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We extend the standard solid-state quantum mechanical Hamiltonian containingonly Coulomb interactions between the charged particles by inclusion of $1/c^2$terms representing (transverse) current-current interaction. For its derivationwe use the classical formulation of Landau-Lifshitz, however consequently inthe Coulomb gauge. Our Hamiltonian does not coincide with the DarwinHamiltonian and we emphasize the mathematical inconsistency in its derivation.We show, that the quantized version of our Hamiltonian is equivalent to thenon-relativistic QED considering only states without photons and retaining onlyterms of order $1/c^2$. The importance of this extended Hamiltonian lies in thepossibility to distinguish external from internal magnetic fields. This aspectmay be relevant for theories of the Meissner effect.

On the Nonrelativistic Quantum-Mechanical Hamiltonian with $1/c^2$ terms. Transverse current-current interaction