Iwo Bialynicki-Birula

Uncertainty relations for information entropy in wave mechanics

New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality.

Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata

Very simply unitary cellular automata on a cubic lattice are introduced to model a discretized time evolution of the wave functions for spinning particles. In each evolution step the undated value of the wave function at a given site depends only on the values at the nearest sites. The discretized evolution is also unitary and preserves chiral symmetry. The case of the spin-1/2 particle is studied in detail and it is shown that every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and is shown to reproduce the retarded propagator in the continuum limit. Generalizations to include massive particles (Dirac theory), spin-1 particles (Maxwell theory), and higher-spin particles are also described.

Gaussons: Solitons of the Logarithmic Schrödinger Equation

Properties of the Schrodinger equation with the logarithmic nonlinearity are briefly described. This equation possesses soliton-like solutions in any number of dimensions, called gaussons for their Gaussian shape. Excited, stationary states of gaussons of various symmetries, in two and three dimensions are found numerically. The motion of gaussons in uniform electric and magnetic fields is studied and explicit solutions describing linear and rotational internal oscillations are found and analyzed.

Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator

Abstract An explicit formula for an arbitrary function of the evolution operator is derived. With its use, the continuous analog of the Baker-Campbell-Hausdorff problem is solved. The application of this result to the quantum theory of scattering leads to a new closed expression for the phase shifts in every order of perturbation theory.

The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism

It is shown that the use of the Riemann–Silberstein (RS) vector greatly simplifies the description of the electromagnetic field both in the classical domain and in the quantum domain. In this review, we describe many specific examples where this vector enables one to significantly shorten the derivations and make them more transparent. We also argue why the RS vector may be considered as the best possible choice for the photon wavefunction.

Tying Knots in Light Fields

We construct analytically, a new family of null solutions to Maxwells equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is shear free, preserves the topology of the knots and links. Our approach combines the construction of null fields with complex polynomials on S3. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.

Entropic uncertainty relations

Abstract New entropic uncertainty relations for angle-angular momentum and position-momentum, derived recently by Partovi, are related to older relations of a similar type, which were proved by Bialynicki-Birula and Mycielski. Significantly improved lower bounds are obtained in both cases.

V Photon Wave Function

Publisher Summary This chapter describes the photon wave function, and explains the basic properties of a well-defined mathematical object—a six-component function of space-time variables—that describes the quantum state of the photon. The most essential property that does not hold for the photon wave function is that the argument of the wave function cannot be directly associated with the position operator of the photon. The position operator for the photon simply does not exist. However, one should remember that for massive particles also, the true position operator exists only in the nonrelativistic approximation. The concept of localization associated with the Newton-Wigner position operator is not relativistically invariant. The photon wave function is not restricted to the wave mechanics of photons. The same wave functions also appear as mode functions in the expansion of the electromagnetic field operators.

Canonical separation of angular momentum of light into its orbital and spin parts

It is shown that the photon picture of the electromagnetic field enables one to determine unambiguously the splitting of the total angular momentum of the electromagnetic field into the orbital part and the spin part.

FERMI ACCELERATOR IN ATOM OPTICS

We study the classical and quantum dynamics of a Fermi accelerator realized by an atom bouncing off a modulated atomic mirror. We find that in a window of the modulation amplitude, dynamical localization occurs in both position and momentum. A recent experiment [A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995)] shows that this system can be implemented experimentally.