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In this work we study the problem of one-dimensional elastic collisions of billiard balls, considered as rigid bodies, in a framework very different from the classical one presented in text books. Implementing the notion of impedance matching as a way to understand efficiency of energy transmission in elastic collisions, we find a solution which frames the problem in terms of this conception. We show that the mass of the ball can be seen as a measure of its impedance and verify that the problem of maximum energy transfer in elastic collisions can be thought of as a problem of impedance matching between different media. This approach extends the concept of impedance, usually associated with oscillatory systems, to system of rigid bodies.

A generalized propagation matrix method is used to study how scattering off local Einstein phonons affects resonant electron transmission through quantum wells. In particular, the parity and the number of the phonon mediated satellite resonances are found to depend on the available scattering channels. For a large number of phonon channels, the formation of low-energy impurity bands is observed. Furthermore, an effective theory is developed which accurately describes the phonon generated sidebands for sufficiently small electron-phonon coupling. Finally, the currentvoltage characteristics caused by phonon assisted transmission satellites are discussed for a specific double barrier geometry.

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