Let
$$\begin{cases} i\frac{\partial}{\partial t} \Psi(x,t) = \Delta \Psi(x,t);\\ \Psi(x,0) = \varphi(x) \end{cases}$$ Why do physicists seek a solution of this equation in the form: $$ \Psi(x,t) = e^{iS(x,t)} a(x,t) $$ where $S(x,t)=\int_{0}^{t} L(x(s),\dot{x} (s))\,{\rm d}s$, such that $L(x(s),\dot{x}(s))$ is the Lagrangian associated to the Hamiltonian which is also associated to $\Delta$ and $a(x,t)$ is the amplitude.
Thank you in advance.