That's really too complicated for a short answer, of a few paragraphs, here. YouTube has some introductory videos about the topic, for example, from a "Dr PhysicsA"; a search on YouTube for "Dr Physics Schrödinger Wave Equation" will let you find them.
equation is a double differential relate to the energy of particle with wave function
1) What is the definition of dielectric permittivity on the basis of Maxwell equations? 2) What is Poisson equation of Electrostatics? Derive the Poisson equation from Maxwell equations. 3) Write the Biot-Savar equation. What is the meaning? 4) Derive a wave equation of a plain electromagnetic wave from Maxwell equation.
The time dependent equation is more general. The time independent equation only applies to standing waves.
Erwin Schrodinger
The intensity of an electromagnetic wave is given by the following equation: I = (ε0c/2)E02, where ε0 is the permittivity of free space, c is the speed of light in a vacuum, and E0 is the maximum amplitude of the electric field. I added a related link below that shows how to derive this.
This is the Schrodinger equation from 1925-1926.
Schrodinger wave equation
equation is a double differential relate to the energy of particle with wave function
No, it is not solvable for any multi-electron system.
For general waves...probably d'Alembert, who solved the one-dimensional wave equation. In quantum it would have to be Schrodinger.
It is also called wave mechanics because quantum mechanics governed by Schrodinger's wave equation in it's wave-formulation.
The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics which Erwin Schrodinger developed in 1926. This model is based on the Schrodinger Equation.
1) What is the definition of dielectric permittivity on the basis of Maxwell equations? 2) What is Poisson equation of Electrostatics? Derive the Poisson equation from Maxwell equations. 3) Write the Biot-Savar equation. What is the meaning? 4) Derive a wave equation of a plain electromagnetic wave from Maxwell equation.
Schrodinger
The equation, as originally written by Erwin Schrodinger, does not use relativity. More complicated versions of his original equation, which do incorporate relativity, have been developed.For more information, please see the related link below.
The time dependent equation is more general. The time independent equation only applies to standing waves.
It is used to find probability distributions (expectation values) of properties of subatomic particles.