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We present a proof of an exact equivalence of the two approaches that are most used in computing conductance in quantum electron and phonon transport: the wave function matching and Green's functions methods. We can obtain all the quantities defined in one method starting from those obtained in the other. This completes and illuminates the work started by Ando (1991 Phys. Rev. B 44 8017) and continued later by Khomyakov et al (2005 Phys. Rev. B 72 035450). The aim is to allow for solving the transport problem with whichever approach fits most the system at hand. One major corollary of the proven equivalence is our derivation of a generalized Fisher-Lee formula for resolving the transmission function into individual phonon mode contributions. As an illustration, we applied our method to a simple model to highlight its accuracy and simplicity.

International audience; A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination of the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons.