Time-dependent Schr\"odinger equations having isomorphic symmetry algebras. II. Symmetry algebras, coherent and squeezed states

Using the transformations from paper I, we show that the Schr\"odinger equations for: (1)systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators, all have isomorphic Lie space-time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is deomonstrated.Comment: LaTeX, 22 pages, new format, edited, with added discussion of the classical motio

Seeking a solution of the Pioneer Anomaly

The 1972 and 1973 launched Pioneer 10 and 11 were the first missions to explore the outer solar system. They achieved stunning breakthroughs in deep-space exploration. But around 1980 an unmodeled force of \sim 8 \times 10^{-8} cm/s^2, directed approximately towards the Sun, appeared in the tracking data. It later was unambiguously verified as not being an artifact. The origin remains unknown (although radiant heat remains a likely cause). Increasing effort has gone into understanding this anomaly. We review the situation and describe programs to resolve the issue.Comment: 7 pages, 1 figure, invited talk at the Fourth Meeting on CPT and Lorentz Symmetry, 8-11 Aug. 2007, held at Indiana Universit

Earth Flyby Anomalies

In a reference frame fixed to the solar system's center of mass, a satellite's energy will change as it is deflected by a planet. But a number of satellites flying by Earth have also experienced energy changes in the Earth-centered frame -- and that's a mystery.Comment: 5 pagea 3 figure

Displacement-Operator Squeezed States. I. Time-Dependent Systems Having Isomorphic Symmetry Algebras

In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schr\"odinger equation for systems with potentials $V(x,\tau)=g^{(2)}(\tau)x^2+g^{(1)}(\tau)x +g^{(0)}(\tau)$. We describe a set of number-operator eigenstates states, $\{\Psi_n(x,\tau)\}$, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, $\Psi_0$, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, $\tau$. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of $V(x,\tau)$ and having isomorphic Lie algebras, will be dealt with in the following paper (II).Comment: 23 pages, LaTe

The organisation and functions of local Ca²⁺ signals

Calcium (Ca2+) is a ubiquitous intracellular messenger, controlling a diverse range of cellular processes, such as gene transcription, muscle contraction and cell proliferation. The ability of a simple ion such as Ca2+ to play a pivotal role in cell biology results from the facility that cells have to shape Ca2+ signals in space, time and amplitude. To generate and interpret the variety of observed Ca2+ signals, different cell types employ components selected from a Ca2+ signalling 'toolkit', which comprises an array of homeostatic and sensory mechanisms. By mixing and matching components from the toolkit, cells can obtain Ca2+ signals that suit their physiology. Recent studies have demonstrated the importance of local Ca2+ signals in defining the specificity of the interaction of Ca2+ with its targets. Furthermore, local Ca2+ signals are the triggers and building blocks for larger global signals that propagate throughout cells