![]() ![]() The homogeneous sphere (see Section 3.4.2) has particularly simple orbits. To illustrate the concepts introduced in the previous subsection, we consider orbital properties in a few simple spherical potentials. Orbits in specific spherical potentials ¶ Radial and azimuthal frequencies are simply equal to \(2\pi\) over the radial and azimuthal period. 2007 for methods for dealing with this situation). Some care is necessary in this numerical calculation, because the integrands integrably diverge at the end-points of the integration interval (see Press et al. All of the quantities \(T_r\), \(\Delta \psi\), and \(T_\psi\) typically need to be computed numerically. Where we need the absolute value of \(\Delta \psi\), because \(\Delta \psi\) can be negative. The general theory of relativity and galaxies Part IV: Galaxy formation and evolution (under construction) Equilibria of elliptical galaxies and dark matter halos Orbits in triaxial mass distributions and surfaces of section ![]() Gravitation in elliptical galaxies and dark matter halos The kinematics and dynamics of galactic rotation Equilibria of collisionless stellar systems Orbits in the isochrone potential and other spherical potentials General properties of orbits in spherical potentials Vertical dashed black lines indicate the value of t and T where the Husimi projections are shown. Horizontal dashed green and blue lines indicate the asymptotic value of each measure in both panels. The selected initial coherent state ρ ̂ x is defined by the phase-space coordinates x = ( 2.894, 0 − 0.4, 0 ) with energy width σ x = 0.693 (units of ε). The bottom rectangular panels show the Rényi occupations L 2 ( A, ρ ̂ x ( t ) ) (solid blue curve) and L 2 ( ε x, ρ ̂ x ( t ) ) (solid green curve) for (a) a pure initial coherent state ρ ̂ x ( t ) and (b) the time-mixed coherent state ρ ¯ x ( T ) in the chaotic-energy region ε x = 1. The top square panels show the Husimi function projected in the atomic coordinate plane ( Q, P ) at different times t and T for both (a) the pure ρ ̂ x ( t ) state and (b) the time-mixed ρ ¯ x ( T ) coherent state. ![]() We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely, one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. ![]()
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