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arXiv:1101.0328 · Journal of Physics A Mathematical and Theoretical

Irreversible behaviour and collapse of wave packets in a quantum system with point interactions

Italo Guarneri

Published Nov 14, 2011• 15 citations
arXivHTMLPDF

Abstract

A system of a particle and a harmonic oscillator, which have pure point spectra if uncoupled, is known to acquire an absolutely continuous spectrum when they are coupled by a sufficiently strong point interaction. Here, the dynamical mechanism underlying this spectral phenomenon is exposed. The energy of the oscillator is proven to exponentially diverge in time, while the spatial probability distribution of the particle collapses into a δ-function at the interaction point. On account of this result, a generalized model with many oscillators which interact with the particle at different points is argued to provide a formal model for the approximate measurement of position and collapse of wave packets.

DOI
10.1088/1751-8113/44/48/485304

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A system of a particle and a harmonic oscillator, which have pure point spectra if uncoupled, is known to acquire an absolutely continuous spectrum when they are coupled by a sufficiently strong point interaction. Here, the dynamical mechanism underlying this spectral phenomenon is exposed. The energy of the oscillator is proven to exponentially diverge in time, while the spatial probability distribution of the particle collapses into a δ-function at the interaction point. On account of this result, a generalized model with many oscillators which interact with the particle at different points is argued to provide a formal model for the approximate measurement of position and collapse of wave packets.
Abstract Some variants of Smilansky’s model of a particle interacting with harmonic oscillators are examined in the framework of scattering theory. A dynamical proof is given of the existence of wave operators. Analysis of a classical version of the model provides a transparent picture for the spectral transition to which the quantum model owes its renown, and for the underlying dynamical behaviour. The model is thereby classified as an extreme case of chaotic scattering, with aspects related to wave packet reduction and irreversibility.
We present a set of exact system solutions to a model we developed to study wave function collapse in the quantum spin measurement process. Specifically, we calculated the wave function evolution for a simple harmonic oscillator of spin \frac{1}{2}, with its magnetic moment in interaction with a magnetic field, coupled to an environment that is a bath of harmonic oscillators. The system's time evolution is described by the direct product of two independent Hilbert spaces: one that is defined by an effective Hamiltonian, which represents a damped simple harmonic oscillator with its potential well divided into two, based on the spin and the other that represents the effect of the bath, i.e., the Brownian motion. The initial states of this set of wave functions form an orthonormal basis, defined as the eigenstates of the system. If the system is initially in one of these states, the final result is predetermined, i.e., the measurement is deterministic. If the bath is initially in the ground state,and the wave function is initially a wave packet at the origin, it collapses into one of the two potential wells depending on the initial spin. If the initial spin is a vector in the Bloch sphere not parallel to the magnetic field, the final distribution among the two potential wells is given by the Born rule applied to the initial spin state with the well-known ground state width. Hence, the result is also predetermined. We discuss its implications to the Bell theorem[1]. We end with a summary of the implications for the understanding of the statistical interpretation of quantum mechanics.
Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a {\it single} harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a "bath" which consists of a {\it single} harmonic oscillator.
Kramers-Kronig or dispersion relations that arise from the analytic behavior of electromagnetic and other classical waves are here adapted to quantum waves. In the framework of theories that view the wave packet (wp) collapse as a time-continuous process, we postulate analytic properties for the component-amplitudes in a wp as functions of a complex time. We then construct a model which embodies the removal of the non-selected components in the aftermath of the measurement, to be ultimately followed by a thermalequilibrium like superposition state of the system. Conjugate relations hold between component-moduli and phases and these show that a non-selected component (one that vanishes in the measurement) acquires within the duration of the collapse a fast oscillating phase factor. Thus, by virtue of mathematical properties, both phase-decoherence and amplitude-decay have to occur in a collapse process, indifferently to the physical mechanism.
It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, such as the Hamiltonian. Resonances, i.e., states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism based on the so-called rigged Hilbert space (RHS). In the RHS, the states with complex energy are denoted as Gamow vectors (GVs), and they model decay processes. We study the GVs of the reversed harmonic oscillator, and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data that are not composed only by a summation of GVs, and we analyze different wave packets belonging to specific function spaces. Our work furnishes support for the idea that irreversible wave propagation can be investigated using rigged-Hilbert-space quantum mechanics and provides insight for the experimental investigation of irreversible dynamics.
The measurement problem of quantum mechanics concerns the question as to under which circumstances coherent wave evolution becomes disrupted to produce eigenstates of observables, instead of evolving superpositions of eigenstates. The problem already needs to be addressed within wave mechanics, before second quantization, because low-energy interactions can be dominated by particle-preserving potential interactions. We discuss a scattering array of harmonic oscillators, which can detect particles penetrating the array through interaction with a short-range potential. Evolution of the wave function of scattered particles, combined with Heisenberg’s assertion that quantum jumps persist in wave mechanics, indicates that the wave function will collapse around single oscillator sites if the scattering is inelastic, while it will not collapse around single sites for elastic scattering. The Born rule for position observation is then equivalent to the statement that the wave function for inelastic scattering amounts to an epistemic superposition of possible scattering states, in the sense that it describes a sum of probability amplitudes for inelastic scattering off different scattering centers, whereas, at most, one inelastic scattering event can happen at any moment in time. Within this epistemic interpretation of the wave function, the actual underlying inelastic scattering event corresponds to a quantum jump, whereas the continuously evolving wave function only describes the continuous evolution of probability amplitudes for scattering off different sites. Quantum jumps then yield definite position observations, as defined by the spatial resolution of the oscillator array.