Chapter 4 Linear Differential Operators

152 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS
Here r is the distance away from the impurity and V (r) is the (spherically
symmetric) impurity potential and χ(r) = √ 4πrψ(r) where ψ(r) is the threedimensional
wavefunction. The impurity attracts electrons to its vicinity. Let
χ 0 k (r) = sin(kr) denote the unperturbed wavefunction, and χk(r) denote the
perturbed wavefunction that beyond the range of impurity potential becomes
sin(kr + η(k)). We fix the 2nπ ambiguity in the definition of η(k) by taking
η(∞) to be zero, and requiring η(k) to be a continuous function of k.
• Show that the continuous-spectrum contribution to the change in the
number of electrons within a sphere of radius R surrounding the impurity
is given by
kf 2
π 0
R
0
|χk(x)| 2 − |χ 0 k (x)|2
dr dk = 1
π [η(kf ) − η(0)]+oscillations.
Here kf is the Fermi momentum, and “oscillations” refers to Friedel oscillations
≈ cos(2(kf R + η)). You should write down an explicit expression
for the Friedel oscillation term, and recognize it as the Fourier transform
of a function ∝ k −1 sin η(k).
• Appeal to the Riemann-Lebesgue lemma to argue that the Friedel density
oscillations make no contribution to the accumulated electron number in
the limit R → ∞.
(Hint: You may want to look ahead to the next part of the problem in
order to show that k −1 sin η(k) remains finite as k → 0.)
The impurity-induced change in the number of unbound electrons in the interval
[0, R] is generically some fraction of an electron, and, in the case of
an attractive potential, can be negative — the phase-shift being positive and
decreasing steadily to zero as k increases to infinity. This should not be surprising.
Each electron in the Fermi sea speeds up as it enters an attractive
potential well, spends less time there, and so makes a smaller contribution
to the average local density than it would in the absence of the potential.
We would, however, surely expect an attractive potential to accumulate a net
positive number of electrons.
• Show that a negative continuous-spectrum contribution to the accumulated
electron number is more than compensated for by a positive number
Nbound =
∞
0
(ρ0(k) − ρ(k))dk = −
∞
0
1 ∂η 1
dk =
π ∂k π η(0).
of electrons bound to the potential. After accounting for these bound
electrons, show that the total number of electrons accumulated near the