Schr\"{o}dinger Equation from Hamilton-Jacobi Equation

The time-dependent Schr\"{o}dinger equation is now considered fundamental and the time-independent Schr\"{o}dinger equation is derived from it for stationary states. Historically, it was the other way around. Schr\"{o}dinger obtained his time- independent equation first and then obtained the time-dependent equation for time-independent potentials. He then postulated it to be valid in general. We use the classical Hamilton-Jacobi equation to obtain both the time-dependent Schr\"{o}dinger equation and the equation of continutiy. We first derive Schr\"{o}dinger's ``Ansatz'' for the action S in terms of the wave function. By this change of variables the Hamilton-Jacobi equation is transformed into a complex equation. The equation of continuity is obtained from the imaginary part. The real part and the remaining imaginary part are combined to give a Schr\"{o}dinger equation with a nonlinear term that is a remanent of the Hamilton-Jacobi equation. When this nonlinear term is dropped the linear time-dependent Schr\"{o}dinger equation is obtained. This approach fills a gap in the development of the time-dependent Schr\"{o}dinger equation. It also shows the intimate connection between classical and quantum mechanics.