The long-term cyclotron dynamics of relativistic wave packets: spontaneous collapse and revival

In this work we study the effects of collapse and revival as well as {\it Zitterbewegung} (ZB) phenomenon, for the relativistic electron wave packets, which are a superposition of the states with quantum numbers sharply peaked around some level $n_0$ of the order of few tens. The probability densities as well as average velocities of the packet center and the average spin components were calculated analytically and visualized. Our computations demonstrate that due to dephasing of the states for times larger than the cyclotron period the initial wave packet (which includes the states with the positive energy only) loses the spatial localization so that the evolution can no longer be described classically. However, at the half-revival time $t=T_R/2$ its reshaping takes place firstly. The behavior of the wave packet containing the states of both energy bands (with $E_n>0$ and $E_n<0$) is more complicated. At short times of a few classical periods such packet splits into two parts which rotate with cyclotron frequency in the opposite directions and meet each other every one-half of the cyclotron period. At these moments their wave functions have significant overlap that leads to ZB. At the time of fractional revival each of two sub-packets is decomposed into few packets-fractions. However, at $t=T_R$ each of the two sub-packets (with positive or negative energy) restores at various points of the cyclotron orbit, that makes it impossible reshaping of initial wave packet entirely unlike the wave packet which consists of states with energies $E_n>0$ only. Obtained results can be useful for the description of electromagnetic radiation and absorption in relativistic plasma on astrophysics objects, where super high magnetic field has the value of the order $10^8-10^9$T, as well as for interpretation of experiments with trapped ions

Harper-Hofstadter problem for 2D electron gas with k{\bf k}-linear Rashba spin-orbit coupling

The Harper-Hofstadter problem for two-dimensional electron gas with Rashba spin-orbit coupling subject to periodic potential and perpendicular magnetic field is studied analytically and numerically. The butterfly-like energy spectrum, spinor wave functions as well as the spin density and average spin polarization are calculated for actual parameters of semiconductor structure. Our calculations show that in two-dimensional electron gas subject to periodic potential and uniform magnetic field the effects of energy spectrum splitting caused by large spin-orbit Rashba coupling can be observed experimentally.Comment: 8 pages, 6 figures. submitted to Europhys. Letter

Space-time evolution of Dirac wave packets

In this work we study the dynamics of free 3D relativistic Gaussian wave packets with different spin polarization. We analyze the connection between the symmetry of initial state and the dynamical characteristics of moving particle. The corresponding solutions of Dirac equation having different types of symmetry were evaluated analytically and numerically and after that the electron probability densities, as well as, the spin densities were visualized. The average values of velocity of the packet center and the average spin were calculated analytically, and the parameters of transient Zitterbewegung in different directions were obtained. These results can be useful for the interpretation of future experiments with trapped ions.Comment: 10 pages, 7 figure

The Problem of Empathy in Ethics and Neuroethics

In the course of the research, the main stages of the study of the problem of empathy and related concepts in the history of ethics and neuroethics were studied. It is established that the first attempts to study the phenomenon of empathy without using this concept were undertaken in the ethical concepts of the Scottish Enlightenment. In the 19th century, a sense of compassion, close to empathy, was actively studied, including in the emerging evolutionary ethics. Experimental psychology, which was born in this historical period, also had a significant impact on the study of moral feelings, including feelings of empathy. In the late XX — early XXI century, the concept of empathy becomes one of the central in neuroethics, largely due to the discovery of “mirror neurons”. The essential theoretical and practical debatable problems of explaining prosocial and moral behavior on the basis of empathy are analyzed.Рассматриваются основные этапы изучения проблемы эмпатии и сопряженных с ней понятий в истории этики и нейроэтике. Установлено, что первые попытки изучения феномена эмпатии без употребления данного понятия были предприняты в этических концепциях шотландского Просвещения. В XIX в. активно изучалось близкое к эмпатии чувство сострадания, в том числе в возникшей эволюционной этике. Зародившаяся в данный исторический период экспериментальная психология также оказала значительное влияние на изучение моральных чувств, в том числе и идеи «вчувствования». В конце ХХ — начале XXI в. понятие эмпатии становится одним из центральных в нейроэтике во многом благодаря открытию «зеркальных нейронов». Анализируются существенные теоретические и практические дискуссионные проблемы объяснения просоциального и нравственного поведения на основе эмпатии.Исследование выполнено в рамках проекта РФФИ № 20-011-00124 «Трансформация нравственной культуры под влиянием нейронаук»