Adiabatic Switching and Time-Ordered series

Abstract

If we can solve a problem with a time independent Hamiltonian H₀, we surely meet many interesting but hard problems with a Hamiltonian

$$ H(t) = H_0 + \hat V(t) $$

((2.1))

where an extra term $\hat V(t)$ appears: sometimes the complication $\hat V(t)$ depends on time, but in other cases it is static. Here, H(t) is in the Schrödinger picture, the one that comes directly from classical physics with $p^x \to \frac{\partial } {{\partial x}}$, and so on, and sometimes we shall write H_S(t) and Ψ_S for extra clarity. So, we have the task of solving

$$ ih--\frac{\partial } {{\partial t}}|\Psi _S \left. {(t)} \right\rangle = H_S (t)|\Psi _S (t) $$

((2.2))

which is notoriously difficult. We can solve formally by introducing the unitary time evolution operator U_S such that

$$ |\Psi _S \left. {(t)} \right\rangle = \left. {U_S (t,t_0 )|\Psi _S (t_0 )} \right\rangle . $$

((2.3))