I am wondering if scientists have tried and succeeded to use the ansatz
$$ \Phi(t,x) = (A(t,x) + \hbar B(t,x) + ...)\exp(i S(t,x) / \hbar) $$ To solve more complex QM equations like for Hydrogen.
Using the ansatz to solve a free particle lead to the correct plane waves. But solving the Hamilton Jacobi Equations (HJE) for hydrogen lead to issues with solutions (as far as I understand) living in planes and not defined in whole of $R^3$. Still the success of the Bohr model is teasing as it suggest that you could indeed succeed with this approach.
The idea is to apply the Schrödinger equation to the ansatz and collect terms with the same order of $\hbar$. The first term yields the classical correspondance principle HJE and for the standard hydrogen atom Hamiltonian the HJE is the same as for a planetary like two body system. All the rest of the equations seam to yield equations with elements on the same order so the expansion on paper look correct. It is an infinite sequence of second order equations and looks complicated. We know that the Bohr model yields a pretty good energy level so we could perhaps simplify and assume $A,B,...$ do not depend on time. The main obstacle to this approach, however, is that the solution to HJE seam to be defined on a plane.
