  We present a geometric perspective on how to quantify the bending and the
twisting of quantum curves traced by state vectors evolving under nonstationary
Hamiltonians. Specifically, relying on the existing geometric viewpoint for
stationary Hamiltonians, we discuss the generalization of our theoretical
construct to time-dependent quantum-mechanical scenarios where both
time-varying curvature and torsion coefficients play a key role. Specifically,
we present a quantum version of the Frenet-Serret apparatus for a quantum
trajectory in projective Hilbert space traced out by a parallel-transported
pure quantum state evolving unitarily under a time-dependent Hamiltonian
specifying the Schrodinger evolution equation. The time-varying curvature
coefficient is specified by the magnitude squared of the covariant derivative
of the tangent vector to the state vector and measures the bending of the
quantum curve. The time-varying torsion coefficient, instead, is given by the
magnitude squared of the projection of the covariant derivative of the tangent
vector to the state vector, orthogonal to the tangent vector and state vector
and, in addition, measures the twisting of the quantum curve. We find that the
time-varying setting exhibits a richer structure from a statistical standpoint.
For instance, unlike the time-independent configuration, we find that the
notion of generalized variance enters nontrivially in the definition of the
torsion of a curve traced out by a quantum state evolving under a nonstationary
Hamiltonian. To physically illustrate the significance of our construct, we
apply it to an exactly soluble time-dependent two-state Rabi problem specified
by a sinusoidal oscillating time-dependent potential...
