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Abstract

In this thesis, we examine a simple case where two nonidentical, spineless bosons scatter elastically from each other, assuming that their interaction is weak and thus only leading-order perturbation theory is valid. We derive two different formulae for the differential cross-sections when the projectile particle is initially in a quantum superposition of two pure states (pure state) and a mixed ensemble of the same two pure states (mixed state). We take into account the relativistic effects fully in principle, although we only focus on the low-energy limit in the formula in one of our examples, which can be interpreted as a non-relativistic approximation. First, the derivation was made assuming that both incoming and outgoing particles are plane waves, which will provide us with an approximation. Then, the differential cross-section was derived assuming that only the initial state of the projectile is a wave packet. Next, we compared the differential cross-sections when the projectile particles are in a pure state and a mixed state in several examples, including the Half-plane waves, waves in the double-slit experiment, and double Gaussian wave packets. Finally, we discussed the limiting cases and performed integrations over target momentum to obtain an inclusive differential cross-section with respect to the momentum of the projectile particle. The results show that decoherence alters the differential cross-section when the incoming projectile has a finite width in at least one dimension.