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.
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.
Quantum and classical mechanics are fundamentally different theories, but the correspondence principle states that quantum particles behave classically in the appropriate limit. For high-energy periodic quantum systems, the emergence of the classical description should be understood in a distributional sense, i.e., the quantum probability density approaches the classical distribution when the former is coarse-grained. Following a simple reformulation of this limit in the Fourier space, in this paper, we investigate the macroscopic behavior of freely falling quantum particles. To illustrate how the method works and to fix some ideas, we first successfully apply it to the case of a particle in a box. Next, we show that, for a particle bouncing under the gravity field, in the limit of a high quantum number, the leading term of the quantum distribution corresponds to the exact classical distribution plus sub-leading corrections, which we interpret as quantum corrections at the macroscopic level.
In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr’s correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum number. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator and the particle in a box, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck’s constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.
We investigate the quantum electrodynamics of many-body systems, and discuss the circumstances under which the results are identical to the results of classical electrodynamics. This classical behavior is shown to occur when the motion of a large number of the particles is correlated. The correlation manifests itself in a form of coherence which is characterized by the many-body quantum state's being very nearly an eigenstate of the current operator. We present several explicit examples of quantum states which exhibit such coherence and show that they give the expected classical results. Nowhere do we assume $h\ensuremath{\rightarrow}0$.
In this review we summarize and amplify recent investigations of coupled quantum dynamical systems with few degrees of freedom in the short-wavelength, semiclassical limit. Focusing on the correspondence between quantum and classical physics, we mathematically formulate and attempt to answer three fundamental questions. (i) How can one drive a small dynamical quantum system to behave classically? (ii) What determines the rate at which two single-particle quantum-mechanical subsystems become entangled when they interact? (iii) How does irreversibility occur in quantum systems with few degrees of freedom? These three questions are posed in the context of the quantum-classical correspondence for dynamical systems with few degrees of freedom, and we accordingly rely on two short-wavelength approximations to quantum mechanics to answer them: the trajectory-based semiclassical approach on the one hand, and random matrix theory on the other hand. We construct novel investigative procedures towards decoherence and the emergence of classicality out of quantumness in dynamical systems coupled to external degrees of freedom. In particular, we show how dynamical properties of chaotic classical systems, such as local exponential instability in phase space, also affects their quantum counterparts. For instance, it is often the case that the fidelity with which a quantum state is reconstructed after an imperfect time-reversal operation decays with the Lyapunov exponent of the corresponding classical dynamics. For related reasons, but perhaps more surprisingly, the rate at which two interacting quantum subsystems become entangled can also be governed by the subsystem's Lyapunov exponents. Our method allows us to differentiate quantum coherent effects (those related to phase interferences) from classical ones (those related to the necessarily extended envelope of quantal wavefunctions) at each stage in our investigations. This makes it clear that all occurrences of Lyapunov exponents we witness have a classical origin, although they require rather strong decoherence effects to be observed. We extensively rely on numerical experiments to illustrate our findings and briefly comment on possible extensions to more complex problems involving environments with many interacting dynamical systems, going beyond the uncoupled harmonic oscillators model of Caldeira and Leggett.
This thesis consists of two parts. Although both are concerned with macroscopic quantum phenomena, I choose to present the two parts quite separately. Macroscopic Quantum Phenomena: Tests of Quantum Mechanics The correctness of quantum mechanics has been verified in numerous situations. A particularly strong test of quantum mechanics was suggested by Bell, who showed that the predictions of all classical theories, based as they are on the assumptions of local realism, contradict those of quantum mechanics. Later work by Greenberger, Horne, and Zeilinger showed that the GHZ phenomena also contradicts all classical theories. Experimental tests of Bells inequality performed to date support quantum mechanics (No experimental test of the GHZ phenomenon has yet been attempted). However, such tests have been so far restricted to microscopic systems in the sense that one particle is at a detector at a time. Our interest here in the first part of this thesis is in tests of quantum mechanics against local realistic theories in macroscopic or mesoscopic systems where there is a significant number of particles incident on each analyser. The states considered in the first two chapters are not simple quantum superpositions of macroscopically distinct states, but we nevertheless show how one can obtain tests of quantum mechanics in situations involving large numbers of particles. A more elementary macroscopic quantum state, and one in which the deviation from classical interpretations is striking, is the Schrodinger Cat state considered by Schrodinger in his famous "Schrodinger Cat" paradox. In Chapter Three we consider a scheme for generating such a state. The Quantum Brownian Motion Master Equation The second section of this thesis involves an examination of various aspects of the Quantum Brownian Motion Master Equation. We begin with a discussion of the recent problem occurring from the study of the quantum Brownian motion (QBM) master equation, which can occur because the QBM master equation is not of the Lindblad form. We examine two specific examples, namely the damped free particle and the two level atom. More specifically, in the case of the evolution of a free particle, we show that the equation of motion for the Wigner function (which is exactly the same as the Fokker-Planck equation for classical Brownian motion) gives unphysical results if the initial position distribution is well localised. In the case of the two level atom we show that unphysical results can also occur even in the steady state. The last chapter of the thesis involves using the quantum Brownian motion master equation to estimate the rate of diagonalisation of the off diagonal elements of a Fermion system. These elements contain the "quantum information" and hence knowledge of how long they exist can determine the type of model required to describe the semiconductor medium. Using a very simplistic approach we can show that there are two main time scales present and for normal parameters one of these is very small. If we neglect electron-electron scattering we can show that the the many electron system solution is simply a product of the one electron solutions.
The quantum description of the microscopic world is incompatible with the classical description of the macroscopic world, both mathematically and conceptually. Nevertheless, it is generally accepted that classical mechanics emerges from quantum mechanics in the macroscopic limit. In this paper, we challenge this perspective and demonstrate that the behaviour of a macroscopic system can retain all aspects of the quantum formalism, in a way that is robust against decoherence, particle losses and coarse-grained (imprecise) measurements. This departure from the expected classical description of macroscopic systems is not merely mathematical but also conceptual, as we show by the explicit violation of a Bell inequality and a Leggett–Garg inequality.