The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

The Physics of Quantum Information

This Review article provides an overview of the fundamental origins and important applications of the main spin–orbit interaction phenomena in modern optics that play a crucial role at subwavelength scales. Light carries both spin and orbital angular momentum. These dynamical properties are determined by the polarization and spatial degrees of freedom of light. Nano-optics, photonics and plasmonics tend to explore subwavelength scales and additional degrees of freedom of structured — that is, spatially inhomogeneous — optical fields. In such fields, spin and orbital properties become strongly coupled with each other. In this Review we cover the fundamental origins and important applications of the main spin–orbit interaction phenomena in optics. These include: spin-Hall effects in inhomogeneous media and at optical interfaces, spin-dependent effects in nonparaxial (focused or scattered) fields, spin-controlled shaping of light using anisotropic structured interfaces (metasurfaces) and robust spin-directional coupling via evanescent near fields. We show that spin–orbit interactions are inherent in all basic optical processes, and that they play a crucial role in modern optics.

Spin-orbit interactions of light

Tables of integrals

It is shown that electromagnetic vortices can act as beam guides for charged particles. The confinement in the transverse directions is due to the rotation of the electric and magnetic fields around the vortex line. A large class of exact solutions describing various types of relativistic beams formed by an electromagnetic wave with a simple vortex line is found both in the classical and in the quantum case. In the second case, the motion in the transverse direction is fully quantized. Particle trajectories trapped by a vortex are very similar to those in a helical undulator.

Particle beams guided by electromagnetic vortices: new solutions of the Lorentz, Schrödinger, Klein-Gordon, and Dirac equations.

New uncertainty relations based on the information entropy are reviewed and contrasted with the traditional uncertainty relations, which were based on the dispersions of the the physical variables. Improved lower bounds are given for the position-momentum and the angle-angular momentum pairs. Novel uncertainty relation for the angular distribution and angular momentum in the three-dimensional space is introduced.

Entropic uncertainty relations in quantum mechanics

Quantum mechanical uncertainty relations for the position and the momentum and for the angle and the angular momentum are expressed in the form of inequalities involving the Renyi entropies. These uncertainty relations hold not only for pure but also for mixed states. Analogous uncertainty relations are valid also for a pair of complementary observables (the analogs of x and p) in N‐level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Renyi entropies. The mathematical proofs of all the inequalities discussed in this paper can be found in Phys. Rev. A 74, No. 5 (2006); arXiv:quant‐ph/0608116.

Rényi Entropy and the Uncertainty Relations

Quantum mechanics as a generalization of Nambu dynamics to the Weyl-Wigner formalism

We show that a free quantum particle in two dimensions with zero angular momentum (s wave) in the form of a ring-shaped wave packet feels an attraction towards the center of the ring, leading first to a contraction followed by an expansion. An experiment to demonstrate this effect is also outlined.

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Iwo Bialynicki-Birula

Papers

Particle beams guided by electromagnetic vortices: new solutions of the Lorentz, Schrödinger, Klein-Gordon, and Dirac equations.

Entropic uncertainty relations in quantum mechanics

Rényi Entropy and the Uncertainty Relations

Quantum mechanics as a generalization of Nambu dynamics to the Weyl-Wigner formalism

In- and outbound spreading of a free-particle s-wave.