1) We understand that a stationary state takes the following form: V[7, t] = VT]e-iot (1) W 7] is an eigenstate, and zo is the associated eigenenergy. (Remember that h = 1 is in natural units.) In the commonly idealized scenario, this represents a system's steady state because it possesses a time-independent probability density. However, in practical situations, all excited states exhibit a finite lifetime, denoted as r, and a more realistic portrayal of the probability density for any excited state can be described as follows: p[t] = e-# (2) Quantum Field Theory is the framework that allows for the decay of these so-called " stationary states." In traditional quantum mechanics, governed by the Schrodinger equation, a practical approach is to instead embrace a more physically plausible representation of excited states: V[T, t] = we-icote 27 (3) Question:Focus on the temporal component of this, T[t] = e-2, and calculate: (a) T = [T], the temporal Fourier transform of T[t]. Call the Fourier frequency E, so that you have T[E], a complex-valued energy spectrum for the wave function. (b) The spectral density, , is defined as = T[e] + T[e]* = 2Re[T ]. Curiously, the inverse Fourier transform of is equal to T[t], thus, the Fourier transform in the domain of real numbers for the time-varying segment of the wave function. - You can use Mathematica for plotting