Variance of the energy of a quantum system in a time-dependent perturbation: Determination by nonadiabatic transition...

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_k(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_k ⁽¹⁾(t) of the excited states to first order in the perturbation. This separates c_k ⁽¹⁾(t) for each state into an adiabatic term a_k ⁽¹⁾(t) and a nonadiabatic term b_k ⁽¹⁾(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_k ⁽¹⁾(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_k(t)|² 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_k(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_k is set equal to |b_k(t)|², but not if the probability of excitation is set equal to |c_k(t)|². We illustrate the differences between the variances calculated with the two different forms of P_k for vibration-rotation transitions of HCl in the gas phase.