Textos de Fisica para professores e estudantes--II
This paper has been withdrawn by the author.
physics.ed-ph↗
arXiv subjects
Oscar Bolina
Publications and source records attributed to Oscar Bolina.
Path integral representations for the spin-pinned quantum XXZ chain
Two discrete path integral formulations for the ground state of a spin-pinned quantum anisotropic XXZ Heisenberg chain are introduced. Their properties are discussed and two recursion relations are proved.
math-ph↗
Textos de f\'ısica para professores e estudantes
This is a series of short teaching papers dealing with specific topics in a standard first-year undergraduate Physics course. ----- Este texto compõe-se de quatro pequenas notas -- independentes entre si -- em que se discutem alguns tópicos espec\'ıficos de f\'ısica para o segundo grau e o primeiro ano universitário.
physics.ed-ph↗
Trotter formula and thermodynamic limits
We discuss the interchangeability of the thermodynamic limit β\to \infty and the infinite limit of the Trotter number n \to \infty when Trotter formula e^{H_{0}+V} =\lim_{n \to \infty} (e^{H_{0}/n} e^{V/n}) is used to calculate partition functions with Hamiltonians of the form H=H_{0}+V.
physics.ed-ph↗
Kinematics on oblique axes
We solve a difficult problem involving velocity and acceleration components along oblique axes, and propose two problems of central force motion to be solved using oblique axes.
physics.ed-ph↗
The gambler's ruin problem in path representation form
We consider the classical one-dimensional random walk of a particle on the right-half real line. We assume that the particle is initially at position x=k, k > 0, and moves to the right with probability p or to the left with probability 1-p. We consider that the particle is absorbed at the origin without fixing the number of steps needed to get there. We calculate the probability P(x=k) that the particles end up at the origin, given that it starts at x=k, by means of a geometric representation of this random walk in terms of paths on a two-dimensional lattice.
math.PR↗
A bubble theorem
Proof of a simple mathematical inequality using elements of surface tension theory and ideal gas law to the formation and coalescence of bubbles.
physics.ed-ph↗
The Precessing Top
I discuss the precession motion of a symmetrical top without the assumption that its precessional velocity is much smaller than its spin angular velocity. I derive the general formula for the precessional velocity in an elementary way and discuss the cases of slow and fast precessions.
physics.ed-ph↗
A continuum approximation for the excitations of the (1,1,...,1) interface in the quantum Heisenberg model
It is shown that, with an appropriate scaling, the energy of low-lying excitations of the (1,1,...,1) interface in the $d$-dimensional quantum Heisenberg model are given by the spectrum of the $d-1$-dimensional Laplacian on an suitable domain.
math-ph↗
Oscar's Physics Phaire: A Collection of Problems
You will find here a number of (mostly) elementary physics problems dealing mainly with uniform motion kinematics. In preparing this collection I have tried to create original situations that could help bring motivation to an introductory course. You are welcome to suggest improvements and ... provide solutions!
physics.ed-ph↗
Finite-volume excitations of the 111 interface in the quantum XXZ model
We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R^2). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization).
math-ph↗
Path Integral Representation for Interface States of the Anisotropic Heisenberg Model
We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e., has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.
math-ph↗
Bounds on Correlation Functions of Quantum Rotators
We derive a McBryan-Spencer bound to the correlation function of a one-dimensional array of quantum rotators in the Villain approximation of the cosine interaction. We obtain the partition function of the system in the gas representation and establish a lower bound on the external charge correlation function. We also discuss the possible existence of a Kosterlitz-Thouless phase for the quantum rotator in the Villain approximation.
math-ph↗
Intrinsic Kinematics
We show how some geometric elements of the path of a particle moving in a plane -- the osculating circle and its radius of curvature -- can be used to construct the parabolic trajectory of projectiles in motion under gravity.
physics.ed-ph↗
Difficult Problems Having Easy Solutions
We discuss how a class of difficult kinematic problems can play an important role in an introductory course in stimulating students' reasoning on more complex physical situations. The problems presented here have an elementary analysis once certain symmetry features of the motion are revealed. We also explore some unexpected directions these problems lead us.
physics.ed-ph↗
A Note on Eye Movement
In a simplified fashion, the motion of the eyeball in its orbit consists of rotations around a fixed point. Therefore, this motion can be described in terms of the Euler's angles of rigid body dynamics. However, there is a physiological constraint in the motion of the eye which reduces to two its degrees of freedom. This paper reviews the basic features of the kinematics of the eye and the laws governing its motion.
physics.bio-ph↗