  Energy is often partitioned into heat and work by two independent paths
corresponding to the change in the eigenenergies or the probability
distributions of a quantum system. The discrepancies of the heat and work for
various quantum thermodynamic processes have not been well characterized in
literature. Here we show how the work in quantum machines is differentially
related to isochoric, isothermal, and adiabatic processes. We prove that the
energy exchanges during the quantum isochoric and isothermal processes are
simply depending on the change in the eigenenergies or the probability
distributions. However, for a time-dependent system in a non-adiabatic quantum
evolution, the transitions between the different quantum states representing
the quantum coherence can affect the essential thermodynamic properties, and
thus the general definitions of the heat and work should be clarified with
respect to the microscopic generic time-dependent system. By integrating the
coherence effects in the exactly-solvable dynamics of quantum-spin precession,
the internal energy is rigorously transferred as the work in the thermodynamic
adiabatic process. The present study demonstrates that quantum adiabatic
process is sufficient but not necessary for thermodynamic adiabatic process.
