Abstract
We extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of the (transverse) current-current diamagnetic interaction starting from the non-relativistic QED restricted to the states without photons and neglecting the retardation in the photon propagator. This derivation is supplemented with a derivation of an analogous result along the non-rigorous old classical Darwin-Landau-Lifshitz argumentation within the physical Coulomb gauge.
Highlights
The standard Hamiltonian of solid-state theory considers a system of electrons and ions interacting only through Coulomb forces
Already 100 years ago Darwin [1] has argued in the frame of the classical electrodynamics of point-like particles, that up to order 1/c2 one might separate the motion of the particles from that of the electromagnetic field
Under this form it looks similar to the result we obtained before and a quantization of the classical Hamiltonian of the point-like charged particles (14) is immediate by second quantization of (16) as it was done in Ref. [5] and one may omit the configuration space quantization either of (14) or (13). This leads to the same Hamiltonian (6) as we deduced from the non-relativistic QED
Summary
The standard Hamiltonian of solid-state theory considers a system of electrons and ions interacting only through Coulomb forces. By restricting the discussion to the states without photons and neglecting the retardation in the photon propagator we give in Section 3 a derivation of the electronic Hamiltonian containing the diamagnetic interaction between the transverse currents (microscopic Biot-Savart law !). This proof (stimulated by the classical paper of Holstein, Norton and Pincus [4] on diamagnetism) was first published in a preprint [5] and included in a recent book [6]. From the same assumption follows that Darwin’s Hamiltonian coincides with the classical version of the Hamiltonian we derived in Section 3 from the non-relativistic QED
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