The exact solutions of the Schrödinger equation for quantum damped oscillator with modified Caldirola-Kanai Hamiltonian are evaluated. We also investigate the cases of under-, over-, and critical damping.

The phonon contribution to the phenomenon of high temperature superconductivity in the cuprates is argued as being masked as polarons generated by the polarization of the charge reservoir accompanying the Jahn-Teller tilting of the apical oxygen. We discuss the Mahan oscillator-spring extension model as an analogy to the charge reservoir-CuO plane -axis polarons. Using the Boltzmann kinetic equation, we show that the polaron dissociates or collapses at a temperature corresponding to the critical temperature of the superconductor.

The crossover of large to small radius polaron is studied in terms of the inverse-relaxation time and temperature. It is found that the small radius polaron exists at higher temperature than the large radius polaron. A formula which relates the inverse-relaxation time to the ratio of arbitrary temperature and Debye temperature of the crystal is derived. The polaron crossover temperatures in NaCl and KBr are found from plotted graphs. The straight line emerging at the Debye temperature T D of a graph reflects the increase of the inverse relaxation time for increasing temperature up to the collapse of the small radius polaron. The relationship between the small and large radius polarons is found and known ratios of the effective and the bare masses of the electrons for the two substances are used to validate our results. The small radius polaron's mass is later compared with the mass obtained from the hopping formula and is found to be approximately equal. Finally, we point out that the crossover temperature is material-specific since it depends on the Debye and the effective dielectric function.

The relativistic symmetries of the Dirac equation within the framework of spin and pseudospin symmetries is investigated for DengFan potential including the Coulomb-like and Hulthen-type potential tensor interaction terms. The energy eigenvalues and the corresponding wave function are obtained using the parametric generalization of NikiforovUvarov method. We have also reported some numerical results and gures to show the eect of the tensor interactions.

We performed a two-variable canonical transformation on the time momentum operator, and without loss of generality we carried out a three-variable transformation on the coordinate and momentum space operators to trivialize the Hamiltonian operator of the system. Fortunately, this operation separates the time-coordinate and space coordinate naturally, and the wave function of the time-dependent Harmonic Oscillator is evaluated via the generator.