Just trying to get my head around undergraduate quantum mechanics and they throw a lot of stuff at you, so some help is appreciated. I understand that the wave function is some abstract vector living in an infinite dimensional hilbert space, and that it's a function. But then the textbook i'm reading goes on to represent the state as superpositions of two states: s=al1>+bl2> I've actually done quite a lot of QM calculations and am very familiar with the maths involved in ket-bra notation and what they represent. But i'm just confused as to how they tie in with the Hilbert space and infinite dimensional vectors and such. They've now represented the hamiltonian as a 4x4 matrix, the state by 2/2 matrix and I'm just confused as to how the functions tie into this at all. Do these states have anything to do with functions, or the wave function. I mean we were specifically told not to think of the wave function or QM in matrix form and just think of it as an abstract vector function, but now all the math we are doing is using superpositions and using vectors to calculate eigenvectors and eigenfunctions of the hamiltonian and so on. It's just all very confusing and overwhelming haha. Thanks for any help.