  In this work we explore the possibility of quantum bound states in a
Schwarzschild gravitational field leveraging the analogy of the elementary
derivation of bound states in the Coulomb potential as taught in an
undergraduate course in Quantum Mechanics. For this we will also need to go
beyond non-relativistic quantum mechanics and utilize the relativistic Dirac
equation for a central potential as taught in an advanced undergraduate or
first year graduate (special) relativistic quantum mechanics course. Finally,
the special relativistic Dirac equation must be extended to the general
relativistic version for curved spacetime. All these disparate component pieces
exist in excellent, very readable textbooks written for the student reader,
with sufficient detail for a curious student to learn and explore. We pull all
these threads together in order to explore a very natural question that a
student might ask: "If the effective $1/r$ radial potential of the
Schwarzschild metric (with angular momentum barrier), as taught in elementary
GR courses for undergraduates, appears Newtonian-like (with a $1/r^3$
correction), then is it possible to derive quantum bound states in the
Schwarzschild spacetime by simply changing the radial potential $V(r)$ from
$V_C(r)=-e^2/r$ to $V_{Schw}=-G M m/r$?"
