  The fundamentals of a quantum heat engine are derived from first principles.
The study is based on the equation of motion of a minimum set of operators
which is then used to define the state of the system. The relation between the
quantum framework and thermodynamical observables is examined. A four stroke
heat engine model with a coupled two-level-system as a working fluid is used to
explore the fundamental relations. In the model used, the internal Hamiltonian
does not commute with the external control field which defines the two
adiabatic branches. Heat is transferred to the working fluid by coupling to hot
and cold reservoirs under constant field values. Explicit quantum equation of
motion for the relevant observables are derived on all branches. The dynamics
on the heat transfer constant field branches is solved in closed form. On the
adiabats, a general numerical solution is used and compared with a particular
analytic solution. These solutions are combined to construct the cycle of
operation. The engine is then analyzed in terms of frequency-entropy and
entropy-temperature graphs. The irreversible nature of the engine is the result
of finite heat transfer rates and friction-like behavior due to
noncommutability of the internal and external Hamiltonian.
