DETERMINISTIC PROPERTIES CONTAINED IN QUANTUM SYSTEMS WITH CONTINUOUS SPECTRUM
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When a quantum system with continuous spectrum has a definite anergy, ordinary quantum mechanics uses a single continuous stationary state to describe its properties, the expectation values for arbitrary physical quantities are therefore either infinity or zero. In this paper we point out in the form of mathematical theorem that the system possesses new and deterministic results with evident physical meanings. Classical results for the system are the classical limit cases of these results, and the ordinarv expectation values are the statistical cases of our results.Keywords:
Continuous spectrum
Stationary state
Physical system
Infinity
Statistical Mechanics
Quantum system
Classical limit
It is discussed the distributions of the wavefunction of the quantum mechanics in the classical limit, the single particle system and the ensemble of the classical mechanics. It proved that the wavefunction of the quantum mechanics can only describe an ensemble and can't describe the motion of the single particle. It is the origin of the statistical character of the quantum mechanics.
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The theoretical treatment of quasiperiodically driven quantum systems is complicated by the inapplicability of the Floquet theorem, which requires strict periodicity. In this work we consider a quantum system driven by a biharmonic driving and examine its asymptotic long-time limit, the limit in which features distinguishing systems with periodic and quasiperiodic driving occur. Also, in the classical case this limit is known to exhibit universal scaling, independent of the system details, with the system's reponse under quasiperiodic driving being described in terms of nearby periodically driven system results. We introduce a theoretical framework appropriate for the treatment of the quasiperiodically driven quantum system in the long-time limit and derive an expression, based on Floquet states for a periodically driven system approximating the different steps of the time evolution, for the asymptotic scaling of relevant quantities for the system at hand. These expressions are tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force.
Floquet theory
Quantum system
Classical limit
Biharmonic equation
Quasiperiodicity
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While ultimately they are described by quantum mechanics, macroscopic mechanical systems are nevertheless observed to follow the trajectories predicted by classical mechanics. Hence, in the regime defining macroscopic physics, the trajectories of the correct classical motion must emerge from quantum mechanics, a process referred to as the quantum to classical transition. Extending previous work [Bhattacharya, Habib, and Jacobs, Phys. Rev. Lett. 85, 4852 (2000)], here we elucidate this transition in some detail, showing that once the measurement processes that affect all macroscopic systems are taken into account, quantum mechanics indeed predicts the emergence of classical motion. We derive inequalities that describe the parameter regime in which classical motion is obtained, and provide numerical examples. We also demonstrate two further important properties of the classical limit: first, that multiple observers all agree on the motion of an object, and second, that classical statistical inference may be used to correctly track the classical motion.
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We analyze the question of separability in a continuously measured quantum system as it approaches the classical limit. We show that the record of position measurements can approach the classical limit even when the system is described by highly nonseparable states. In particular, in systems with a chaotic classical limit, chaos can work to enhance the entanglement in the system in the classical regime. This coexistence of nonclassical states and classical dynamics can be understood by analyzing the conditioned evolution of the measured system and the conditions for the quantum-to-classical transition. PACS Nos.: 03.65.Ta, 03.65.Ud, 03.67.Mn, 05.45.Mt, 03.67.–a
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Starting from the quantum theory of identical particles, we show how to define a classical mechanics that retains information about the quantum statistics. We consider two examples of relevance for the quantum Hall effect: identical particles in the lowest Landau level, and vortices in the Chern-Simons Ginzburg-Landau model. In both cases the resulting classical statistical mechanics is shown to be a nontrivial classical limit of Haldane's exclusion statistics.
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Classical limit
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Based on the Hamilton-Jacobi equation and the Bohm's quantum potential,it is pointed out that the classical limit of the quantum mechanics is the classical statistical mechanics.Through analyzing the Gaussian packet from two approaches,the perspective of the classical limit via the H-J equation is shown to be more profound than that via the Ehrenfest theorem.
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