1. The problem statement, all variables and given/known data An electron in the Coulomb Field of the proton is in the state: |ψ> = (4/5)|1, 0, 0> + (3i/5)|2, 1, 1> with |n, l, m> as the quantum numbers defining the state a) What is <E> for this state? What are <L2> and <Lz>? b) What is |ψ(t)>? Which expectation values from a) vary with time? c) Show that the expectation value <x> will be time dependent provided the matrix element <ψ100|xˆ|ψ211 ≠ 0. d)Show that this matrix element is indeed nonzero. You don’t need to calculate a value for the matrix element, although you can if you want. 2. Relevant equations Basically laid out in the problem statement 3. The attempt at a solution So I completed part a) and part b) except I haven't shown whether or not the Energy commutes with the hamiltonian. The reason I haven't is I haven't been able to find the representation of the energy (in order to commute it with the hamiltonian) anywhere. Part of me thinks that the energy operator has the Hamiltonian as the generator of time translations (right?) and as such since the energy operator involves the Hamiltonian when it is commutated with the actual Hamiltonian operator it would be, in effect, like commuting the Hamiltonian with itself, which obviously commutes. Can anyone comment on this line of thinking and let me know where I went astray or if it is indeed logical thinking? Also if anyone could provide the representation of the Energy operator I would be very thankful. Moving on I need help with parts c) and d). Part c) asks to show <x> is time dependent (which means it doesn't commute with H) given that a particular matrix element is nonzero.. I imagine that when you commute H with x these matrix elements (<100|x|100>, <211|x|100>, <100|x|211> and <211|x|211>) play a role in the calculation and I'm assuming since we're tasked to show one particular element is nonzero than that means the other three are zero and this one term is what keeps x from commuting with H (and thus making it time dependent) can anyone comment on this as well and maybe give me some advice moving forward? I would truly appreciate it. Thanks!