Notes on the Kondo Problem to Heavy Fermions - From a Rookie's Perspective.
The eigenstate of
the Hamiltonian gives
solve this equation with the conduction electron Green's
function In the continuous -representation we have which is the Lippmann-Schwinger
equation of scattering theory. To obtain the scattered wave affected by
incoming Bloch wave of
momentum , we have
the Green's function of the following form and further define the eigenstate becomes therefore by iterating these two equations we have where the matrix element Consider the plane waves of a box of volume with periodic boundary conditions, the
Green's function is using the sphere coordination we have Integrating out the angle contribution of this integral, consider the eigenvalue , therefore the Green's function is where , and this expression has two pole at . Perform the
contour integration in the upper plane, we have With far field asymptotic approximation of and consider the
Taylor's expansion , we have where and the exponential phase is simplified by
Hence the scattered wave function becomes where the scattering amplitude The differential scattering cross-section is given by The integration form of this expression gives the total
scattering cross-section :
Using the relation the transition rate becomes Remind the perturbation form of the T-matrix with the operator identity we have taking the diagonal elements, Consider the matrix elements and the representation of the delta function the total scattering cross-section becomes which can be expressed in terms of the imaginary part of the
forward scattering amplitude . This is known as optical theorem
of scattering theorem. For a spherically symmetric scattering, the
potential
depends only on thee angle
between and . Expand the term with Legendre
polynomials ,
where is a real
function, and is the
phase shift of the th partial
wave. The factor comes from
the independence of the scattering on the -component of the angular momentum . Using the orthogonality relation of
the Legendre polynomials, we have The total scattering cross-section in terms of phase shift is
Expand the plane wave in terms of partial waves, where is a
spherical Bessel function of order . Substituting this term back to the
scattered wave function and using the asymptotic form, we have