Thermodynamic work and heat for a quantum process: Approach by Hamiltonian decomposition

The separation of internal energy into heat and work in quantum thermodynamics is a controversial issue for a long time, and we revisit and solve this problem in this work. It is shown that the Hamiltonian plays dual roles for a quantum system, and by decomposing the interaction Hamiltonian between system and environment accordingly, an ``effective Hamiltonian" for an open quantum system can be proposed. The explicit expression of the effective Hamiltonian is obtained systematically, and as a consequence, the internal energy of an open quantum system can be well defined, leading to the reasonable definitions of work and heat for a general quantum process.