differential equation describes the time-evolution of the coefficients, but now in terms of matrix elements of . To arrive at the adiabatic theorem, neglect the right hand side. This is valid if the rate of change of the Hamiltonian is small and there is a finite gap between the energies. This is known as the adiabatic approximation. Under the adiabatic approximation, which integrates precisely to the adiabatic theorem with the phases defined in the statement of the theorem. The dynamical phase is real because it involves an integral over a real energy. To see that the geometric phase is purely imaginary, differentiate the normalization of the eigenstates and use the product rule to find that Thus, is purely imaginary, so the geometric phase is purely real. |} {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Adiabatic approximation |- | Proof with the details of the adiabatic approximation We are going to formulate the statement of the theorem as follows: For a slowly varying Hamiltonian in the time range T the solution of the schroedinger equation with initial conditions where is the eigenvector of the instantaneous Schroedinger equation can be approximated as: where the adiabatic approximation is: and also called Berry phase And now we are going to prove the theorem. Consider the time-dependent Schrödinger equation with Hamiltonian We would like to know the relation between an initial state and its final state at in the adiabatic limit First redefine time as : At every point in time can be diagonalized with eigenvalues and eigenvectors . Since the eigenvectors form a complete basis at any time we can expand as: where The phase is called the dynamic phase factor. By substitution into the Schrödinger equation, another equation for the variation of the coefficients can be obtained: The term gives , and so the third term of left side cancels out with the right side, leaving Now taking the inner product with an arbitrary eigenfunction , the on the left gives , which is 1 only for m = n and otherwise vanishes. The remaining part gives For the will oscillate faster and faster and intuitively will eventually suppress nearly all terms on the right side. The only exceptions are when has a critical point, i.e. . This is trivially true for . Since the adiabatic theorem assumes a gap between the eigenenergies at any time this cannot hold for . Therefore, only the