The Schrödinger equation is be pretty basic stuff but it is pretty cool. This is how it comes from the energy conservation law:
We start out with de Broglie (and the theory of relativity)
and we know that all particles are also waves, a typical wave equation being
Just for the fun of it we make the derivative
and we find out that we can get the momentum through deriving the function. Who would have thought that?
Next we define the momentum operator
and we assume that for every possible wave function the following is true (this and the definition of the position operator are the axioms of quantum mechanics):
With this knowledge we can look at the energy conservation law of mechanics (T ... kinetic energy, V ... potential energy, E ... total energy)
The kinetic energy can be rewritten
We can multiply this equation by Ψ
Finally we replace the momentum by the momentum operator
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