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Tunneling Hamiltonian

For the description of the transport of electrons across a quantum dot, which is tunnel coupled to leads at different chemical potentials, it is usual to assume that the total Hamiltonian of the composite system of the leads and the quantum dot is the sum of three contributions: That of the leads (noninteracting electrons), that of the quantum dot and a third one, the "tunneling Hamiltonian", which reflects the possibility that electrons can move from the leads to the quantum dot or vice versa. The text aims at a mathematically clear derivation of such a separation. I will start the discussion with the total Hamiltonian of the system acting on a many-electron wave function, including the attractive interaction between nuclei and electrons as well as the repulsive Coulomb-interaction between different electrons. Indeed, a natural separation of the total Hamiltonian in the described form will be obtained. An analysis of the tunneling Hamiltonian shows that the electron-electron interaction yields contributions to it which represent the correlated tunneling of two electrons at the same time. For the derivations it was useful to introduce a map called "antisymmetric product". In an appendix I show possible exact representations of the total Hamiltonian (with arbitrary potential V(r)) obtained by the use of the antisymmetric product.