Calculate the standard deviation of the energy for a particle in a state, which is a superposition of two stationary states with coefficients c1 and c2. Do this calculation in two ways: (i) using the wave function of this state and a standard deviation of quantum mechanical averages, and (ii) using the probabilistic interpretation of the coefficients c1 and c2. Did you get the same results?
I assume that your teacher has in mind the probabilistic interpretation in which the probabilities are p1 = |c1|^2 and p2=|c2|^2.
A stationary state is an eigenstate of the energy.
If the stationary states 1 and 2 have the same energy E, the standard deviation of the energy is 0 in both cases (i) and (ii), since in this case <H^2> = E^2 = <H>^2
Therefore we only have left the case of stationanry states with different energies, E1 not= E2.
In this ...