Macroscopic Quantum Behaviour of Periodic Quantum Systems

A Martín-Ruiz,Adrián Carbajal-Domínguez,J Bernal

doi:10.4236/jmp.2014.51007

Abstract

In this paper we introduce a simple procedure for computing the macroscopic quantum behaviour of periodic quantum systems in the high energy regime. The macroscopic quantum coherence is ascribed to a one-particle state, not to a condensate of a many-particle system; and we are referring to a system of high energy but with few degrees of freedom. We show that, in the first order of approximation, the quantum probability distributions converge to its classical counterparts in a clear fashion, and that the interference effects are strongly suppressed. The harmonic oscillator provides a testing ground for these ideas and yields excellent results.

Highlights

The superposition principle lies at the heart of quantum mechanics, and it is one of its features that most distinctly marks the departure from classical concepts [1]
In this paper we introduce a simple procedure to compute the high energy regime of a general density matrix for periodic quantum systems
To this end we extend the Fourier expansion in Equation (2) to the spatial components of the matrix density (7)

Summary

Introduction

The superposition principle lies at the heart of quantum mechanics, and it is one of its features that most distinctly marks the departure from classical concepts [1]. A widely accepted explanation nowadays for the appearance of classical like features from an underlying quantum world is the environment induced decoherence approach [3,4,5,6]. This problem has been considered before by Cabrera and Kiwi [7] In this work, they use purely quantum-mechanical results to analyze (by inspection) the amplitude of the oscillations and the spatial autocorrelation function for large quantum numbers. They use purely quantum-mechanical results to analyze (by inspection) the amplitude of the oscillations and the spatial autocorrelation function for large quantum numbers They conclude that even for arbitrarily high quantum numbers, a superposition of (a few) eigenstates retains quantum effects.

General Procedure

Macroscopic Density Matrix of the Harmonic Oscillator

Final Remarks