Time-Dependent Hamiltonians

Abstract

For many time-dependent problems, most notably in spectroscopy, we often can partition the time-dependent Hamiltonian into a time-independent part that we can describe exactly and a time-dependent part H H V t + () = 0 (2.1) Here H 0 is time-independent and V t () is a time-dependent potential, often an external field. Nitzan, Sec. 2.3., offers a nice explanation of the circumstances that allow us to use this approach. It arises from partitioning the system into internal degrees of freedom in H 0 and external degrees of freedom acting on H 0 . If you have reason to believe that the external Hamiltonian can be treated classically, then eq. (2.1) follows in a straightforward manner. Then there is a straightforward approach to describing the time-evolving wavefunction for the system in terms of the eigenstates and energy eigenvalues of H 0 . We know H n E = n . (2.2) 0 n The state of the system can be expressed as a superposition of these eigenstates: c t n (2.3) $ψ$ (t) = ∑ n () n The TDSE can be used to find an equation of motion for the expansion coefficients c t = k $ψ$ (t) (2.4) k () Starting with ∂ $ψ$ −i = H $ψ$ (2.5) ∂t ∂ c t k () = − i k H t (2.6) $ψ$ () ∂t i inserting ∑ n n = 1 = − ∑ k H n c t (2.7) n () n n