In their paper, the physicists developed a new way to obtain the Schrödinger equation starting from a mathematical identity using classical statistical mechanics based on the Hamilton-Jacobi equation. ... In quantum mechanics, both amplitude and phase depend on each other, and this makes the quantum wave equation linear."

Why can Schrödinger equation be derived?

The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian. The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions.

How was wave equation discovered?

Using Newton's recently formulated laws of motion, Brook Taylor (1685–1721) discovered the wave equation by means of physical insight alone [1]. ... From this early work connecting the wave equation and the Fourier transform, much of engineering mathematics of wave motion and transformations has been developed.