Emissions from relativistic electrons travelling in periodic electrostatic fields were observed by Smith and Purcell (1953) and Doucas et at. (1992) as extraordinarily (e.g. 1018 times) stronger than any emission that can be conceived with classical electrodynamics under any equivalent condition. The mechanism is identified as the free-electron two-quantum Stark (FETQS) emission generated by the axial uniform motion, which cannot be radiated in the axial direction. From the excellent agreement between the theoretical result for FETQS emission driven by the axial uniform motion, and the experimental observations and many emission phenomena in plasmas, it is concluded that a high-energy electron follows a quantum-mechanically derived formula (without taking the classical limit ←0), although this diverges in the classical limit. From the extraordinarily large FETQS emission due to macroscopic motion, we speculate that even the FETQS emission generated by electron spin can be macroscopically observable. The spin-generated FETQS emission in an arbitrary direction is calculated using relativistic quantum mechanics. It is found that the total power of this emission scales as γ2 times the emission power in the equivalent magnetic wiggler, where γ is the Lorentz factor of the electron, and the emission is practically confined in a cone of angle 1/γ about the axial direction.