term will remain in the limit . In order to show this more rigorously we first need to remove the term. This can be done by defining We obtain: This equation can be integrated: or written in vector notation Here is a matrix and is basically a Fourier transform. It follows from the Riemann-Lebesgue lemma that as . As last step take the norm on both sides of the above equation: and apply Grönwall's inequality to obtain Since it follows for . This concludes the proof of the adiabatic theorem. In the adiabatic limit the eigenstates of the Hamiltonian evolve independently of each other. If the system is prepared in an eigenstate its time evolution is given by: So, for an adiabatic process, a system starting from nth eigenstate also remains in that nth eigenstate like it does for the time-independent processes, only picking up a couple of phase factors. The new phase factor can be canceled out by an appropriate choice of gauge for the eigenfunctions. However, if the adiabatic evolution is cyclic, then becomes a gauge-invariant physical quantity, known as the Berry phase. |} {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Generic proof in parameter space |- | Let's start from a parametric Hamiltonian , where the parameters are slowly varying in time, the definition of slow here is defined essentially by the distance in energy by the eigenstates (through the uncertainty principle, we can define a timescale that shall be always much lower than the time scale considered). This way we clearly also identify that while slowly varying the eigenstates remains clearly separated in energy (e.g. also when we generalize this to the case of bands as in the TKNN formula the bands shall remain clearly separated). Given they do not intersect the states are ordered and in this sense this is also one of the meanings of the name topological order. We do have the instantaneous Schrödinger equation: And instantaneous eigenstates: The generic solution: plugging in the full Schrödinger equation and multiplying by a generic eigenvector: And if we introduce the adiabatic approximation: for each We have and where And C is the path in the parameter space, This is the same as the statement of the theorem but in terms of the coefficients of the total wave function and its initial state. Now this is slightly more general than the other proofs given we consider a