OK so I am looking for a definitive answer. I am very confused and don't know why! Any help would be much appreciated, I will try and convey the source of my confusion below. Firstly what exactly is psi(x) which crops up everywhere? Does it mean the probability amplitude of measuring the particle at x, or is it "the state of the system", or are these the same things? I ask because I have seen the dirac ket psi a lot, e.g. H|psi>=ih (d/dt)|psi> But I also see: <x|psi> = psi(x) which is described as the probability amplitude of measuring the particle at x when the system has state |psi>. However I also see H psi(x) = ih (d/dt)psi(x) Then I see psi(x) = 2^(-0.5) (phi1(x) + phi2(x)) Here the phi are described as "energy states", and they are eigenfunctions of the TISE. So here I guess psi(x) is the state of the system: are the 2^(-0.5) the amplitudes of measuring E1 and E2, or is phi(x) part of the amplitude?