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The objectives of this work are to purpose to investigation of number of quantum mechanical properties of defect induced anharmonic crystals, keeping the methods of calculation as simple as possible. Chapter I is devoted to the survey of the literature, work done in the past (historic prospects and retrospects), objectives and state of the art methodology of the work. In Chapter II the quantum dynamics of electrons and phonons is developed with the help of a newly formulated Hamiltonian. The Green's functions for electrons and phonons are developed in order to obtain the shifts and widths for electrons and phonons to describe the lifetimes. Chapter III is devoted to the theory of thermal conductivity of crystalline solids and quantum wells. Various models for thermal conductivity have been reported in this chapter. Chapter IV is devoted to calculate the free energy of an isotopically disordered anharmonic crystal using thermodynamic Green's functions. The graphics presented for low and high temperature studies of Helmholtz free energy based on the present theory reveal that the different terms involved in the formulation contain a linear and non linear temperature dependence.