2. Why do we need creation and annihilation operators?
Main point is that a particle can be created by creation operator and destroyed by the annihilation operator. But to destroy a particle is to change from one vector in Hilbert space to another, right? So these operators act on states and actually change them. In a definite way. So why do we need them? We can just write the states as we want...if I know I want a state with two particles of definite momenta I can just write it, why use these operators to create particles? Obviously, I am missing something here. What is the real purpose of creation and annihilation operators? Is it because through these operators, fields are represented better than through momentum or energy operators? So, combinations of creation and annihilation operators give us our observables?