1,721,006 research outputs found
Classical behaviour in quantum systems: the case of straight tracks in a cloud chamber
No full textThe aim of this paper is to discuss in a pedagogical way the problem of the emergence of classical behaviour in certain physical systems which, in principle, are correctly described by quantum mechanics. It is stressed that the limit (h) over bar -> 0 is not sufficient and the crucial role played by the environment must be taken into account. In particular, recalled is the old problem raised by Mott in 1929 (Proc. R. Soc. Lond. A 126 79) concerning the straight tracks observed in a cloud chamber, produced by an alpha-particle emitted by a source in the form of a spherical wave. The conceptual relevance of the problem for a clearer understanding of the classical limit is discussed in a historical perspective. Moreover, a simple mathematical model is proposed, where the result of Mott is obtained in a rigorous mathematical way
A mathematical primer on quantum mechanics
No full textThis book offers a rigorous yet elementary approach to quantum mechanics that will meet the needs of Master’s-level Mathematics students and is equally suitable for Physics students who are interested in gaining a deeper understanding of the mathematical structure of the theory. Throughout the coverage, which is limited to single-particle quantum mechanics, the focus is on formulating theory and developing applications in a mathematically precise manner. Following a review of selected key concepts in classical physics and the historical background, the basic elements of the theory of operators in Hilbert spaces are presented and used to formulate the rules of quantum mechanics. The discussion then turns to free particles, harmonic oscillators, delta potential, and hydrogen atoms, providing rigorous proofs of the corresponding dynamical properties. Starting from an analysis of these applications, readers are subsequently introduced to more advanced topics such as the classical limit, scattering theory, and spectral analysis of Schrödinger operators. The main content is complemented by numerous exercises that stimulate interactive learning and help readers check their progress.
Quadratic forms for the fermionic unitary gas model
No full textWe consider a quantum system in dimension three composed by a group of N identical fermions, with mass 1/2, interacting via zero-range interaction with a group of M identical fermions of a different type, with mass m/2. Exploiting a renormalization procedure, we construct the corresponding quadratic form and define the so-called Skornyakov-Ter-Martirosyan extension H α, which is the natural candidate as a possible Hamiltonian of the system. It is shown that if the form is unbounded from below then H α is not a self-adjoint and bounded from below operator, and this in particular suggests that the so-called Thomas effect could occur. In the special case N = 2, M = 1 we prove that this is in fact the case when a suitable condition on the parameter m is satisfied. © 2012 Polish Scientific Publishers
Scattering of a light particle by a system of harmonic oscillators
No full textWe discuss the asymptotic wave function of a quantum system in R^3 composed by heavy and light particles, in the case where the light particles are in scattering states and no interaction
is assumed among particles of the same kind. We first review a recent result concerning the case
of K heavy and N light particles, where the one-particle potential acting on each heavy particle
decays at infinity. Then we consider the case of one light particle interacting with a system of
harmonic oscillators and prove the same kind of result following, with some modification, the
proof of the previous case. A possible application to the analysis of the scattering of a light
particle from condensed matter is also outlined
From quantum to classical world: emergence of trajectories in a quantum system.
Full text linkThis note deals with models of quantum systems where the emergence of a classical behavior can be concretely analyzed. We first briefly review some well known difficulties arising in the classical limit of quantum mechanics according to the Copenhagen interpretation. Then we discuss the seminal contribution by Mott (1929) on the tracks observed in a cloud chamber, where the problem can be approached in a particularly transparent way. Finally, we propose a model Hamiltonian, with interaction described by spin dependent point interactions, where Mott’s analysis can be rephrased and the result can be rigorously formulated
A simple model for decoherence
No full textThe meaning of decoherence as a (practically) irreversible
process in Quantum Mechanics is discussed. It is also introduced a simple two-particle model consisting of a heavy (the system) and a light (the environment) particle and it is explicitely computed the decoherence effect on the heavy particle due to the presence of the light one
Asymptotic expansion for the wave function in a one-dimensional model of inelastic interaction
No full textWe consider a two-body quantum system in dimension one composed by a test particle interacting with a harmonic oscillator placed at the position a > 0. At time zero the test particle is concentrated around the position R(0) with average velocity+/-v(0) while the oscillator is in its ground state. In a suitable scaling limit, corresponding for the test particle to a semiclassical regime with small energy exchange with the oscillator, we give a complete asymptotic expansion of the wave function of the system in both cases R(0) a. (C) 2011 American Institute of Physics. [doi:10.1063/1.3549587
Semiclassical wave-packets emerging from interaction with an environment
No full textWe study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of N harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e., a highly correlated continuous superposition of states with well localized position and momentum, and the oscillators are in the ground state. Furthermore, we assume that the positions of the oscillators are not collinear with the center of the spherical wave. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schrodinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in L-2(R3(N+1)) and describing rather different situations. In the first one, all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly deformed spherical wave. The second one consists of a sum of N terms where in each term there is only one excited oscillator and the test particle is correspondingly described by the free evolution of a wave packet, well concentrated in position and momentum. Moreover, the wave packet emerges from the excited oscillator with an average momentum parallel to the line joining the oscillator with the center of the initial spherical wave. Such wave packet represents a semiclassical state for the test particle, propagating along the corresponding classical trajectory. The main result of our analysis is to show how such a semiclassical state can be produced, starting from the original spherical wave, as a result of the interaction with the environment. (C) 2014 AIP Publishing LLC.We study the quantum evolution in dimension three of a system composed by a test
particle interacting with an environment made of N harmonic oscillators. At time zero
the test particle is described by a spherical wave, i.e., a highly correlated continuous
superposition of stateswith well localized position and momentum, and the oscillators
are in the ground state. Furthermore, we assume that the positions of the oscillators
are not collinear with the center of the spherical wave. Under suitable assumptions on
the physical parameters characterizing the model, we give an asymptotic expression
of the solution of the Schr ̈odinger equation of the system with an explicit control
of the error. The result shows that the approximate expression of the wave function
is the sum of two terms, orthogonal in L2(R3(N+1)) and describing rather different
situations. In the first one, all the oscillators remain in their ground state and the test
particle is described by the free evolution of a slightly
Revisiting Quantum Mechanical Zero-Range Potentials
No full textInthiscontributionwemakeabriefoverviewofhistoryandrecentresults in the theory of many quantum particles interacting via zero-range forces. We recall the regularisation mechanism suggested by several authors in the past in order to avoid the “fall to the center” problem in three-body systems. Following those suggestions a family of three-body point interaction Hamiltonians bounded from below were made available recently. We conclude showing that a similar kind of ultraviolet problem is already present in the theory of point interaction Hamiltonians in one-body Quantum Mechanics. A careful look to the entire family of many center point interaction Hamiltonians shows that the great majority of them do not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, those Hamiltonians appear to be renormalised by default as opposed to the “local” point interaction Hamiltonians usually considered in the literature since the early days of Quantum Mechanics. The renormalization mechanism turns out to be very similar to the one used in the three-body problem
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