# Katharine Hunt

### Research

## Quantum Theory

Quantum systems in time-dependent fields — We have derived new theoretical results
for transition probabilities in quantum systems in timedependent electromagnetic fields.
These results challenge Dirac’s expression for the transition probability, which depends
on the norm-square of the coefficient c_{k}(t) for the excited state | k_{0} 〉 of the original unperturbed Hamiltonian H_{0}, in the time-dependent wave function. By integrating by parts, Landau and Lifshitz
separated ck(t) into an adiabatic term a_{k}(t) that follows the adiabatic theorem of Born and Fock, and a nonadiabatic term b_{k}(t) that depends on the time-derivative of the perturbation up to time t. The adiabatic
term describes the adjustment of the initial state to the perturbation without actual
transitions, while the transition probability is given by the norm-square of bk(t).
Our work reinforces this statement and goes beyond the results of Landau and Lifshitz.
We proved that the energy separates into adiabatic and nonadiabatic terms. The nonadiabatic
term in the energy is given by the sum over excited states of | b_{k}(t) |2 times the transition energy (E_{k} – E_{0}), corrected for the perturbation. We proved that the power absorbed by a molecule
from an electromagnetic field is equal to the time-derivative of the nonadiabatic
term in the energy. Further, the vibrational wave packets associated with a_{k}(t) and b_{k}(t) evolve on different electronic potential energy surfaces. Our results for transition
probabilities differ significantly from the results of Dirac’s theory, during short
perturbing pulses with frequencies that are off-resonant from the transition frequency.
For a cosine wave of frequency ω in a Gaussian envelope, | b_{k}(t) |^{2} is larger than | c_{k}(t) |^{2} when ω > ω_{k0}, the transition frequency to state k; while the opposite is true when ω < ω_{k0}. These results are independent of the phase of the oscillating wave relative to the
peak of the Gaussian envelope. The differences are also quite stark for a perturbing
pulse that rises to a level plateau, and later falls off. While the perturbation is
constant, the nonadiabatic transition probability is constant as required physically,
since a static perturbation cannot induce transitions; in contrast Dirac’s form of
the transition probability continues to oscillate while the perturbation is constant,
as shown in Figure 1.

**Collision-induced spectroscopic processes -** Spectroscopic processes that are forbidden for single molecules are observed in dense
gases and liquids, because the electronic charge distorts during molecular collisions.
Our work has focused on collision-induced absorption in the IR by H_{2} gas, H_{2}/H and H_{2}/He mixtures, N_{2} gas and O_{2} gas, with applications in astrophysics and in atmospheric profiling. Interaction-induced
absorption affects the radiative profiles of gases in star-forming nebulae; very old,
very cool white dwarf stars; the outer planets and exoplanets that are termed “hot
Jupiters” and “warm Neptunes,” and the atmospheres of satellites of the outer planets.
My research group calculates the total dipole moments ab initio and then expresses
the results in spherical-tensor form for subsequent line-shape calculations. Dipole
surfaces for H_{2}-H are shown in Figure 2. This research area involves collaborations with Lothar Frommhold
and Martin Abel (University of Texas, Austin), Magnus Gustafsson (Luleå, Sweden),
Tijs Karman, Gerrit Groenenboom, and Ad van der Avoird (Nijmegen, the Netherlands),
and Richard Dawes (Missouri University of Science and Technology).