  We provide a unique and novel extension of da Costa's calculation of a
quantum mechanically constrained particle by analyzing the perturbative back
reaction of the quantum confined particle's eigenstates and spectra upon the
geometry of the curved surface itself. We do this by first formulating a two
dimensional action principle of the quantum constrained particle, which upon
wave function variation reproduces Schr\"odinger's equation including da
Costa's surface curvature induced potentials. Given this action principle, we
vary its functional with respect to the embedded two dimensional inverse-metric
to obtain the respective geometrodynamical Einstein equation. We solve this
resulting Einstein equation perturbatively by first solving the da Costa's
Schr\"odinger equation to obtain an initial eigensystem, which is used as
initial-input data for a perturbed metric inserted into the derived Einstein
equation. As a proof of concept, we perform this calculation on a two-sphere
and show its first iterative perturbed shape. We also include the back reaction
of a constant external magnetic field in a separate calculation. The
geometrodynamical analysis is performed within a two dimensional dilation
gravity analog, due to several computational advantages.
