system
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human
What does the Schrodinger equation mean and what does it tell us?

I am beginning quantum mechanics and we have been studying the Schrodinger equation pretty extensively. But I am confused about what this equation represents and why can't we derive it?
gpt
The time-dependent Schrodinger equation tells you how a quantum state vector evolves with time. It says that the Hamiltonian operator is the generator of translations in time. If you know the state vector at time t = 0, the Schrodinger equation and the Hamiltonian tell you what the state vector will be for all time.

You can't derive the Schrodinger equation because it's a **postulate** of quantum mechanics. You can either postulate linear and unitary time evolution, and the TDSE follows, or you can just postulate the TDSE directly. Either way you're making an assumption about how quantum states evolve with time. This is a fundamental assumption to quantum mechanics, hence why you can't "derive" it.