1549380323-Statistical Mechanics Theory and Molecular Simulation

570 Quantum time-dependent statistical mechanics

d. Calculate the energy spectrumQ(ω) forω >0. Interpret your results,

and in particular, explain how the allowed absorptions and emissions

are manifest in your final expression. Plot the absorption part of your

spectrum. Where do you expect the peak intensity to occur?

Hint: Consider using a convergence factor, exp(−ǫ|t|), and letǫgo to 0

at the end of the calculation.

e. Based on your results from parts a–d, plot the spectrum three-dimensional

rigid rotor, for which the energy eigenvalues areElm= ̄h^2 l(l+ 1)/ 2 Iand

m=−l,...,lis the quantum number for thez-component of angular

momentum. Where do you expect the peak intensity to occur in the 3-

dimensional case?

14.7. Derive a discrete path-integral representation for the Kubo-transformed quan-

tum time correlation functionKAB(t) defined in eqn. (14.6.11).

14.8. Consider two spin-1/2 particles at fixed points in space a distanceRapart

and interacting with a magnetic fieldB= (0, 0 ,B). The particles carry charge

qand−q, respectively. The Hamiltonian of the system is

Hˆ=−γB·Sˆ−q

2

R

,

whereSˆ=Sˆ 1 +Sˆ 2 is the total spin, andγis the spin gyromagnetic ratio.

a. What are the allowed energy levels of this system?

b. Suppose that a time-dependent perturbation of the form

Hˆ 1 (t) =−γb·Sˆe−t^2 /τ^2 ,

whereb= (b, 0 ,0) is applied att=−∞. Att=−∞, the system is in its

unperturbed ground state. To first order in perturbation theory, what is

the probability, ast−→∞, that the system will make a transition from

its ground state to a state with energy−q^2 /R?

14.9. Thedensity of vibrational states, also known as thepower spectrum orspec-

tral density, is the Fourier transform of the velocity autocorrelation function:

I(ω) =

1

√

2 π

∫∞

−∞

dt e−iωtCvv(t).

I(ω) encodes information about the vibrational modes of a system; however,

it does not provide any information about net absorption intensities. For

the two model velocity autocorrelation functions in Problem 13.1, calculate

the density of vibrational states and interpret them in terms of the physical

situations described by these two model correlation functions.