generic set of parameters, and we see that the Berry phase acts as a local geometric quantity in the parameter space. Finally integrals of local geometric quantities can give topological invariants as in the case of the Gauss-Bonnet theorem. In fact if the path C is closed then the Berry phase persists to Gauge transformation and becomes a physical quantity. |} Example applications Often a solid crystal is modeled as a set of independent valence electrons moving in a mean perfectly periodic potential generated by a rigid lattice of ions. With the Adiabatic theorem we can also include instead the motion of the valence electrons across the crystal and the thermal motion of the ions as in the Born–Oppenheimer approximation. This does explain many phenomena in the scope of: thermodynamics: Temperature dependence of specific heat, thermal expansion, melting transport phenomena: the temperature dependence of electric resistivity of conductors, the temperature dependence of electric conductivity in insulators, Some properties of low temperature superconductivity optics: optic absorption in the infrared for ionic crystals, Brillouin scattering, Raman scattering Deriving conditions for diabatic vs adiabatic passage We will now pursue a more rigorous analysis. Making use of bra–ket notation, the state vector of the system at time can be written where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the position operator It is instructive to examine the limiting cases, in which is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change). Consider a system Hamiltonian undergoing continuous change from an initial value , at time , to a final value , at time , where . The evolution of the system can be described in the Schrödinger picture by the time-evolution operator, defined by the integral equation which is equivalent to the Schrödinger equation. along with the initial condition . Given knowledge of the system wave function at , the evolution of the system up to a later time can be obtained using The problem of determining the adiabaticity of a given process is equivalent to establishing the dependence of on . To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started. Using bra–ket notation and using the definition , we have: We can expand In the perturbative limit