The simplified Fermi accelerator in classical and quantum mechanics

We review the simplified classical Fermi acceleration mechanism and construct a quantum counterpart by imposing time-dependent boundary conditions on solutions of the free Schrödinger equation at the unit interval. We find similiar dynamical features in the sense that limiting KAM curves, respectively purely singular quasienergy spectrum, exist(s) for sufficiently smooth wall oscillations (typically ofC ² type). In addition, we investigate quantum analogs to local approximations of the Fermi map both in its quasiperiodic and irregular phase space regions. In particular, we find pure point q.e. spectrum in the former case and conjecture that random boundary conditions are necessary to model a quantum analog to the chaotic regime of the classical accelerator.     

A Velocity Field and Operator for Spinning Particles in (Nonrelativistic) Quantum Mechanics

Starting from the formal expressions of the hydrodynamical (or local) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce—in terms of the ordinary tensorial language—a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin-1/2 particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the socalled Zitterbewegung, which is the spin internal motion observed in the center-of-mass from (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spinvector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schrödinger theory), also a component which is parallel to the curl of the spin vector.     

On the Role of Spin in Quantum Mechanics

From the invariance properties of the Schrödinger equation and the isotropy of space we show that a generic (non-relativistic) quantum system is endowed with an external motion, which can be interpreted as the motion of the centre of mass, and an internal one, whose presence disappears in the classical limit. The latter is caused by the spin of the particle, whatever is its actual value (different from zero). The quantum potential in the Schrödinger equation, which is responsible of the quantum effects of the system, is then completely determined from the properties of the internal motion, and its unusual properties have a simple and physical explanation in the present context. From the impossibility to fix the initial conditions relevant for the internal motion follows, finally, the need of a probabilistic interpretation of quantum mechanics.     

Semi-classical asymptotics in solid state physics

This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization. For the time-dependent Schrödinger equation this construction yields wave packets which follow the trajectories of the semiclassical model. For the time-independent equation one gets asymptotic eigenfunctions (or quasimodes) for the energy levels predicted by Onsager's relation.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Non-Hamiltonian dynamical systems

The class of dynamical systems is considered, which are described by several mutually noncommuting Hamiltonian currents, in particular, relativistic bi-Hamiltonian systems, the evolution of which is described by a pair of 4-momenta p and p The examination is conducted in classical and quantum realizations. The evolution equations are derived of relativistic bi-Hamiltonian systems in the Heisenberg and Schrödinger pictures. It is shown that the quantum theory of relativistic bi-Hamiltonian systems is not compatible with the unitary condition and is nonunitary. A physical interpretation is given of nonunitary quantum theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 5–12, October, 1990.     

Physical foundations of quantum theory: Stochastic formulation and proposed experimental test

The time-dependent Schrödinger equation has been derived from three assumptions within the domain of classical and stochastic mechanics. The continuity equation isnot used in deriving the basic equations of the stochastic theory as in the literature. They are obtained by representing Newton's second law in a time-inversion consistent equation. Integrating the latter, we obtain the stochastic Hamilton-Jacobi equation. The Schrödinger equation is a result of a transformation of the Hamilton-Jacobi equation and linearization by assigning the arbitrary constant =2mD. An experiment is proposed to determine and to test a hypothesis of the theory directly. A mathematical apparatus is formulated from the Jacobian formalism to derive physical parameters from (x, t) and to obtain operators for the boundary cases of the theory. The operator formalisms are compared by means of a well-known solution in the quantum theory.     

Resonances en limite semiclassique et exposants de Lyapunov

We determine the width of resonance-free domains in the complex plane for the semiclassical Schrödinger operator –h ²+V(x) whenh0, in terms of Lyapunov exponents for the associated classical flow.     

A New Estimate for the Groundstate Energy of Schrödinger Operators

A new estimate for the groundstate energy of Schrödinger operators on L²(ⁿ) (n 1) is presented. As a corollary, it is shown that the groundstate energy of the Schrödinger operator with a scalar potential V is more than the classical lower bound ess.inf_x__V(x) if V is essentially bounded from below in a certain manner (enhancement of the groundstate energy due to quantization). As an application, it is proven that the groundstate energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).     

Quantum theory of single events: Localized De Broglie wavelets,Schrödinger waves,and classical trajectories

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps (x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and (x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves is also discussed.     

Green Functions of the Stationary Schrödinger Equation and Tomographic Representation of Quantum Mechanics

Invertible maps of operators of quantum observables onto functions of c-number arguments and their associative products are reviewed. In particular, the symplectic tomography map is discussed and an expression connecting an arbitrary operator and its tomographic symbol is written down. This formula is applied to obtain explicit expressions for tomographic symbols, which are symplectic tomograms of Green functions of stationary and nonstationary Schrödinger equations written for the case of harmonic oscillator. The connection between the so-called classical propagator (X,,,t,X,,,0) and the tomographic symbol of the evolution operator of nonstationary Schrödinger equation is found. The spin tomography is presented as a map of operators acting in spinor space onto functions of c-arguments. As an example, the spin located in a magnetic field is considered and the tomographic symbol of resolventa is obtained. Tomographic symbols of hermitian conjugate operators are shown to be complex conjugate functions.     

On the extreme relativistic limit of arbitrary spin wave equations

The extreme relativistic limit (E-representation) of the wave equation in the Schrödinger formi/t =H describing particles and anti-particles of spin s and non-zero rest mass m is presented here. As the wave function has just the minimum number of 2(2s+1) components, the necessity of avoiding redundant components by auxiliary conditions does not arise. Relevant expressions are given for the infinitesimal generators of the Poincaré group and for the operators representing the observables in this representation.     

The nonrelativistic Schrödinger equation in “quasi-classical” theory

The author has recently proposed a quasi-classical theory of particles and interactions in which particles are pictured as extended periodic disturbances in a universal field (x, t), interacting with each other via nonlinearity in the equation of motion for . The present paper explores the relationship of this theory to nonrelativistic quantum mechanics; as a first step, it is shown how it is possible to construct from a configuration-space wave function (x ₁,x ₂,t), and that the theory requires that satisfy the two-particle Schrödinger equation in the case where the two particles are well separated from each other. This suggests that the multiparticle Schrödinger equation can be obtained as a direct consequence of the quasi-classical theory without any use of the usual formalism (Hilbert space, quantization rules, etc.) of conventional quantum theory and in particular without using the classical canonical treatment of a system as a crutch theory which has subsequently to be quantized. The quasi-classical theory also suggests the existence of a preferred absolute gauge for the electromagnetic potentials.     

Fisher information and the complex nature of the Schrödinger wave equation

We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrödinger wave equation. Complexity, in particular, traces from an efficiency scenario (demanded by MFI) where the two components of p(x) are so separated that their informations add.     

Relativistic Quantum Mechanics as a Telegraph

A derivation by Fröhner of non-relativistic quantum mechanics via Fourier analysis applied to probability theory is not extendable to relativistic quantum mechanics because Schrödinger's positive definite probability density * is lost (Dirac's spin 1/2 case being the exception). The nature of the Fourier link then changes; it points to a redefinition of the probability scheme as an information carrying telegraph, the code of which is Born's as extended by Dirac and by Feynman. Hermitian symmetry of the transition amplitude between Dirac representations expresses reciprocity of preparation and measurement (the quantal coding and decoding), two equally active interventions of the physicist; as the measurement perturbs the system retrodiction implies retroaction evidenced in delayed choice. Reciprocity of knowledge and organization vindicates Wigner's claim that reciprocal to the action of matter upon mind there exists a direct action of mind upon matter: psychokinesis, branded by Jaynes as a psychiatric disorder of the Copenhagen school. As for factlike irreversibility, it is expressed by the enormity of the change rate from information to negentropy: while gain in knowledge is normal psychokinesis is paranormal. Stapp's recent discussion of psychokinesis in a quantum context should be resumed in association with an EPR correlation; an experimental test is proposed.     

Quantum mechanical states as attractors for Nelson processes

In this paper we reconsider, in the light of the Nelson stochastic mechanics, the idea originally proposed by Bohm and Vigier that arbitrary solutions of the evolution equation for the probability densities always relax in time toward the quantum mechanical density ¦¦² derived from the Schrödinger equation. The analysis of a few general propositions and of some physical examples show that the choice of the L¹ metrics and of the Nelson stochastic flux is correct for a particular class of quantum states, but cannot be adopted in general. This indicates that the question if the quantum mechanical densities attract other solution of the classical Fokker-Planck equations associated to the Schrödinger equation is physically meaningful, even if a classical probabilistic model good for every quantum stale is still not available. A few suggestion in this direction are finally discussed.Written in honor of J.-P. Vigier.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

Singular perturbations of discrete spectrum

Even solutions of the Schrödinger equation with retaining potential x² are constructed for singular perturbation potentials |x|^–. It is shown that the perturbation automatically entails an induced point potential, taking account of which the perturbation matrix elements and Rayleigh-Schrödinger series may be constructed when 1 < < 3/2. In the opposite case (3/2 2), although the solutions are analytic with respect to , not even diverging series can be obtained for the energy solutions without solution of the Schrödinger equation. The analogy with quantum field theory is explored.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 58–64, March, 1988.