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I am taking my first Quantum Mechanics course. It is a graduate level course (I am at the end of my Junior year as an undergrad) and I have not taken any introduction courses to Quantum Mechanics before. I am about a month in and I am doing fine in it so far. Although I am doing fine, I still have some confusion about some main concepts. This past month has been entirely mathematics with some examples of spin 1/2 systems thrown in. At my level of understanding it all seems extremely abstract. My main problem is relating it to real life scenarios.

For example, I know that an observable is measurements made in real life to some quantity and when we take these measurements, the system is thrown into an eigenstate of the observable. But this concept of observable still seems very ambiguous to me. How would one obtain the information on an observable? And by that I mean, how would these "measurements" translate into a matrix?

Right now I am being thrown so much mathematics, and it is hard to take it all in without some grip on how to apply it or where I could use it.

My other question is, where did the Schrodinger equation come from? My professor says it is a fundamental postulate of quantum mechanics and cannot be derived, but what was the logic used to construct it?

Thanks