (half of the eigenstates are in the
Bethe form
Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills
An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as 'resource-based,' as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students' work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students
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