  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 Ando[Ando T 1991 Phys. Rev. B
44 8017] and continued later by Komyakov et al.[Khomyakov P A, Brocks G, Karpan
V, Zwierzycki M and Kelly P J 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.
