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In this work, we estimate the relativistic corrections to the energy spectrum of hydrogen resulting from the full one-photon-exchange interaction using a highly precise numerical method. In the frame of the effective Schrodinger-like equation, which is derived exactly from the Bethe-Salpeter equation in quantum electrodynamics, we express the effective potential corresponding to the full one-photon-exchange interaction in terms of eight scalar functions. Unlike the usual calculations performed in the references, where the effective potential is expanded in terms of momenta order by order, we retain the exact momentum dependence in the effective potential to estimate its corrections to the energy spectrum of hydrogen using a highly precise numerical method. We also use the same numerical method to double-check the results in the non-relativistic case to ensure accuracy. We discuss the comparison of the numerical results with those obtained using the usual bound-state perturbative theory. Our calculations suggest that it is possible to accurately account for all the relativistic contributions using this method. It would be interesting to extent these calculations to positronium, muonic hydrogen, and cases involving nuclear structure and radiative corrections., Comment: 2 figures

Smith-Purcell radiation (SPR) occurs when an electron skims above a spatial grating, but the fixed momentum compensation from the static grating imposes limitations on the emission wavelength. It has been discovered that a temporally periodic system can provide energy compensation to generate light emissions in free space. Here, we introduce temporal SPR (t-SPR) emerging from a time grating and propose a generalized t-SPR dispersion equation to predict the relationship between radiation frequency, direction, electron velocity, modulation period, and harmonic orders. Compared to conventional SPR, t-SPR can: 1) Provide a versatile platform for manipulating SPR emission through temporal modulation (e.g., period, amplitude, wave shape). 2) Exhibit strong robustness to the electron-grating separation, alleviating the constraints associated with extreme electron near-field excitation. 3) Introduce additional energy channels through temporal modulation, enhancing and amplifying emission., Comment: 6 pages, 3 figures

Recently, metasurfaces have experienced revolutionary growth in the sensing and superresolution imaging field, due to their enabling of subwavelength manipulation of electromagnetic waves. However, the addition of metasurfaces multiplies the complexity of retrieving target information from the detected fields. Besides, although the deep learning method affords a compelling platform for a series of electromagnetic problems, many studies mainly concentrate on resolving one single function and limit the research's versatility. In this study, a multifunctional deep neural network is demonstrated to reconstruct target information in a metasurface targets interactive system. Firstly, the interactive scenario is confirmed to tolerate the system noises in a primary verification experiment. Then, fed with the electric field distributions, the multitask deep neural network can not only sense the quantity and permittivity of targets but also generate superresolution images with high precision. The deep learning method provides another way to recover targets' diverse information in metasurface based target detection, accelerating the progression of target reconstruction areas. This methodology may also hold promise for inverse reconstruction or forward prediction problems in other electromagnetic scenarios.