prove the Schrodinger's equation ??
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Answer:
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Answer:
Andrew's answer is a perfectly good justification for why the Schrodinger equation is plausible (once you're familiar with classical physics), but as a derivation, it's backwards. Classical mechanics is an approximation to quantum mechanics, so the task should really be to derive the former from the latter. Presumably someone will attempt this here:
As far as I'm aware, there is no derivation of quantum mechanics that doesn't assume the basic axioms that the state of a system is described by a vector in a Hilbert space, and the expectation values of operators are given by matrix elements between states. The Schrodinger equation is just a way of rewriting the additional axiom that time evolution acts linearly on states.
This makes it seem like the Schrodinger equation doesn't have much content, but there's a more subtle reason that physicists make such a big deal about it. This reason is just as easily illustrated with Newton's law of motion F=ma, so I'll talk about that instead. If your physics teacher walks up to you and says "F=ma", it's not very helpful because one could read the equation as simply the definition of Force. However, your teacher is also making an implicit promise that they will tell you what F is in different physical situations, from which you can calculate a.
The Schrodinger equation is similar. It is basically already an axiom of quantum mechanics (linear time evolution), so its derivation is a little uninteresting. What's interesting is that one can independently specify the Hamiltonian corresponding to various physical situations (like the Hydrogen atom, or a collision between elementary particles in the Standard Model), and consequently deduce what happens.