By the Principle of Superposition, a state vector can be defined as pic.01 also, the state vector can represent a wave function in a continuous case as pic.02 My (1) question is, in pic.03, why the state vector can be pulled out from the integral??? I have an idea but I think it should be wrong: ------------ As shown in pic.02, the state vector is a set of values that each element is the value of the wave function, e.g. phi(0), phi(0.002), etc. Thus, it can be considered as a constant since every elements will never change Therefore, it can be pulled out from the integral ------------- However, this idea is fail to explain in the observable case. Considering, in pic.04, question (2), Q is an operator which contain a momentum component, a differential operator. If the state vector is constant, it should give zero value when a momentum operator act on it and the Dirac representation will be failed. So, if it is not a constant, how can it be pulled out from the integral???????? ======================== And my (2) question is that, in pic.04, step 4 to step 5, why the operator Q can skip the bra(x) vector and act on the state vector??????? ========================= Thank you so much as I'm new in Quantum Mechanics!