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 #2 Spring 2009 Due Date: March 3, 2009 1. Magnetic Dipole and Electric Quadrupole Transitions In the light-matter interaction Hamiltonian ()=− qiA p iVt ∑⋅ i mi =ε −(⋅ −ωt)+ . . A A 0 ˆ exp ⎣⎡i k r ⎦⎤cc the leading term in the long wavelength expansion eik r⋅=1+⋅+Lik r leads to the electric dipole transitions. The second term leads to magnetic dipole and electric quadrupole transitions, which we will describe here. We can use the identity ˆ ⋅ ⋅ =εˆ ⋅ pr ⋅kεp k r()() () = 1 ε(pr − rp k )+1ˆ +)ˆ ε(pr rp k 2 2 to separate Vt() into two distinct light-matter intereaction terms where the first gives rise to magnetic dipole transitions, and the second leads to electric quadrupole transitions. For the notation above, pr represents an outer product (tensor product p: r ), so that ⎛prpr pr ⎞kxxxy xxz ⎛⎞ ⎜ ⎟⎜⎟() ⋅k=(ε ε ε )pr pr pr⎟kyεˆ ⋅pr xy z ⎜yx yy yz ⎜⎟⎜ ⎟⎜⎟pzxr zy pr kz⎝ pr zz⎠⎝⎠ This expression is meant to imply that the component of r that lies along k can influence the magnitude of p along ε. Alternatively this term could be written ε( ) .∑ab xyz , = , , a pa b rkb a) From the first term, express the magnetic dipole interaction in terms of the magnetic field B and the orbital angular momentum L . What is the form of the magnetic dipole moment? A useful vector identity is ()AB CD ( )=(ACBD )⋅ − ⋅ ADBC ⋅×⋅× ⋅()( )( ) b) From the second term, derive the matrix elements for electric quadrupole transitions in analogy to the electric dipole moment, and show that it is a tensor quantity.5.74, Problem Set #2 Page 2 2. The infrared rotational-vibrational spectrum of a diatomic Let’s study the correlation function for the rotational-vibrational infrared absorption spectrum of a heteronuclear diatomic molecule. We are considering the transitions induced by infrared light that is resonant with a vibrational transition with frequency much greater than kT, and the frequency of the infrared light is much higher frequency than the splitting between rotational levels. We will use the harmonic oscillator and rigid rotor Hamiltonians to describe the vibrational and rotational degrees of freedom: H0 = Hvib + Hrot H = p2 + 1 kq 2 = hΩ⎛ n + 1 ⎞ vib 2MR 20 ⎜⎝ 2 ⎟⎠ L22H = I = MRrot R2I Here where MR is the reduced mass and q is the vibrational coordinate. I is the moment of inertia, which is related to the rotational constant by B = h /4π Ic . So, the state of the system will be described by three quantum numbers nJM . The diatomic will interact with the light through the transition dipole moment. Since we assume that there are no interactions between the vibrational and rotational degrees of freedom, we will also partition the dipole operator into two parts: μ=μˆ m where μˆ is a unit vector oriented along the transition moment (and along the vibrational coordinate in this case), and m is the magnitude of the dipole moment, which we take to be a weak function of the nuclear coordinate: ⎛∂m ⎞ mm= 0 +⎜ ⎟ q⎝∂q ⎠q0 = m0 + mq 1 m0 is the magnitude of the permanent dipole moment and m1 is the amplitude of the transition dipole moment.5.74, Problem Set #2 Page 3 For the following, the system will interact with an electric field oriented along the εˆ = zˆaxis of the laboratory. Remember that in describing the probability of occupying an initial state, we need to consider both probability of occupying the initial state at temperature T and the probability of having a particular orientation in the laboratory frame. The initial state of the system will always be isotropically distributed, i.e. equal probability of finding a dipole oriented in any direction. a) Evaluate the dipole matrix element kεˆ⋅μ l for the interactions between an initial state ′′′l = nJM and a final state k =nJM using the separation of orientational and vibrational variables. Calculate this only to the leading order expansion coefficient for m. b) Evaluate the correlation function for a gas phase ensemble of molecules at μ(t)μ(0) pressures low enough to neglect interactions between molecules. c) Plot the correlation function for temperatures such that B k/ T >> 1, B= kT, and B k/ T<< 1 (but neglect temperatures high enough that you need to consider n=2). Be sure to point out the important features of the time-dependent behavior. You will probably want to plot the rotational, vibrational and combined contributions separately. For the purpose of plotting results, you can use Ω0 = 2000 cm-1 and B= 5 cm-1. d) Calculate the analytical form of the spectrum corresponding to the correlation functions in (b) and plot the lineshapes. Point out how the frequency resolved features in the spectrum were manifested in the time-dependent behavior in (b). 3. Numerical calculation of correlation function from a trajectory The file cf.dat on the 5.74 web site contains a trajectory of instantaneous transition frequencies ωi(t) for the vibration of a molecule in a fluctuating environment. The file is a set of roughly 32000 consecutive frequencies in cm-1 for points separated by 10 femtoseconds (10 x 10-15 sec). We will calculate the correlation function using the time-averaging expression for a single molecule valid for an ergodic system. = () 0a) Express the integral version of the time-correlation function Ct() δω tδω( )as an explicit summation over all pairs of points for a finite discrete data set. b) Calculate ω0 and plot the trajectory δω(t) . c) Plot the correlation function for the frequency fluctuations using the expression you found in (a). What is the correlation timeτcfor these fluctuations? How does this value compare to an exponential fit? d) Make a histogram for occurrences ofδω and compare to a Gaussian distribution with a standard deviation ()0C