B. Graded exercise 7 (original exercise not based on Griffiths' book) Consider a particle moving in ID subject to the action Of a Dirac-delta potential V (x) = 06 You have studied the corresponding Hamiltonian = / 2m + V (i) in Quantum Mechanics I as an example Of a Hamiltonian with a spectrum containing both discrete and continuous domains. In its eigenbasis, you found 1-10 = + dk (20) There is a single bound state IgbGS) with = wo and corresponding wave function = (xli,bGs) = V/eAIxI with = ma/h2. The scattering eigenstates have a.'(k) = hk2 / 2m, and their corresponding eigenfunctions are a bit annoying to write unless we impose some boundary conditions. In particular, let us care Only about stationary states with particles "corning from the left" (that is, terms eikx vanish for 'x > O), for which you found + B (kleikx for > O k) = (xl+(k)) = A(k) x for x < O il(k) B(k) = F(k) = rn.a er(k) = (21) (22a) (22b) (22c) A(k) is a normalization factor that is quite involved except in the high-energy limit, as you will prove shortly. In this exercise we are going to let the system start at time zero in the ground state, = and then apply the sinusoidal perturbation Of frequency wp W (t) = W cos(wpt), with = (23) where g > O is the perturbation rate. We will then analyze probability Of transitioning into the continuum, using Fermi's golden rule as presented in class. l. [5] Argue that in the high-energy limit k/ A i 00 (equivalent to a.'/wo + 00) a constant A(k) = normalizes the states up to Order A/ k or 2. [8] Show that the matrix elements V(k) Of the perturbation are given by (A2 + k2)2 (24) I suggest you work in the position representation, where you'll have to work out simple integrals Of exponentials multiplied by polynomials (we have handled such integrals many times in previous exercises, so don't forget to show me how you do them!). 8 3. [5] In class we denoted the labels Of the states Of the continuum by y'3 p) where '30 is the label for the eigenenergies and any extra labels required to identify the eigenstates. For the Hamiltonian (20) Of this exercise, identify 130 and p, and find the density Of states p(w) that relates dO with dw. 4. [5] Suppose < What's the estimation made by Fermi's Golden rule for transition probability 5. [121 Suppose now that Wp -F with w hk2/2rn > O. Using again Fermi's Golden rule, argue that (t) r (w)t and find out the functional form Of the rate l'(w). Specifically, you should be able to S -4 co nti write it in the following form I -F (WO/w)E5 (25) Identify the values Of the exponents Show then that in the high-energy limit w wo, the rate becomes a monotonically decreasing function Of w. In particular, show that increasing the energy by a factor f > l, the ratio between the transition rates follows a power law r fK, identifying the power K.