In this study, we show how a static magnetic field can control photon-induced electron transport through a quantum dot system coupled to a photon cavity. The quantum dot system is connected to two electron reservoirs and exposed to an external perpendicular static magnetic field. The propagation of electrons through the system is thus influenced by the static magnetic and the dynamic photon fields. It is observed that the photon cavity forms photon replica states controlling electron transport in the system. IF the photon field has more energy than the cyclotron energy, then the photon field is dominant in the electron transport. Consequently, the electron transport is enhanced due to activation of photon replica states. By contrast, the electron transport is suppressed in the system when the photon energy is smaller than the cyclotron energy.

We present analytical method to calculate single particle matrix elements used in atomic and nuclear physics. We show seven different formulas of matrix elements of the operator f (r)d m r where f (r) = r µ , r µ jJ (qr), V (r) corresponding to the Gaussian and the Yukawa potentials used in nuclear shell models and nuclear structure. In addition, we take into account a general integral formula of the matrix element n ′ l ′ |f (r) d (m) r |n l that covers all seven matrix elements obtained analytically.

The Coulomb form factors for the elastic and inelastic electron-nucleus scatterings have been calculated for 12C and 20Ne nuclei in the ground and excited states with the same parity. We use a microscopic theory involving the effects from high configurations outside the model space, which are called the Core Polarization (CP) effects. For the core polarization matrix elements, the realistic Michigan sum of the three-range Yukawa (M3Y) interaction and the Modified Surface Delta Interaction (MSDI) are used as the two-body interactions. Additionally, the Harmonic Oscillators (HO) potential is applied to calculate wave functions. In the final step, a comparison has been made between the theoretical calculations of Coulomb form factors based on (M3Y) and (MSDI) interactions and the available experimental data. It is noticed that the Coulomb form factors for the (M3Y) interaction give a sensible delineation of the measured data.

The text of this paper covers a generalization of the radial two-body nuclear potentials and then derivation of a new formula to evaluate their matrix element. A new formula was derived and applied successfully to calculate the single-particle matrix elements of the spin-isospin dependent and spin-isospin independent potentials by using a mathematica code.

The computations of the elastic and inelastic Coulomb form factors for the electron-nucleus scattering of Beryllium nucleus Be9 have performed with Core Polarization (CP) effects including the realistic Michigan sum of Three Range Yukawa (M3Y) Interaction, and the other residual interaction which is Modified Surface Delta Interaction (MSDI). In addition to mean square root charge density and charge radii for the ground state. The perturbation theory was adopted to compute the Core Polarization by using the Harmonic Oscillators (HO) potential to calculate single-particle radial wave functions.    In the comparison between the theoretical calculations of Coulomb form factors by (MSDI) interaction, realistic (M3Y) interaction, and the experimental results that measured before, it noticed that the Coulomb form factors for the (M3Y) interaction gave a reasonable depiction of the measured data.