G25.2651: Statistical Mechanics

Problem set #8

Due: May 5, 2003

1.

The quantum time correlation function corresponding to an absorption process is

where Q is the canonical partition function. Consider a more ``symmetric'' correlation function

where is a complex time variable. Show that the Fourier transforms and and give a relationship between and . Finally, give an expression for the absorption probability spectrum in terms of

2.

Consider the free rotational motion of a rigid heteronuclear diatomic molecule of (fixed) bond length R and moment of inertia , where is the reduced mass, about an axis through its center of mass perpendicular to the internuclear bond axis. The molecule is constrained to rotate in the x-y plane only. One of the atoms carries a charge q and the other a charge -q.

a.

Ignoring center of mass motion, write down the Hamiltonian, for the molecule.

b.

Find the eigenvalues and eigenvectors of .

c.

If the molecule is exposed to spatially homogeneous, monochromatic radiation with an electric field given by

write down the perturbation Hamiltonian .

d.

Calculate the energy spectrum for . 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, , and let to go 0 at the end of the calculation.

e.

Based on your results from parts (a)-(d), plot the spectrum 3-dimensional rigid rotor, for which the energy eigenvalues are and m=-l,...,l is the quantum number for the z-component of angular momentum. Where do you expect the peak intensity to occur in the 3-dimensional case?

3.

In order to describe a system in a non-cubic box, such as might occur under the action of a directional shearing force, one often introduces a box matrix , whose columns contain the three cell vectors , and that describe a parallelpiped:

  
Figure 1:

a.

We know from elementary geometry that the volume of the box is given by . Show that the volume is also given by

b.

In the canonical ensemble, , with a Cartesian Hamiltonian

In order to make the dependence on the box matrix explicit, introduce the transformation

where is a momentum conjugate to the scaled coordinate . Here, and index the three spatial directions, x, y, and z. Show that this transformation preserves the phase space measure:

Show, therefore, that the pressure tensor estimator discussed in class can be deduced from the thermodynamic relation

where are indices that run over the three cartesian directions x,y,z.

c.

The isothermal-isobaric (NPT) ensemble is particularly useful for determining the bulk viscosity of a substance. Recall the partition function of this ensemble is

where is the external applied pressure.

d.

Show that the linear response formula is preserved is is chosen to the the isothermal-isobaric distribution function .

e.

Next, consider coupling a system to an external compression field described by the equations of motion

where and index the three spatial directions, x, y, and z. Show that the equations of motion satisfy the incompressibility condition.

f.

Consider the specific choice:

where is the compression rate. The coefficient of bulk viscosity, is given by a generalization of Newton's law of viscosity:

where means average over the equilibrium NPT distribution function and is the full nonequilibrium average. Using the linear response formula to evaluate , derive the appropriate Green-Kubo expression for .

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