Dragomir M. Davidović

The K-shell ionisation of atoms by relativistic protons

The K-shell ionisation is investigated using an extension of the first-order time-dependent perturbation-theory treatment of Moller (1932) taking Dirac plane waves for the description of the incident and scattered protons and the Darwin approximation for the relativistic wavefunctions of the K-shell atomic electrons and the ejected electron. The differential cross sections delta 2 sigma / delta k delta q and d sigma /dk, where k is the wavenumber of the ejected electron and hq is the momentum change of the proton, are calculated. The double differential cross section for collisions in which a change of spin of the atomic electron occurs is less sharply peaked for small q, and has a broader overall distribution with respect to q, than the double differential cross section for collisions without spin change, although being much less in magnitude than the latter for small q. This leads to total K-shell ionisation cross sections sigma with significant contributions coming from collisions with spin change of the atomic electron for large atomic numbers at high energies of impact.

The libration limits of the elastic pendulum

The motion of the elastic pendulum is ruled by coupling between the pendulum mode and the harmonic oscillator mode. The bob of the pendulum executes trajectories in a domain of the vertical plane, the shape and size of which are set by the initial conditions and the total energy of the pendulum. The limits of this libration, in exact resonance, are studied using parabolic coordinates. Other Hamiltonians, with similar interaction terms, exhibit similar libration limits.

The K-shell ionization of atoms by relativistic electrons

The theoretical investigation carried out by Arthurs and Moiseiwitsch (1958) has been extended to higher energies and to atoms with greater nuclear charge Z. Allowance for the difference between the theoretical and experimental ionization energies arising from the potential field of the electrons in the outer shells is made using a similar procedure to that due to Perlman (1960). The authors have calculated K-shell ionization cross sections as functions of the electron impact energy for C, Al, Ni, Cu, Ag, Sn, Au and compared their theoretical curves with the available experimental data, the accordance being somewhat variable.

Dynamical time versus system time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system time operator, i.e., for the free one-dimensional particle, is obtained.

The Doppler interpretation of Ro/mer’s method

Ro/mer’s method for the determination of the speed of light, based on the observation of the immersions and emersions of Jupiter’s satellites, can be interpreted in the spirit of the Doppler effect. Although this statement appears to be plausible, textbooks very seldom adopt this point of view. Granted the tabular value of the satellite’s period, this interpretation permits one to determine the speed of light in about a week, as contrasted to half a year in the original method. It is suggested to include the measurement as a student experiment in the optics curriculum. Results of actual observations are given.

A theoretical analysis of reflection of X‐rays from water at energies relevant for diagnostics

The reflection of X‐rays from a semi‐infinite water target, for energies used in X‐ray diagnostics, is treated by the analog Monte Carlo simulation. In the developed procedure it was possible to calculate separately contributions of photons scattered, before reflection, fixed number of times with target electrons. It turned out that multiple collision type of reflection dominates at all energies investigated, whenever the absorption is small. The same process was also treated analytically as the classical albedo problem for isotropic scattering without energy loss. Very good agreement of results of the two approaches is obtained.

An electrostatic interpretation of the weighted flow theorem of electromagnetic flowmeters

ContentsThe weighted flow theorem of electromagnetic flowmeters links the output voltage of an electromagnetic flowmeter with the spatial velocity distribution of a fluid and the weight function. The existing derivations of the theorem are based on a formal Greens function treatment of the underlying partial differential equation describing flowmeter operation. The main aim of this paper is to provide a physically transparent and mathematically rigorous picture of the origin of the flowmeter output voltage.We show that electrostatic charges build up both in the volume of the meter and on its surfaces as a weakly conductive fluid enters the magnetic field of the meter. The flowmeter output voltage is completely determined by these volume and surface charges. Departing from the physical conditions in the fluid and the fluid boundaries we establish relations satisfied by those charge densities and then derive the analytical expression for the output voltage by standard electrostatic methods. The resulting expression is transformed into a volume integral of the scalar product of fluid velocity and the weight vector which in turn is equal to the vector product of the magnetic induction and a virtual current, as introduced by Bevir for the first time. We further show that this latter quantity may be interpreted as that part of the normalized current density of the operating flowmeter the contours of which are closed outside of the fluid. We also give a simple interpretation of the results based on conservation of energy.We hope that the proposed electrostatic approach could provide a more intuitive insight into flowmeter operation to electrical engineers than the general Greens function formalism.ÜbersichtDas Wertigkeitstheorem der induktiven Durchflußmesser verbindet die Ausgangsspannung des induktiven Durchflußmessers mit der räumlichen Verteilung der Geschwindigkeit der Flüssigkeit und der Wertigkeitsfunktion. Die bisherigen Ableitungen dieses Theorems stützen sich auf die Benützung der Greenschen Funktion für die grundsätzliche partielle Differentialgleichung des Durchflußmessers. Das Ziel der vorliegenden Arbeit ist, eine physikalisch klare und mathematisch strenge Vorstellung vom Ursprung der Ausgangsspannung des induktiven Durchflußmessers zu geben.Es wird gezeigt, daß elektrostatische Ladungen im Volumen und an den Oberflächen des induktiven Durchflußmessers entstehen, wenn eine schwach leitende Flüssigkeit in das Magnetfeld des Meßgeräts eintritt. Die Ausgangsspannung des Durchflußmessers ist durch diese Volumen-und Oberflächenladungen vollständig bestimmt und kann aus bekannten elektrostatischen Beziehungen ermittelt werden. Der resultierende Ausdruck wird in ein Volumenintegral des Skalarproduktes der Flüssigkeitsgeschwindigkeit und des Wertigkeitsvektors transformiert. Der Wertigkeitsvektor ist dem Vektorprodukt der magnetischen Induktion und einer virtuellen Stromdichte gleich, die erstmals von Bevir eingeführt wurde. Es wird weiter gezeigt, daß diese Größe auch als diejenige Komponente der normalisierten Stromdichte aufgefaßt sein kann, deren Stromlinien sich über den Außenkreis schließen. Wir geben auch eine einfache energetische Deutung der Resultate an.Wir hoffen, daß die vorliegende elektrostatische Deutung für Elektroingenieure besser geeignet und anschaulicher ist als die formelle Benützung der Greenschen Funktion.

Subtle inconsistencies in the straightforward definition of the logarithmic function of annihilation and creation operators and a way to avoid them

Using the resolution of identity spanned by coherent states of the harmonic oscillator, any entire function of the creation and the annihilation operators and its action on a vector in Hilbert space can be defined directly and simply. We show that such a direct approach applied to non-entire functions In a and In â † , present in the literature, may lead to errors and contradictions. We elucidate their roots and propose a way to avoid them. We discuss the obtained results.

Coherent states on von Neumann lattice and some of their applications

Coherent states on von Neumann lattice, which form complete but not overcomplete set, are orthogonalized using Löwdin procedure and corresponding coefficients arising in orthogonalization are tabulated. So obtained states may be interpreted as eigenstates of operators of optimal unsharp measurement of coordinate and momentum, in a slightly modified von Neumann sense. Using the obtained results the most important statistical characteristics of these operators are calculated on coherent states and compared with corresponding values of standard quantum mechanical operators of coordinate and momentum.