New sum and product uncertainty relations, containing variances of up to five observables, but not containing explicitly their covariances, are derived. New inequalities for three observables, ...especially for the angular momentum and spin-1/2 operators, are also presented.
Rotating quantum Gaussian packets Dodonov, V V
Journal of physics. A, Mathematical and theoretical,
10/2015, Volume:
48, Issue:
43
Journal Article
Peer reviewed
Open access
We study two-dimensional quantum Gaussian packets with a fixed value of mean angular momentum. This value is the sum of two independent parts: the 'external' momentum related to the motion of the ...packet center and the 'internal' momentum due to quantum fluctuations. The packets minimizing the mean energy of an isotropic oscillator with the fixed mean angular momentum are found. They exist for 'co-rotating' external and internal motions, and they have nonzero correlation coefficients between coordinates and momenta, together with some (moderate) amount of quadrature squeezing. Variances of angular momentum and energy are calculated, too. Differences in the behavior of 'co-rotating' and 'anti-rotating' packets are shown. The time evolution of rotating Gaussian packets is analyzed, including the cases of a charge in a homogeneous magnetic field and a free particle. In the latter case, the effect of initial shrinking of packets with big enough coordinate-momentum correlation coefficients (followed by the well known expansion) is discovered. This happens due to a competition of 'focusing' and 'de-focusing' in the orthogonal directions.
I consider a possibility of parametric amplification of the microwave vacuum field in a reentrant cavity enclosing a nonlinear crystal whose refractive index is modulated by periodic high-intensity ...short laser pulses. The main result is that the total number of created 'Casimir quanta' depends neither on the laser beam shape, nor on the duration or power of individual pulses, but it depends on the total energy of all the pulses, provided the duration of each pulse is much shorter than the period of field oscillations in the selected resonant mode. The scheme can be feasible if reliable materials with high nonlinear coefficients can be found.
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential ...A=B(t)-y(1+α),x(1-α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.
Full text
Available for:
IZUM, KILJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK
We study the influence of the non-equidistancy of the frequency spectrum on the Dynamical Casimir effect in a rectangular cavity with a harmonically oscillating ideal wall. The transition from the ...linear growth of the mean photon number of photons created from vacuum in the cavity with equidistant spectrum to the exponential growth in a weakly non-equidistant case is shown explicitly.
► We consider the dynamical Casimir effect in different rectangular cavities. ► One-dimensional models predict the linear growth of the number of photons. ► Three-dimensional models predict the exponential growth. ► We show how one model is transformed continuously into the other. ► The key parameter is the degree of non-equidistancy of the frequency spectra.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
We study the evolution of the most general initial Gaussian packet with nonzero correlation coefficient between the coordinate and momentum operators in the presence of a repulsive delta-potential ...barrier, using the known exact propagator of the time-dependent Schrödinger equation. For the initial packet localized far enough from the barrier, we define the transmission coefficient as the probability of discovering the particle in the whole semi-axis on the other side of the barrier. It appears that the asymptotical transmission coefficient (calculated in the large time limit) depends on two dimensionless parameters: the normalized ratio of the potential strength to the initial mean value of momentum and the ratio of the initial momentum dispersion to the initial mean value of momentum. For small values of the second parameter, the result is reduced to the well-known formula for the transparency of the delta barrier, obtained in the plane-wave approximation by solving the stationary Schrödinger equation. For large values of the second parameter, the transmission coefficient can be much larger than that calculated in the plane-wave approximation. For a fixed initial spread of the packet in the coordinate space, the initial correlation coefficient influences the transparency of the barrier only indirectly, through the increase in the initial momentum dispersion.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Seventy five years ago, three remarkable papers by Schroedinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and ...a particle moving in uniform constant electric and magnetic fields. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the 'standard citation package' used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. I include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated in the late 1980s/early 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and 'nonclassical' states commenced. The 1990s are described in a more 'squeezed' manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists. (Author)
Abstract
The motion of an artificial Earth satellite with constant absolute value of the acceleration is considered. This requirement is equivalent to imposing a second-order nonlinear nonholonomic ...constraint or a third-order linear nonholonomic constraint. Two theories of motion of nonholonomic systems with high-order constraints are used for solving this problem. According to the first theory, a consistent system of differential equations is constructed with respect to the generalized coordinates and the Lagrange multipliers; the second theory is based on the application of the generalized Gauss principle. The results are different, although the constraints are satisfied in both theories. It turns out that infinitely many solutions can be built, but using these theories one can find two specific solutions. The question of the difference of these two solutions from the set of all other possible solutions is raised. We also simplify the previously obtained differential equations. The transition to dimensionless variables is made. Three parameters of motion prior to imposition of the constraint are single out, which control the motion after the application of the constraint. The solutions obtained from these theories of motion of nonholonomic systems are compared.
Exact and approximate formulas for the upper bound of the relative energy difference of two Gaussian states with a fixed fidelity between them are derived. The reciprocal formulas for the upper bound ...of the fidelity for the fixed value of the relative energy difference are also obtained. The bounds appear higher for pure states than for mixed ones, and their maximal values correspond to squeezed vacuum states. In particular, to guarantee the relative energy difference less than 10%, for quite arbitrary Gaussian states, the fidelity between them must exceed the level 0.998866.
The new conditions for the implementation of economic activity at the national and international levels require the accelerated formation of sovereignty in critical areas, one of which is financial. ...The formation of full-fledged financial sovereignty is relevant for all states of Greater Eurasia, and therefore it is advisable to implement this task through joint and coordinated efforts within the framework of a broad Eurasian partnership.
Aim
. Find out the most promising areas for the formation of financial sovereignty within the framework of a large Eurasian partnership.
Methods
. A comprehensive analysis of the causes and conditions for the dependence of national economies on external financial systems, a comparative analysis of the factors of Western components dominance in the global financial architecture.
Results
. The main directions of the formation of the financial sovereignty for the countries of Greater Eurasia on the basis of increasing the level of independence of national financial systems from the Western infrastructure of the financial sector are identified. The most promising methods for achieving this goal are proposed based on the creation of new types of reserve assets within the framework of a large Eurasian partnership, which are an alternative to debt obligations of the United States and European countries, as well as the formation of an integrated system for financial instruments trading by the countries of Greater Eurasia.
Conclusions
. The growing economic confront ation between the West and a number of the largest economies of Greater Eurasia is extremely relevant to the problem of the formation of real financial sovereignty, the achievement of which requires new approaches, the creation of a number of specialized institutions and large-scale cooperation between the leading economic centers of the continent in order to deepen the integration of national financial systems and markets.
PDF
