Mandal Anirban, Hunt Katharine L C
Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA.
J Chem Phys. 2020 Mar 14;152(10):104110. doi: 10.1063/1.5140009.
For a quantum system in a time-dependent perturbation, we prove that the variance in the energy depends entirely on the nonadiabatic transition probability amplitudes b(t). Landau and Lifshitz introduced the nonadiabatic coefficients for the excited states of a perturbed quantum system by integrating by parts in Dirac's expressions for the coefficients c (t) of the excited states to first order in the perturbation. This separates c (t) for each state into an adiabatic term a (t) and a nonadiabatic term b (t). The adiabatic term follows the adiabatic theorem of Born and Fock; it reflects the adjustment of the initial state to the perturbation without transitions. If the response to a time-dependent perturbation is entirely adiabatic, the variance in the energy is zero. The nonadiabatic term b (t) represents actual excitations away from the initial state. As a key result of the current work, we derive the variance in the energy of the quantum system and all of the higher moments of the energy distribution using the values of |b(t)| for each of the excited states along with the energy differences between the excited states and the ground state. We prove that the same variance (through second order) is obtained in terms of Dirac's excited-state coefficients c(t). We show that the results from a standard statistical analysis of the variance are consistent with the quantum results if the probability of excitation P is set equal to |b(t)|, but not if the probability of excitation is set equal to |c(t)|. We illustrate the differences between the variances calculated with the two different forms of P for vibration-rotation transitions of HCl in the gas phase.
对于处于含时微扰中的量子系统，我们证明能量的方差完全取决于非绝热跃迁概率幅b(t)。朗道和栗弗席兹通过在狄拉克关于激发态系数c(t)的表达式中分部积分，将微扰展开到一阶，引入了受扰量子系统激发态的非绝热系数。这将每个态的c(t)分离为一个绝热项a(t)和一个非绝热项b(t)。绝热项遵循玻恩和福克的绝热定理；它反映了初始态对微扰的调整而无跃迁。如果对含时微扰的响应完全是绝热的，能量的方差为零。非绝热项b(t)表示实际偏离初始态的激发。作为当前工作的一个关键结果，我们利用每个激发态的|b(t)|值以及激发态与基态之间的能量差，推导出量子系统能量的方差以及能量分布的所有高阶矩。我们证明，就狄拉克的激发态系数c(t)而言，能得到相同的方差（到二阶）。我们表明，如果将激发概率P设为等于|b(t)|，则方差的标准统计分析结果与量子结果一致，但如果将激发概率设为等于|c(t)|，则不一致。我们举例说明了对于气相中HCl的振转跃迁，用两种不同形式的P计算出的方差之间的差异。
