  One of the fundamental questions in the emerging field of quantum
thermodynamics is the role played by coherence in energetic processes that
occur at the quantum level. Here, we address this issue by investigating two
different quantum versions of the first law of thermodynamics, derived from the
classical definitions of work and heat. By doing so, we find out that there
exists a mathematical inconsistency between both scenarios. We further show
that the energetic contribution of the dynamics of coherence is the key
ingredient to establish the consistency. Some examples involving two-level
atomic systems are discussed in order to illustrate our findings.
