Ever since the formulation of quantum laws governing the microworld, the study of quantum scattering in one dimension (1D) and three dimension (3D) has been indispensable in understanding the physics and interactions governing the world of molecules, atoms, nuclei and sub-atomic particles. Recently there is a renewed interest in the study of 1D quantum transmission and tunneling across heterostructures like barrier, well and its combination. This is because these structures are quite important for the fabrication of short-wavelength light-emitting diodes and diode lasers, and for other optoelectronic applications based on resonant tunneling. High quality heterostructures is possible by growth techniques like molecular beam epitaxy and metalorganic chemical vapor deposition. They posses predesigned potential profiles and impurity dis- tributions with dimensional control close to interatomic spacing. If the energy of the incident particle coincides with the resonance states generated by the heterostructure then the transmission gets enhanced and hence they play quite important role in the transport of charges in electronic devices. In view of this a systemic study of 1D scattering which is of pedagogical and academic interest is undertaken in this thesis. Analytical S matrix theory of 3D potential scattering provides a unified framework for the identification of bound and resonance states generated by the potential. Both these types of states are represented by the poles of S -matrix in complex energy or momentum planes. Bound states generated by the scattering potential are normalizable negative energy states with zero width. On the other hand resonance states are complex eigenstates, the real part corresponds to the resonance energy Er and imaginary part corresponds to its width Γr. If the resonance is very sharp, it corresponds to long lived quasibound (QB) state and within the interaction domain the QB state wave function behaves like a bound state wave function.
Dr. Umamaheswari A., “Study of resonances and absorption in one dimensional quantal scattering”, 2011.