Adiabatic and nonadiabatic contributions to the energy of a system subject to a time-dependent perturbation: Complete separation and physical interpretation.

When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state |0> of an unperturbed Hamiltonian H0, the wave function acquires excited-state components |k> with coefficients ck(t) exp(-iEkt/h), where Ek denotes the energy of the unperturbed state |k>. It is well known that each coefficient ck(t) separates into an adiabatic term ak(t) that reflects the adjustment of the ground state to the perturbation - without actual transitions - and a nonadiabatic term bk(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of ck(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH′(t). The nonadiabatic term is a sum over excited states |k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates |k′(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state |k′(t)> is given by λ2 |bk(t)|2. At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule. [ABSTRACT FROM AUTHOR]