Quantum harmonic oscillator presented in a nonorthodox approach

An example of nonequivalent but s-equivalent Lagrange function for a one-dimensional harmonic oscillator are presented and discussed. The quantum approach displays a quite different structure from the usual standard one.     

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.     

Quantum mechanics as applied mathematical statistics

Basic mathematical apparatus of quantum mechanics like the wave function, probability density, probability density current, coordinate and momentum operators, corresponding commutation relation, Schrödinger equation, kinetic energy, uncertainty relations and continuity equation is discussed from the point of view of mathematical statistics. It is shown that the basic structure of quantum mechanics can be understood as generalization of classical mechanics in which the statistical character of results of measurement of the coordinate and momentum is taken into account and the most important general properties of statistical theories are correctly respected.     

Quantum mechanics as an integrable system

Basic dynamical equations of quantum mechanics, including the Schrödinger, Pauli, Dirac, and Klein-Gordon equations, are bi-hamiltonian systems with an infinite number of conservation laws.     

半空间谐振子的量子力学和半经典近似

对于半空间谐振子,给出了包括位置、动量算符及其平方的矩阵元,定态上的不确定关系等量子力学的基本结果．由于位置矩阵元的结果较复杂,借助Heisenberg对应原理给出了它的一个很好的近似表达式．     

Quantum mechanics as a matrix symplectic geometry

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraM _N (C) generated by the Schwinger matrices.     

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 mechanics and geometric analysis on manifolds

A central idea of modern geometric analysis is the assignment of a geometric structure, usually called thesymbol, to a differential operator. It is known that this operation is closely related to quantum mechanics. For a class of linear operators, including the Dirac operator, a geometric structure, called aco-Riemannian metric, is assigned to such symbols. Certain other topics related to the geometric structure of quantum mechanics, e.g., the symplectic structure of the projective space of Hilbert space, are briefly treated.     

Quantum statistical mechanics as a construction of an embedding scheme

The aim of the present paper is to show that the formalism of equilibrium quantum statistical mechanics can fully be incorporated into Ludwig's embedding scheme for classical theories in many-body quantum mechanics. A construction procedure based on a recently developed reconstruction procedure for the so-called macro-observable is presented which leads to the explicit determination of the set of classical ensembles compatible with the embedding scheme.     

Quantum mechanics as the limit of a symmetric local theory

It is shown that the quantum-mechanical spin-correlation function of the photon pairs emitted in atomic cascade experiments can be obtained by a limiting process from a local realistic theory which is symmetric between the analyzers.     

Quantum mechanics for two-timers

Extensions of standard quantum mechanics with joint probability distributions for position coordinates and momenta have been proposed in the literature. Time is assumed to be one-dimensional in these studies. In view of recent interest in two-dimensional time, the construction is extended to this situation and found to satisfy the necessary consistency conditions.     

Quantum mechanics as demanded by the special theory of relativity

We present a new approach on the interpretation of the quantum mechanism. The derivation is phenomenological and incorporates an energetic vacuum which interacts with elementary particles. We consider a classical ensemble average for the square of 4-velocities of identical elementary particles with the same initial conditions in Minkowski space. The relativistic extension of a result in Brownian motion allows the variance to be identified with Bohm's quantum potential. A simple relation between 4-velocities and 4-momenta at a specific 4-position with given proper time leads to one of two statistical equations that constitute our quantum theory, the other being the continuity equation. The Klein-Gordon equation is a consequence of these two statistical equations.     

Quantum mechanics of inflation

We derive the Harrison-Zeldovich spectrum by perturbatively solving the Wheeller-DeWitt equations for an inflating universe coupled to a scalar field in 2 + 1 and 3 + 1 dimensions.     

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 mechanics of seeing

A human being viewing a defocused television tube with sweep voltages turned off will see point scintillations at sufficiently low intensities. We show that quantum mechanics predicts these scintillations. Furthermore, by assuming a response of the human nervous system of a type not inconsistent with experiment, measurement theory is used to show that these scintillations will be distributed in proportion to the magnitude squared of the electron wave function incident upon the television tube screen. This nervous system response is to break up the wave incident upon a spot on the retina into a number of similar waves transmitted by different nerves to the brain. The number of these waves is proportional to the incident energy density. Since the theory itself predicts the proper probability distribution, it is unnecessary to introduce a postulate for it.     

Quantum mechanics of leptogenesis

Thermal leptogenesis is an attractive mechanism that explains in a simple way the matter-antimatter asymmetry of the universe. It is usually studied via the Boltzmann equations, which describes the time evolution of particle densities or distribution functions in a thermal bath. The Boltzmann equations are classical equations and suffer from basic conceptual problems and they lack to include many quantum phenomena. We show how to address leptogenesis systematically in a purely quantum way, by describing non-equilibrium excitations of a Majorana particle in the Kadanoff-Baym equations with significant emphasis on the initial and boundary conditions of the solutions. We apply our results to thermal leptogenesis, computing analytically the asymmetry generated, comparing it with the semiclassical Boltzmann approach. The non-locality of the Kadanoff-Baym equations shows how off-shell effects can have a huge impact on the generated asymmetry. The insertion of standard model decay widths to the particles excitations of the bath is also discussed. We explain how with a trivial insertion of these widths we regain locality on the processes.     

Quantum mechanics versus realism

Quantum theory accepts the point-like indivisible (classical) character of a particle as a mere product of a measuring process, or what has become known as a collapse. Following the notion of empty waves, which accepts the particle as a real existent entity without regard to the measurement process, we propose an experiment that may shed some light on the reality of the particle and the consequences of that reality.     

Quantum mechanics from self-interaction

We explore the possibility thatzitterbewegung is the key to a complete understanding of the Dirac theory of electrons. We note that a literal interpretation of thezitterbewegung implies that the electron is the seat of an oscillating bound electromagnetic field similar to de Broglie's pilot wave. This opens up new possibilities for explaining two major features of quantum mechanics as consequences of an underlying physical mechanism. On this basis, qualitative explanations are given for electron diffraction, the existence of quantized radiationless states, the Pauli principle, and other features of quantum mechanics.