Born-Oppenheimer theory is based on the separation in timescales between the nuclear and electron dynamics implied by the electron-to-nuclear mass ratio. This makes it naturally fit into a multiscale analysis. It is shown that a fully dynamical Born-Oppenheimer theory follows from a multiscale ansatz on the wave function and a Taylor expansion in the mass ratio. Allowing for a larger spatial scale of electron motion yields an understanding of boson, fermion, and more complex excitations that involve quasi-particles with an effective mass not equal to that of the electron. The theory involves a unified asymptotic expansion in a mass and length scale ratio, and preserves all many-body effects via accounting for the full strength of the interparticle forces. A novel mean-field theory emerges based on the fact that long-scale migration allows each electron to interact with many others on the space-time scale relevant to the coarse-grained equation. Implications for computational methods and applications to quantum nanosystems such as quantum dots, nanowires, superconducting nanoparticles, and liquid He droplets are discussed.
Quantum mean-field asymptotics and multiscale analysis
