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

作者：Mandal Anirban^*; Hunt Katharine L C

来源：Journal of Chemical Physics, 2012, 137(16): 164109.

DOI：10.1063/1.4750045

When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state vertical bar 0 %26gt; of an unperturbed Hamiltonian H-0, the wave function acquires excited-state components vertical bar k %26gt; with coefficients c(k)(t) exp(-iE(k)t/(h) over bar), where E-k denotes the energy of the unperturbed state vertical bar k %26gt;. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation - without actual transitions - and a nonadiabatic term b(k)(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 Schrodinger equation via Dirac%26apos;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 c(k)(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 lambda H%26apos;(t). The nonadiabatic term is a sum over excited states vertical bar k %26gt; of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates vertical bar k%26apos;(t)%26gt; 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 lambda, the probability of finding the system in state vertical bar k%26apos;(t)%26gt; is given by lambda(2) vertical bar b(k)(t)vertical bar(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%26apos;s golden rule.

• 出版日期2012-10-28