a later stage - the concept of Berry Phase. Consider a general Hamiltonian $H({\bf R})$, which is a function of several parameters represented as a vector ${\bf R} = (R_1,R_2,..)$. At any instant, for a fixed ${\bf R}$, the solutions can be obtained by using the time-independent Schroedinger equation as, $$\begin{aligned}
H({\bf R}) \ket{n(\bf R)} = E_n({\bf R}) \ket{n(\bf R)}\end{aligned}$$ Now as ${\bf R}$ changes in the parameter space (starting from ${\bf R}(t=0)$) along some path $\mathcal{C}$, we are interested in knowing how the state changes when the system is initially prepared in the state $\ket{n({\bf R}(t=0))}$. There is a very useful theorem known as the [*Adiabatic theorem*]{} that states - for a [*slowly*]{} varying Hamiltonian, a system initially in the eigenstate will always remain in its [*instantaneous*]{} eigenstate at any later time. From this theorem we have got half the answer to our question. We now know starting from $\ket{n({\bf R}(0))}$, the system will evolve to $\ket{n({\bf R}(t))}$ which is the instantaneous eigenstate of $H({\bf R}(t))$ with slowly varying ${\bf R}$. But what about the phase? We can in general write the evolved state at $t$ as $\ket{\psi(t)} = e^{-i\theta(t)}\ket{n({\bf R}(t))}$. This state will follow the Schroedinger equation, $$\begin{aligned}
