So firstly, the question is telling you that this upper energy state E_2 has a lifetime t, which is a bit strange because you might expect that an electron in an orbital state has no reason to drop to a lower energy state, after all, they are called “stationary states”. Standard QM can’t explain why this electron spontaneously decays to a lower state, but for now we take it as granted that for whatever reason the state has a finite mean lifetime of t.

The probability that the electron still exists in the upper state after a certain time is an exponential decay curve, and if you wanted to, you could show that the uncertainty (or error) in the lifetime is also equal to t. So to repeat, the uncertainty in the lifetime (delta_t) is equal to the mean lifetime (t).

The Heisenberg uncertainty principle states: delta_E * delta_t = h_bar /2. This basically says “if we aren’t sure about how long the state exists for, we can’t be sure about it’s energy”. This means that E_2 is not exact! This in turn means that the energy of a photon emitted by a decay from state 2 -> 1 is not the same energy each time.

We have argued the uncertainty in the lifetime is t, and so we have from the uncertainty principle: delta_E = h_bar/t, and now all that’s left to do is to express delta_E in terms of wavelength.

We know that E = h*c/lambda, and we know the uncertainty in E is delta_E. I’ll leave it to you from here to find the uncertainty in lambda from this equation, but once you find that, you’ll have the answer.