Origin of the Schrodinger equation by L. D. Landau and E. M. Lifshitz

In the book "Quantum Mechanics" by L. D. Landau and E. M. Lifshitz, it is mentioned that,

"The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behavior at all subsequent instants is determined".

Then it continues and says,

"The mathematical expression of this fact is that the value of the derivative $\frac{\partial \psi}{\partial t}$ of the wave function with respect to time at any given instant must be determined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. In the most general form, we can write $i\hbar \frac{\partial \psi}{\partial t}=H \psi$, for some linear operator $H$.

I want to understand on some physical grounds as how is it that the mathematics of the fact that the wave function Ψ completely determines the state of a physical system in quantum mechanics is contained in the equation $i\hbar\frac{\partial \psi}{\partial t}=H \psi$.

• It's a first order differential equation, so only the initial state is needed to determine all future states. – DanielSank Sep 27 '17 at 20:29
• @DanielSank thank you. I am not getting the role of that operator there. Had $H$ been simply a number, I would have no hesitation in accepting your argument. – Seeker Sep 27 '17 at 20:41
• An operator is "something that acts on what it has on its right side (namely, $\psi$) But in QM it is a LINEAR hermitian operator. It can only contain linear operations like derivatives or integrals, so it will always be a differential equation, no matter how difficult it is. – FGSUZ Sep 27 '17 at 20:59
• @FGSUZ Heisenberg's matrix mechanics formalism of QM has infinite dimensional matrices representing these operators, not spatial differentials. – hyportnex Sep 27 '17 at 22:24