Dilatonic Geometrodynamics of a Two-Dimensional Curved Surface Due to a Quantum Mechanically Confined Particle

Gravity theory has wide applications to solve geometry-sensitive problems in condensed matter physics. In this work, we provide a unique and novel extension of da Costa's calculation of a quantum mechanically constrained particle (Phys. Rev. A 23, 1982) by analyzing the perturbative back reaction of the quantum confined particle's eigenstates and spectra upon the geometry of the curved surface. We do this by first formulating a two-dimensional action principle of the quantum-constrained particle, which upon wave function variation reproduces Schrodinger's equation according to da Costa. 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 proceed to solve this resulting Einstein equation perturbatively by first solving da Costa's Schrödinger equation to obtain an initial eigen-system, which is used as initial input data for a perturbed metric inserted into the derived Einstein equation. We perform this calculation for a perturbed two-sphere. 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.