In this dissertation the energy of the ground and excited state of symmetric andrnasymmetric parabolic double quantum wells (DQWs) are investigated by using ourrnformulated trial wave functions and the Hamiltonian. For symmetric DQWs, calculationrnof the frequency corresponding to the ground state energy splitting describesrnthat tunneling takes place in the duration of short time when the frequency of thernphoton is in tera hertz (THz) region. However the tunneling time for excited staternenergy splitting is shorter than that of the ground state energy splitting due to therndecrement of barrier width for parabolic DQWs.rnThe asymmetric DQWs with resonant levels (the ground state energy level in onernwell coincides with the first excited state energy in another well) is analyzed. Thernsplitting of these levels and the tunneling times are calculated. If the typical liferntime of the excited state is much smaller than the tunneling time between wells, therncharged particle can radiate as the result of quantum transition from the excited staternto the ground state. In the opposite case, the asymmetric DQWs can be treated asrna metastable excited nano-system regardless of that the dipole transition from thernexcited state to the ground state is permitted. The life time of this metastable staterncan be considerably reduced by putting it in the resonant cavity. The possibility ofrncoherent radiation of an ensemble of asymmetric DQWs is discussed.rnMoreover the analytical expressions of absorption coefficients and refractive indexrnchanges for parabolic asymmetric DQWs have been determined using the compactrndensity matrix approach. The calculation shows that at some frequency range thernasymmetric DQWs become left handed media with negative values of total refractivernviirnindex. This behavior of the asymmetric DQWs at some frequency range is of greatrnimportance for metamaterial science.rnThe low lying energy levels of 3D two electron QDs with parabolic confinement andrnrigid confinement (the wave functions vanish at the surface of the QD) are obtained byrnthe variational method and perturbation techniques. The quantum states of the QDrnare divided in to the para- and ortho- states like in the theory of helium atom. Ourrnnumerical calculations show that the energy differences between the ground and firstrnexcited states of the para- dots are practically linear function of the coupling constantrn . This allows one to propose the phenomenological formulas for the energy levelsrnas a function of a typical size of QD. The spin function of the para-dot is singletrnwith total spin zero. One of the spin function of the ortho-dot is not factorizablernsymmetric combination. The two functions relate to the entanglement states, whichrnare important for quantum information processing.rnThe lowest transition energy of one and two electron GaAs QDs are calculatedrnby solving the corresponding SchrÂ¨odinger equation. For the two electron GaAs QDrnwe calculated the real and imaginary parts of the dielectric constant, the refractivernindex, and the absorption coefficients as functions of the incident photon energy. Thernmaxima of these quantities show a blue shift comparing with those of the one electronrnQD due to the Coulomb and exchange interaction between electrons.rnReducing the size of quantum dots results in blue shift of the maxima of the abovernmentioned quantities for both one and two electron QDs. But the blue shift in therncase of two electron QD is larger than that one electron QD due to the energies ofrnCoulomb and exchange