By saying totally different I mean , while deriving wave equation for entities like electric and magnetic field we do consider electric and magnetic field oscillating , but as same as this why didn't Schrodinger derived wave equation for electron physically waving ?
In Schrodinger's wave equation, the probability distributions representing clouds of uncertainty or indeterminacy are called wave functions. This is another way of stating that Schrodinger's wave equation is solvable for the wave functions instead of being solvable for the precisely determinable variables of classical physics.
The key distinction between Schrodinger's equation and other wave equations is that Schrodinger's equation is first order in time, with the 'i' taking on the role of the second time derivative. Then, ...
4 One can surely consistently derive the stationary state Schrodinger equation straight from the Dirac-von Neumann axioms. They are The observables of a quantum system are defined to be the (possibly unbounded) self-adjoint operators $\mathcal {O}$ on the Hilbert space. A state $\psi$ of the quantum system is a unit vector in Hilbert space.
10 I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics". However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary.
The equation $\frac {\partial^2 f} {\partial t^2} = c^2\nabla^2 f$, despite being called " the wave equation," is not the only equation that does this. If you plug the wave solution into the Schroedinger equation for constant potential, using $\xi = x - vt$