Nonstationary quantum mechanics. III. Quantum mechanics does not incorporate classical physics

It is shown that disagreement between the prediction of classical and conventional quantum mechanics about momentum probabilities exists in the case of a quasiclassical motion. The discussion is based on the detailed consideration of two specific potentials:U(x)=x and the oscillatory potentialU(x)=m ² x ²/2. The results of the present Part III represent a further development of the idea in Todorov (1980) about the possible inefficiency of conventional theory in the case of potentials swiftly varying with time.     

Quantum mechanics as a deformation of classical mechanics

Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.Work supported in part by the National Science Foundation.     

Discrete gradients in discrete classical mechanics

A simple model of discrete classical mechanics is given where, starting from the continuous Hamilton equations, discrete equations of motion are established together with a proper discrete gradient definition. The conservation laws of the total discrete momentum, angular momentum, and energy are demonstrated.     

Quantum mechanics as a classical diffusion process

A direct construction is provided showing that the classical expectation value for the energy of a particle executing a classical diffusion process is equivalent to the quantum mechanical form with a Hamiltonian structured like that for a charged particle in an electromagnetic field.     

Quantum approach to classical statistical mechanics

We present a new approach to study the thermodynamic properties of d-dimensional classical systems by reducing the problem to the computation of ground state properties of a d-dimensional quantum model. This classical-to-quantum mapping allows us to extend the scope of standard optimization methods by unifying them under a general framework. The quantum annealing method is naturally extended to simulate classical systems at finite temperatures. We derive the rates to assure convergence to the optimal thermodynamic state using the adiabatic theorem of quantum mechanics. For simulated and quantum annealing, we obtain the asymptotic rates of T(t) approximately (pN)/(k(B)logt) and gamma(t) approximately (Nt)(-c/N), for the temperature and magnetic field, respectively. Other annealing strategies are also discussed.     

Quantum mechanics in discrete space and angular momentum

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.     

Scalar Quantum Electrodynamics on the lattice as classical statistical mechanics

Wilson's lattice approximation allows us to apply classical statistical mechanics ideas to the study of Scalar Quantum Electrodynamics. Our main tools are Griffiths-Kelly-Sherman inequalities, the transfer matrix formalism and exponential bounds. Our main result is the existence of the infinite volume limit for every value of the coupling parameters.     

Quantum discrete sine-Gordon model at roots of 1: Integrable quantum system on the integrable classical background

The quantum discrete sine-Gordon model at roots of 1 is studied. It is shown that this model provides an example of an integrable quantum system in an integrable classical background. In particular, the spectrum of quantum integrals of motions in this model depends only on the values of integrals of motion of a background classical system.Dedicated to L.D. Faddeev on his 60th birthday     

Quantum and classical correlations in a classical dephasing environment

We calculate the non-Markovian dynamics of quantum correlations, including entanglement and discord, and classical correlation in a two-qubit system with Ornstein-Uhlenbeck noise and discuss the relations in these correlations. It?s found that the initial-state parameters and the non-Markovian properties play an important role in these dynamics behaviors.