  We present a unifying variational calculus derivation of Groverian geodesics
for both quantum state vectors and quantum probability amplitudes. In the first
case, we show that horizontal affinely parametrized geodesic paths on the
Hilbert space of normalized vectors emerge from the minimization of the length
specified by the Fubini-Study metric on the manifold of Hilbert space rays. In
the second case, we demonstrate that geodesic paths for probability amplitudes
arise by minimizing the length expressed in terms of the Fisher information. In
both derivations, we find that geodesic equations are described by simple
harmonic oscillators (SHOs). However, while in the first derivation the
frequency of oscillations is proportional to the (constant) energy dispersion
of the Hamiltonian system, in the second derivation the frequency of
oscillations is proportional to the square-root of the (constant) Fisher
information. Interestingly, by setting these two frequencies equal to each
other, we recover the well-known Anandan-Aharonov relation linking the squared
speed of evolution of an Hamiltonian system with its energy dispersion.
Finally, upon transitioning away from the quantum setting, we discuss the
universality of the emergence of geodesic motion of SHO type in the presence of
conserved quantities by analyzing two specific phenomena of gravitational and
thermodynamical origin, respectively.
