Problem 1: Lifetime and Energy Uncertainty We know that a stationary state is of the form 4[ř, t] = [7]. (1) Here [F] is an eigenstate and & is the associated eigenenergy. (Remember that h = 1 in natural units.) In the idealized setting normally considered, this is a steady state of the system since it has a time-independent proba- bility density. In practice, though, all excited states have a finite lifetime, t, and a more realistic representation of the probability density for any excited state is p[t] = e (2) It is only through QFT that the decay of such "stationary states" are possible. With standard Schrödinger equa- tion quantum mechanics, a pragmatic expedient is to simply adopt a more physically reasonable excited state representation: t 4[7, t] = [√] 1 ∞ t 2t Τ (3) t [a] Focus on the temporal component of this, T[t]:= 21, and calculate the following: (i) † := F[7], the temporal Fourier transform of T[t]. Call the Fourier frequency ε, so that you have Ť[ɛ], a complex-valued energy spectrum for the wave function. (ii) The spectral density, D, is defined as D[ɛ] := †[ɛ] + †[ɛ]* = 2 Re[7[×]]. Interestingly, the inverse Fourier transform of D[ɛ] is equal to T[t], so all we have really done is found a real-valued Fourier transform of the time-varying portion of the wave function. If you want, you can test this for yourself by calculating F¨¹[D]. [b] Plot the spectral density, D[ɛ], for a fixed value of ε and several different values of t. (Put these all on the same plot, carefully labeling everything.) These plots should make clear that a finite lifetime, r, implies a finite line width--i.e. a spectrum that peaks at ε but is spread out in a Gaussian-like distribution around this energy as shown below. D max δε Dmax 2 Verify that the width of the distribution, 6, is equal to 1/7 at the half-height of the peak. This is typically inter- preted as the range of energies that you might expect to measure in an experiment. It implies that τ δε = 1. (4) This is called the Lifetime Broadening Relation, and it should call to mind the time-energy uncertainty relation. Explain what the LB Relation says about the certainty with which you can know the excited state energy as a function of the lifetime of the excited state. Look at the two extremes, zero lifetime and infinite lifetime, to help elucidate the physics. As an aside, excited state decay is often referred to as being the result of having a small imaginary component to the energy. This comes from looking at 4[r, t] = 4[7]e%% t e The effective energy of the "stationary" state is then &eff = ε0 21 (-1) (5) [c] All excited states have experimentally measurable line widths. Find one example of this in the literature, provide a link, and provide a screen shot to show the broadening data that you have found. The point you should now have is that the line width itself, an observable, allows you to estimate the lifetime of that excited state.