Through continual advancement in laser pulse technology. experimentalists now have al their disposal higher intensities (> 1015 W cm!) and shon.er pulse durations « 10 fs) than ever before. Using sllch technology it is possible to probe and manipulate the electronic and nuclear Illotions in even the smallest fastest-moving diatomic molecules. The first pan. of this thesis presents a simplified theoretical model that allows one to adequately treat the nuclear vibrational and photodissociation dynamics of the O2 ' molecular ion, when subjected to typical infrared wavelengths. Direct comparison between the predictions of this theoretical model and recent experimental observations is provided. The same model is then used as a basis for the proposal of a number of novel techniques that utilize ultrashort laser pulses to l;:ontrol both the dissociation pathway and the population ofthe bound vibrational levels. One of the main areas of theoretical and experimental advancement in recent decades has been the area of cold atom trapping and manipulation. The second part of this thesis considers the theoretical treatment of two interacting particles, confined in various one-dimensioDal potentials. These systems represent a fundamental building block that has been made accessible through recent developments in the field of ultracold atomic physics: The numerical scheme for the treatment of the two-particle system is described and results are presented for two-paJ1icles in a harmonic trap, a o-split harmonic trap. and a double-well potential. Propel1ies of the two-particle ground state and low-energy excited states are examined including the energy spectra, eigenfunctions, reduced single-particle density matrices, momentum distributions and entanglement. ]n particular, focussing upon how these quantities depend upon the two parameters, particle-particle interaction strength and barrier height. In this way, the present work relates to the scope of quantum state control, in such systems, through variation of 'experimentally accessible' control parameters.