Emergence of a Macroscopic Phase in an Isolated Electron-Phonon Superconductor*

Within the number-conserving formalism introduced by Ambegaokar, we constructed a variational wave function of electron-phonon superconductors. The many-body correlations originated from electron-phonon coupling are incorporated in our variational wave function. Moreover, this wave function gives rise to a superposition over the number of condensed particles and the Cooper-pair wave function with a macroscopic phase can be defined naturally. Therefore, we conclude dynamical exchanges of phonons induce a well-defined macroscopic phase with emergence of superconductivity in an electron-phonon superconductor.