For free particles with complete translation symmetry, any energy is allowed. for bound states with no translationsymmetry, only quantized energies are allowed. in solids, it is often found that there are bands of allowed energiesseparated by bands of energy with no allowed states. in crystals, the size of the gap between allowed energy bandsdetermines the characteristics of the solid in many ways. a) what symmetry is responsible for these solutions with energy bands? b) write the form of the solutions giving the energy bands. what characterizes the two factors in these solutions? c) what band structure do insulators have? d) what band structure do intrinsic semiconductors have? e) what band structure do metals have? f) how are quantum metals expected to be different from the classical theory of metals?