This velocity is so nearly that of light, that it seems we have strong reasons to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. They can have any amplitude E 0 (with B 0 depending on E 0 as will be shown later), any wavelength λ, and be retarded or advanced by any phase φ, but they can only travel through empty space at one wave speed c. Given Maxwell's four equations (which are based on observation) we have shown that electromagnetic waves must exist as a consequence. We'll use c for this one since it's the first letter in the Latin word for swiftness - celeritas. Set these two experssions equal to one another and watch. Find the second space and time derivatives of the electric field… ∇ 2 E = −Īnd substitute them back into the second order partial differential equation. Let's examine our possible solution in more detail. Such a solution is an electromagnetic wave. These changes then propagate away at a finite speed. The interesting ones have electric and magnetic fields that change in time. The system size is j = 30.This particular example is one dimensional, but there are two dimensional solutions as well - many of them. 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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