Lifetime and energy uncertainty

The lifetime of a molecule in a certian electronic state is 10e-10 s. What is the uncertianty in energy of this state? Give the answer in J and in J/mol.

How to find value of integral.

Suppose that Yi and Yj are wave functions of A with eigenvalues ai and aj, respectively, and ai does not equal aj. If [A,B] =0, what are the values of the integral? Show reasoning.

How to set-up and solve the problem.

Prove the relationship [x,px]=ih, where px =-ihd/dx. Namely, show that applying the commutator [x,px] to any function f(x) is equal to multiplying ih to f(x).

Under what conditions does Y became an eigenfunction of A?

Suppose that O1 and O2 are eigenfunctions of a linear operator A with eigenvalues a1 and a2, respectively, and that we construct a linear combination Y=C1O1 + C2O2. Under what conditions does Y became an eigenfunction of A? Show reasoning.

Quantum states of ions

Please see the attached file. According to the Schrodinger atomic model, each quantum state of an atom or ion can be labelled using two quantum numbers n and l. Write down a formula for the energy of each of the quantum states of the hydrogen like boron ion B4+. For a given value of n, state how the number of quantum states depends on the value of the second quantum number l.