Question

prove schrodinger equation​

Answer 1

Answer:

Explanation:

E=ℏω (energy is proportional to frequency, the Einstein-Planck relationship)

and

p=ℏk where k is the wavenumber. This is de Broglie's relationship.

Now, there are a few ways to formulate Classical physics, in terms of forces (a'la Newton), in terms of Lagrangians and in terms of Hamiltonians. I imagine Schroedinger thought that the Hamiltonian picture was the simplest to use on this wave-particle picture. A justification for this choice would be that it represents mechanics coming from the principle of optics (i.e. waves), so it does seem like a good choice for physics of matter based on waves!

He actually tried a relativistic form of his equation first, but didn't like it (it is now known as the Klein-Gordon equation) - so reverted to classical mechanics.

I would imagine he started by writing down the basic Hamiltonian equation combining it with the Einstein & de Broglie work to create something like

H=E=ℏω=ℏk22m+V.

This is of course merely saying that the energy (from the Einstein-Planck relation) is equal to the kinetic energy (mv22=p22m ) plus the potential energy.

Now if matter really is a wave there must be some kind of wave equation, such as sin(ωt+kx)+icos(ωt+kx). Which I imagine he played around with. Once you have the solution to that, you can rewrite the equation in terms of operators rather than constants - and you pretty much have the Schroedinger equation. From there it is childs play.

Answer 2

${\huge{\underline{\green{Solution:}}}}$

Erwin Schrödinger, in 1926, proposed an equation called Schrödinger equation to describe the electron distributions in space and the allowed energy levels in atoms. This equation incorporates de Broglie’s concept of wave-particle duality and is consistent with Heisenbergun uncertainty principle. When Schrödinger equation is solved for the electron in a hydrogen atom, the solution gives the possible energy states the electron can occupy [and the corresponding wave function(s) (ψ) (which in fact are the mathematical functions) of the electron associated with each energy state].