MIT OpenCourseWarehttp://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.74, Problem Set #3 Spring 2009 Due Date: Mar. 10, 2003 1. Relaxation from a Prepared State The linear response function R ()τ describes the time-dependent behavior of a system subject to an impulsive perturbation from equilibrium. You can also imagine applying a force that holds the system away from isolated equilibrium state, and then suddenly releasing this force. The relaxation back to equilibrium is dictated by the step response function S(τ), which we will investigate. The Hamiltonian for the system will reflect an applied constant force prior to time zero: HH0 − f ()A f ()t =⎨⎧⎩ f 00 tt ≥< 00 ⎬⎫⎭ = f0 Θ− t= t ( ) () is the unit step function. We have applied the force at a time in the distant past, so that the Θ tsystem at t = 0 has come to equilibrium under the Hamiltonian H. At long times, t = +∞, the system is at equilibrium under H0. We will use the eigenstates of H0 as the basis to solve this problem. a) Give expressions for the expectation value of A at time t = 0, A , and at time t = +∞, A ∞.0 e−βHATo help evaluate ensemble averages such as , you can make use of the classical linear response approximation: e−β(H0 − fA )≈ e−βH0 (1+βfA ) ( fA << H0) Explain why a quantum evaluation is more complicated. b) Given that the system initiates at A at t =0 , what is the time-development of A for t > 0 ,0 where the system evolves only under H0? To solve this, first prepare the system in the state HH − fA at t = 0 . To do this imagine starting with the equilibrium system under H0 at t == 0 0 −∞, gradually ramping up the extra force at a rate η, and then abruptly shutting it off at t =0 : f ()=f ηt ( t).t 0e Θ− Obtain an expression describing the time-evolution of the internal variable A for t > 0 for the limit that η→0. It may be helpful to make use of the Fourier transform relationship that relates the response function to the susceptibility. c) Given that the time correlation function for the fluctuations in A are given by 2 −iωmntC t A t 0 =∑pn A eAA ()= I () ( ) AI mn ,nm Show that the step response in (b) can be differentiated to give the impulse response and that the step response can be expressed as an expansion in cosines, i.e., functions that are even in time.5.74, Problem Set #3 Page 2 2. Harmonic Oscillators The following questions refer to harmonic oscillators with a Hamiltonian 2 2 2 †1H = 1 p +1 mω0 q = hω(aa +2 )0 02m 2 a) Show that the thermally averaged occupation number n for a harmonic oscillator of frequency ω0 is given by n =(exp(βω)−1)−1.h 0 b) Obtain expressions for the time evolution of the operators p(t) and q(t) in terms of raising and lowering operators. c) Calculate the correlation function for the harmonic oscillator coordinate: Cqq= qtq .() (0) 3. Displaced Harmonic Oscillator Model Work through the correlation function description of the electronic absorption spectrum for a transition coupled to nuclear motion, using the displaced harmonic oscillator model discussed in class. H0 = G H G G + E H E E †H=H + E = hω aa +1 )+ EG g g 0 ( 2 g H = H + E = e−ipd / h H e ipd / h + EE e e g e a) Assuming that the system is in the ground electronic state at equilibrium, and making the Condon approximation, show that t μ = μ 2 −ωi egtF (), wheree tμ() ()0 eg iH t / iH t / hFt()= eg he− e . b) If the system is only in the ground state of the nuclear potential at equilibrium, show Ft()=exp d ⎡ 2 (e−ωit0 −1)⎤where dd= ⎣~ ⎦~ 0 2 mω h .5.74, Problem Set #3 Page 3 4. Transformation between Hamiltonians. Motivated by the previous problem, imagine that you wish to express the time-evolution of a wavefunction under one Hamiltonian He in terms of another Hg. We also define the difference Hamiltonian: Heg = He − Hg . This suggests that we find a way of expressing the time evolution under He as: H =H + He geg U =UU e geg Show that we can obtain a transformation between Ue and Ug analogous to what we did with the interaction picture Hamiltonian: U = exp ⎡−i tdτ H τ⎤ H τ = UHU eg +⎢h ∫0 eg ()⎥ eg () g† eg g⎣ ⎦ †This also means that U =UU .eg g