Exact quantum states of a general time-dependent quadratic system from classical action

A generalization of driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization which gives a general quadratic Hamiltonian system does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous and one particular solutions. The kernels are then used to find wave functions which satisfy the Schr\"{o}dinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization which gives a rather complicated quadratic Hamiltonian is simply interpreted as acting an unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.Comment: Submitted to Phys. Rev.

Transition Metals Doped Nanocrystals: Synthesis, Characterization, and Applications

Doping is a technique that makes it possible to incorporate substitutional ions into the crystalline structure of materials, generating exciting properties. This book chapter will comment on the transition metals (TM) doped nanocrystals (NCs) and how doping and concentration influence applications and biocompatibility. In the NCs doped with TM, there is a strong interaction of sp-d exchange between the NCs’ charge carriers and the unpaired electrons of the MT, generating new and exciting properties. These doped NCs can be nanopowders or be embedded in glass matrices, depending on the application of interest. Therefore, we show the group results of synthesis, characterization, and applications of iron or copper-doped ZnO nanopowders and chromium-doped Bi2S3, nickel-doped ZnTe, and manganese-doped CdTe quantum dots in the glass matrices