Thermodynamics, Statistical Physics, and Quantum Mechanics

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5.68 Rayleigh Scattering (Tennessee)

Rayleigh scattering is theelasticscattering ofphotons. Assumethere is a

matrix element which describes thescattering from to It

has the dimensions of

Derive an expression for thedifferential cross section for

Rayleigh scattering. Ignore thephoton polarization.

Assume the specificform for thematrixelement

a)

b)

where is the polarizability tensor and are the polarization

vectors of the photons. What is the result if theinitial photons are

unpolarized and the final photon polarizations are summedover? As-

sume the polarizability is isotropic: where is the unit tensor.

5.69 Scattering from Neutral Charge Distribution

(Princeton)

Consider thenonrelativistic scattering of a particle of massmand charge e

from a fixed distribution of charge Assumethat thecharge distribu-

tion is neutral: it issphericallysymmetric; and thesecond

moment, is defined as

Use the Born approximation to derive thedifferential crosssection

for the scattering of a particle ofwavevectork.

Derive theexpression forforwardscattering

Assume that is for a neutralhydrogenatom in its ground state.

CalculateAin thiscase. Neglect exchangeeffects andassumethat

the target does notrecoil.

a)

b)

c)

General

5.70 Spherical Box with Hole (Stony Brook)

A particle is confined to a spherical box of radiusR. There is a barrier
in the center of the box, whichexcludes theparticle froma radius So
the particle is confined to the region Assume that thewave
functionvanishes at both and and derive an expression for the
eigenvalues andeigenfunctions of stateswithangularmomentum

QUANTUM MECHANICS