Final answer:
The student is asking to find eigenfunctions and eigenvectors of a differential operator with boundary conditions. However, the problem contains elements that relate to both Sturm-Liouville theory and quantum mechanics, and the relationship between the operator and these contexts is not clear.
Step-by-step explanation:
The student has asked to find all eigenfunctions and eigenvectors of the operator L = d²/dt² + b, where b is a real number, on the interval [0,π], given the derivative boundary conditions x(0) = x(π) = 0. To solve this problem, we need to consider a differential equation of the form Lx = λx, where λ is the eigenvalue. However, this case does not appear to correspond to a standard Sturm-Liouville problem due to the presence of the term bx without the proportionality to x. Moreover, quantum mechanics concepts such as wave functions and normalization conditions are referenced, but these pertain to physics rather than the given mathematical equation.
Therefore, it seems there might be a misunderstanding in the question as presented, as the mathematical operator doesn't align with usual quantum mechanics problems. For an exact solution, a proper formulation of the problem is essential with clear connections to either differential equations or quantum mechanics context.