Search

Rationale Social identification with the people in one's neighbourhood has a wide variety of benefits for individual and community health and wellbeing. In particular, previous research shows that residents' social identification with their neighbourhood is protective of mental health. However, researchers are only just beginning to design and evaluate interventions that directly target social identification on health grounds. Objective This longitudinal study evaluated a whole-of-community intervention at scale (Neighbour Day, 2019), in which Australian residents were encouraged to build social connections in their local community. Neighbour Day is a campaign that seeks to raise public awareness of the importance of connecting with neighbours and had a reach of approximately 300,000 people in 2019. Methods Participants were 437 hosts of neighbourhood events held across 276 diverse suburbs across Australia. Participants were surveyed at three-time points; before and after Neighbour Day, as well as at six-month follow up. Results Hosting a Neighbour Day event led to a significant increase in neighbourhood social identification, which was sustained six months later. This increase in social identification predicted increased social cohesion, reduced loneliness and improved wellbeing. Conclusions This study provides evidence that neighbourhood identification is an effective target mechanism to curb loneliness and social fragmentation in the community. Implications are discussed with a focus on how social identity-building interventions can be effectively implemented in community settings to benefit public health.

We complete our study on the uncertainties in position and momentum associated with the semiclassical Hagedorn wave packets by first filling in a technical gap in the dynamics of bound states for isochronous potentials. We then consider scattered states and show that, if the packet is reflected from the potential or transmitted through a symmetric potential, then a minimal uncertainty “initial” state cannot in general lead to a “final” state with minimal uncertainty, and we give an explicit relationship for the difference in terms of a characteristic time associated with classical trajectories. We also characterize the behavior of the uncertainty product in the case where the underlying classical dynamics lead to capture by the potential.We complete our study on the uncertainties in position and momentum associated with the semiclassical Hagedorn wave packets by first filling in a technical gap in the dynamics of bound states for isochronous potentials. We then consider scattered states and show that, if the packet is reflected from the potential or transmitted through a symmetric potential, then a minimal uncertainty “initial” state cannot in general lead to a “final” state with minimal uncertainty, and we give an explicit relationship for the difference in terms of a characteristic time associated with classical trajectories. We also characterize the behavior of the uncertainty product in the case where the underlying classical dynamics lead to capture by the potential.

We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode .¯ for the Hamiltonian H. ¯ when the energyE and Planck's constanth satisfy the appropriate Bohr-Sommerfeld conditions. This means that .¯ h/ is an approximate solution of the Schr ¨ odinger equation in the sense that k(H. ¯ E) .¯k6C¯ 3=2 k .¯ h/k: It follows that H. ¯ h/ has some spectrum within a distance C¯ 3=2 of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.

We study the time evolution of the uncertainties Δx and Δp in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as P1(t)+tP2(t), where P1 and P2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product ΔxΔp achieves its minimum of ħ/2 at arbitrarily large times.

An explicit and computable asymptotic integral representation is obtained for the time‐dependent Wigner distribution associated with the initial quantum state ψ(x,0)=f(x) eiS(x)/ℏ in the semiclassical (ℏ→0) limit. The approximations are valid to arbitrarily high order in ℏ over any finite time interval. The leading order term is further analyzed to obtain a classically determined phase space function which is related to a classical probability density on phase space. The results hold for a large class of time‐dependent potentials.

An explicit integral representation is obtained in the E→0 limit of the solution of the wave equation utt=c(x)2Δu on Rn (n≤3) with initial data of the form u(x,0)=f(x)eiS(x)/E. Estimates are given with respect to the energy norm and the representation is valid for any finite time interval. Away from caustics, the integral is asymptotically evaluated and it is found that the solution is determined by time‐dependent geometrical optics.

We consider the time-dependent Schrodinger equation in one spatial dimension with a time-dependent quadratic Hamiltonian and, under appropriate assumptions on the coefficient functions in the Hamiltonian, construct solutions that approach minimal uncertainty states for large times.

Certain rigorous asymptotic solutions to the wave equation with spatially dependent wave speed are constructed. These solutions have the form of inhomogeneous Gaussian pulses, allowing for the determination of the so-called inhomogeneous rays from the solution of an underlying system of ordinary differential equations. Inhomogeneous ray traces are numerically generated for several examples.

The ℏ→0 limit of the quantum dynamics determined by the Hamiltonian H(ℏ) =−(ℏ2/2m)Δ+V on L2(Rn) is studied for a large class of potentials. By convolving with certain Gaussian states, classically determined asymptotic behavior of the quantum evolution of states of compact support is obtained. For initial states of class C10 the error terms are shown to have L2 norms of order ℏ1/2−e for arbitrarily small positive e.