Problem: Polarons

   

A ``polaron" is formed when an electron has a bound state in the potential created by a distorted lattice, and the lattice distortion is due to the electron itself. To elucidate the microscopic mechanism of polaron formation, consider a single electron in one dimension. The crystal lattice is represented by a continuous medium, and the potential felt by the electrons is , where u(x) is the displacement of the atoms from their original position and is the electron-phonon coupling constant. The energy density associated with the deformation is , where B is the bulk modulus. Show that in one dimension the coupled electron-phonon system is always unstable against the formation of polarons. Calculate the size of the polaron. (Assume that the deformation is in the form of , where and are variable parameters; is the size of the polaron.)