arXiv subjects
Publications and source records attributed to Zofia Bialynicka-Birula.
Sharp uncertainty relations restricting the values of variances in the position space and in the momentum (wavevector) space are derived. They have the same form $ΔrΔk\ge 5/2$ in the classical theory of light beams, in the quantum theory of coherent light beams, and in the quantum theory of individual photons.
In relativistic theories the principle of microscopic causality states that ``information cannot travel faster than the speed of light'' \cite{kaku}. In the present work we show that the time evolution of relativistic wave functions violates this principle. We consider here the wave functions of massless and massive particles. In the case of massless particles the wave functions which violate the microscopic causality have an analytic form while in the case of massive particles we have to rely on numerical calculations. In both cases the wave functions which are strictly localized at $t=0$, at later times do not vanish {\it outside} the future light cone. \end{abstract}
In this work we extend the Zeldovich formula, which was originally derived for the free electromagnetic field and was interpreted as the number of photons. We show that our extended formula gives a universal dimensionless measure of the overall strength of electromagnetic fields: free fields and fields produced by various sources, in classical and in quantum theory. In particular, we find that this number (the Zeldovich number) for macroscopic systems is huge, of the order of $10^{20}$. For the hydrogen atom in the ground state it is equal to 0.025 and for the xenon atom it is around 50.
We derive new solutions of the Schrödinger equation which describe the motion of particles in the Penning trap. These solutions are direct counterparts of classical orbits. They are obtained by injection of classical trajectories into the wave functions of stationary solutions.
We derive new solutions of the Schrödinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an illustration of the meaning of the Ehrenfest theorem in relativistic quantum mechanics.
We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearly seen. In this work we describe backflow in relativistic theories but this phenomenon can occur in the solutions of all kinds of wave equations: quantum and classical.
Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.
It is shown that in the greatly simplified model of the mutually interacting electron-positron pairs and the electric field time-crystal structures can spontaneously form. For a special choice of parameters we find a periodic modulation of the pair number and the electric field.
It is shown that the interpretation of the electron wave function as a classical field is untenable because the so called energy-density defined in \cite{seb} takes on negative values in some regions. The claim that the velocity of the electron never exceeds the speed of light is also invalid. The velocity, as defined by the author, becomes even infinite at some points.
We show that the standard method of introducing the quantum description of the electromagnetic field -- by canonical field quantization -- is not the only one. We have chosen here the relativistic quantum mechanics of the photon as the starting point. The treatment of photons as elementary particles merges smoothly with the description in terms of the quantized electromagnetic field but it also reveals some essential differences. The most striking result is the appearance of various measures of fidelity for quantum states of photons. These measures are used to characterize the localization of photons.
Berry phase is a very general concept. It is applied here to families of solutions of the Dirac equation with different values of spin. The value of the Berry phase in the spin space is given by the same expression as was found before in the momentum space.
In our Comment we question the validity of the claim made by the authors of \cite{cc} that their solutions of the Dirac equation in an external {\em time-dependent} electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has {\em infinite} time duration, there is a continuous electron-positron pair creation and annihilation. Without the proper accounting for these processes, the mathematical solutions of the Dirac equation are not directly applicable to realistic physical situations. In particular, the time evolution of the average values $\langle x\rangle$ and $\langle y\rangle$ does not describe the electron trajectory but the motion of some combination of the electron and positron charge distributions with pathological properties (zitterbewegung).
Analytical solutions of the Dirac equation in an external electromagnetic field are found such that according to the field-theoretic interpretation electron-positron pairs are trapped for a period of time. The naive one-particle interpretation of the Dirac wave function fails in this case completely. Simple electromagnetic field which produces this effect was undeniably concocted and may look artificial but the phenomenon of time traps seems real.
The purpose of this article is to show that the standard method of introducing the quantum description of the electromagnetic field -- by canonical field quantization -- is not the only one. We have chosen instead as the starting point the relativistic quantum mechanics of photons. Our present understanding of the nature of photons significantly differs from what has been known years ago when the concept of a photon has only been emerging. We show how the description of photons treated as elementary particles merges smoothly with the classical description of the electromagnetic field and leads finally to the full theory of the quantized electromagnetic field.
The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound $3/2\hbar$ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for photons.
All known solutions of the Dirac equation describing states of electrons endowed with angular momentum are very far from our notion of the electron as a spinning charged bullet because they are not localized in the direction of propagation. We present here analytic exact solutions, eigenstates of the total angular momentum component $M_z$, that come very close to this notion. These new solutions of the Dirac equation have also intricate topological properties similar to the hopfion solutions of the Maxwell equations.
It is shown that the description of light beams in terms of the corresponding photon quantum numbers elucidates the properties of these beams. In particular, this description shows that the helicity quantum number plays the fundamental role. This mode of description is applied to twisted and knotted electromagnetic waves. We concentrate on the cases where photon wave functions are eigenfunctions of one component of angular momentum. We discovered that for knotted waves the eigenvalue of the angular momentum determines the topology of knots.
Quantum mechanics of photons is derived from the theory of representations of the Poincaré group developed by Wigner. This theory places helicity as the most fundamental property; angular momentum and polarization are secondary characteristics. The properties of the beams of light are shown to be fully determined by the quantum states of the photons. Polarization of light beams is explained as the freedom to chose an arbitrary combination of the helicity states. Quantum mechanics of photons enables one to give a precise meaning to the concept of wave-particle duality.