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

### Abstract

When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state |0⟩ of an unperturbed Hamiltonian H_{0}, the wave function acquires excited-state components |k⟩ with coefficients c_{k}(t) exp(−iE_{k}t/ℏ), where E_{k} denotes the energy of the unperturbed state |k⟩. 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 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 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 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 λ^{2} |b_{k}(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.

© 2012 American Institute of Physics

Received 22 June 2012
Accepted 21 August 2012
Published online 26 October 2012

Article outline:

I. INTRODUCTION
II. PERTURBED ENERGY THROUGH SECOND ORDER
III. PERTURBED ENERGY AT THIRD ORDER
IV. TRANSITIONS TO EIGENSTATES OF THE INSTANTANEOUS HAMILTONIAN H(t)
V. SUMMARY AND DISCUSSION

/content/aip/journal/jcp/137/16/10.1063/1.4750045

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/content/aip/journal/jcp/137/16/10.1063/1.4750045

2012-10-26

2016-02-13

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