The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde

Homogeneously derived transit timings for 17 exoplanets and reassessed TTV trends for WASP-12 and WASP-4

We homogeneously analyse ∼3.2 × 105 photometric measurements for ∼1100 transit lightcurves belonging to 17 exoplanet hosts. The photometric data cover 16 years 2004–2019 and include amateur and professional observations. Old archival lightcurves were reprocessed using up-to-date exoplanetary parameters and empirically debiased limb-darkening models. We also derive self-consistent transit and radial-velocity fits for 13 targets. We confirm the nonlinear TTV trend in the WASP-12 data at a high significance, and with a consistent magnitude. However, Doppler data reveal hints of a radial acceleration about ( − 7.5 ± 2.2) m/s/yr, indicating the presence of unseen distant companions, and suggesting that roughly 10 per cent of the observed TTV was induced via the light-travel (or Roemer) effect. For WASP-4, a similar TTV trend suspected after the recent TESS observations appears controversial and model-dependent. It is not supported by our homogeneus TTV sample, including 10 ground-based EXPANSION lightcurves obtained in 2018 simultaneously with TESS. Even if the TTV trend itself does exist in WASP-4, its magnitude and tidal nature are uncertain. Doppler data cannot entirely rule out the Roemer effect induced by possible distant companions

Effects of BeneficialMutations in pykF Gene Vary over Time and across Replicate Populations in a Long-Term Experiment with Bacteria

The fitness effects of mutations can depend on the genetic backgrounds in which they occur and thereby influence future opportunities for evolving populations. In particular, mutations that fix in a population might change the selective benefit of subsequent mutations, giving rise to historical contingency. We examine these effects by focusing on mutations in a key metabolic gene, pykF, that arose independently early in the history of 12 Escherichia coli populations during a long-term evolution experiment. Eight different evolved nonsynonymous mutations conferred similar fitness benefits of ∼10% when transferred into the ancestor, and these benefits were greater than the one conferred by a deletion mutation. In contrast, the same mutations had highly variable fitness effects, ranging from ∼0% to 25%, in evolved clones isolated from the populations at 20,000 generations. Two mutations that were moved into these evolved clones conferred similar fitness effects in a given clone, but different effects between the clones, indicating epistatic interactions between the evolved pykF alleles and the other mutations that had accumulated in each evolved clone. We also measured the fitness effects of six evolved pykF alleles in the same populations in which they had fixed, but at seven time points between 0 and 50,000 generations. Variation in fitness effects was high at intermediate time points, and declined to a low level at 50,000 generations, when the mean fitness effect was lowest. Our results demonstrate the importance of genetic context in determining the fitness effects of different beneficial mutations even within the same gene