Relaxation and Localization in Condensed Phases.

This thesis consists of two parts. In the first part, relaxation dynamics and line broadening of a quantum mechanical system in condensed phases are studied. In the second part, we focus on the study of critical phenomena is disordered quantum mechanical systems. In Chapter 2, we consider a two-level system (TLS) linearly and off-diagonally coupled to a harmonic bath. The equations of motion for the reduced TLS density matrix are obtained using projection operator techniques. Here, the factorization assumption for the initial density matrix is avoided. From a perturbation calculation to fourth -order in the TLS/bath coupling, we find that the density matrix shows non-Markovian behavior at short times. After this transient period, the Bloch equations of motion are recovered. For the complex Ohmic-Lorentzian model, we found that the phase relaxation time, T_2 , can be greater than twice the population relaxation time, T_1, for a range of parameters. In Chapter 3, we calculate the absorption spectrum for the TLS, finding that the linewidth of the lineshape function can be smaller than 1/T_1, which seems to violate the energy-time uncertainty principle. In Chapter 4, we study localization in energetically disordered systems. A standard diagonally disordered tight -binding Hamiltonian is used. The critical disorder and critical exponents are determined using a numerical renormalization group method. By comparing the numerical results of the inverse participation ratio exponent and its generalizations, we show that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the inverse participation ratio, the correlation length exponent nu = 0.99 +/- 0.04 is obtained, which is in agreement with experiments on compensated or amorphous doped semiconductors.