Determination of the Wave Function:
We have seen how state vectors and operators are used to perform calculations relevant to the physics of particles. Let's now discuss the logically prior question: how is the state vector determined for a given particle? The answer is the Schrodinger Equation. In its simplest form, the Schrodinger equation looks like
H|S> = E|S>
The operator H is called the "Hamiltonian" of the particle or system, and E is total energy operator. The Hamiltonian operator is simply the sum of the kinetic energy operator (which always has the form we saw above) and the potential energy operator (which can be different depending on the system under study). In this form, the Schrodinger equation doesn't seem like much. All it seems to say is that total energy of a particle is the sum of its kinetic and potential energies. Nothing too surprising or even useful. If we project it into the position basis, however, we get something we can sink our teeth into. It looks like
[-(h_bar2 / 2m) grad2 + V(x,y,z)] S(x,y,z,t) = i h_bar dS(x,y,z,t)/dt
I used V(x,y,z) to denote the potential energy operator, which generally is a function of the operators corresponding to spatial positioning (x,y, and z). The right-hand side shows the position-space representation of the total energy operator (i h_bar d/dt). This is similar to the momentum operator except the differentiation is with respect to time rather than spatial variables. This may look pretty nasty, and it will only get worse when we decide on the form of V(x,y,z) for any given system. However, it has the form of a Partial Differential Equation. The good news is that as insane as it looks, it's something we can solve through standard methods of calculus. And when we do solve it, we obtain the particle's wave function S(x,y,z,t)!
Of course, what we wanted was |S>, the quantum state vector. Instead we got the wave function S(x,y,z,t), i.e. the projection of |S> onto the position basis in Hilbert space. Fortunately, this is good enough. For once the wave function is known, we can use other mathematical transformations to determine other representations within Hilbert space if we need them. For example, we can use the Fourier Transform to go from the position-space to the momentum-space. And other representations are also possible and easily accessible should we require them. All possible representations in Hilbert space are equally valid expressions of |S>. No particular representation is objectively better or truer, as they all contain a complete description of the state in the terms of their corresponding basis.
This is getting really long, and I'm only like halfway through what I wanted to say on the topic. I gotta run for now, but I'll try to sit down and crank out the rest later today. If this seems really dry, technical, and abstract, I'm sorry. I'm getting to the point where I can describe some of the really amazing consequences of the theory, so please bear with me.