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Register a new userDiscrete wavelet structure and discrete energy of a classical plane light wave
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In this letter, the wavelet transform is used to decompose the classical linearly polarized plane light wave into a series of discrete Morlet wavelets. It is found that the energy of the light wave can be discrete, associated with its discrete wavelet structure.It is also found that the changeable energy of a basic plane light wave packet or wave train of wave vector $mathord{buildrel{lower3pthbox{$scriptscriptstylerightharpoonup$}}over k} $ and with discrete wavelet structure can be with the form of ${H_{0k}} = n{p_{0k}}omega$ $(n = 1,2,3,...)$, where $n$ is the parameter of discrete wavelet structure, $omega $ is the idler frequency of the light wave packet or wave train, and ${p_{0k}}$ is a constant to be determined.This is consistent with the energy division of $P$ portions in Planck radiation theory, where $P$ is an integer. Finally, the random light wave packets with $n=1$ are used to simulate the Mach-Zehnder interference of single photons, showing the wave-particle duality of light.
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We consider the long-term dynamics of nonlinear dispersive waves in a finite periodic domain. The purpose of the work is to show that the statistical properties of the wave field rely critically on the structure of the discrete resonant manifold (DRM). To demonstrate this, we simulate the two-dimensional MMT equation on rational and irrational tori, resulting in remarkably different power-law spectra and energy cascades at low nonlinearity levels. The difference is explained in terms of different structures of the DRM, which makes use of recent number theory results.
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Classical oscillator differential equation is replaced by the corresponding (finite time) difference equation. The equation is, then, symmetrized so that it remains invariant under the change d going to -d, where d is the smallest span of time. This symmetric equation has solutions, which come in reciprocally related pairs. One member of a pair agrees with the classical solution and the other is an oscillating solution and does not converge to a limit as d goes to 0. This solution contributes to oscillator energy a term which is a multiple of half-integers.
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