By a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg-Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wave functions and closed analytic expressions for time-evolution operators.

The origin and transport of the IAA responsible for rooting was studied in carnation (Dianthus caryophyllus L.) cuttings obtained from secondary shoots of the mother plants. The presence of mature leaves in the cuttings was essential for rooting. Removal of the apex and/or the youngest leaves did not reduce the rooting percentage as long as mature leaves remained attached. Removal of mature leaves inhibited rooting for a 24-day period during which the basal leaves grew and reached maturity. After this period rooting progressed as in intact cuttings. Auxin (NAA + IBA) applied to the stem base of defoliated cuttings was about 60% as effective as mature leaves in stimulating rooting. Application of NPA to the basal internode resulted in full inhibition of rooting. The view, deduced from these results, that auxin from mature leaves is the main factor controlling the rooting process was reinforced by the fact that mature leaves contained IAA and exported labelled IAA to the stem. The distribution of radioactivity after application of (5-3H)-IAA to mature leaves showed that auxin movement in the stem was basipetal and sensitive to NPA inhibition. The features of this transport were studied by applying 3H-IAA to the apical cut surface of stem sections excised from cuttings. The intensity of the transport was lower in the oldest node than in the basal internode, probably due to the presence of vascular traces of leaves. Irrespective of the localization of the sections and the carnation cultivar used, basipetal IAA transport was severely reduced when the temperature was lowered from 25 to 4 degrees C. The polar nature of the IAA transport in the sections was confirmed by the inhibition produced by NPA. Local application of IAA to different tissues of the sections revealed that polar auxin transport was associated with the vascular cylinder, the transport in the pith and cortex being low and apolar. The present results strongly support the conclusion that IAA originating from the leaves and transported in the stem through the polar auxin transport pathway was decisive in controlling adventitious rooting.

The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the "failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of "constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring "anomalous" operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively.

Different families of states, which are solutions of the time-dependent free Schrödinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in [1]. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.

to the corresponding non-relativistic Hermite polynomial as c -> oo . particular power of the vacuum reducing to unity in this limit and a polynomial leading weight function (the vacuum), which leads to the Gaussian one in the c -» oo limit, a