28664197 Semiconductor Physics and Devices 3rd Solution Manual
3rd Exercise Sheet Many-Body Physics - uni-muenchen.de · Lehrstuhl f ur Theoretische Nanophysik...
Transcript of 3rd Exercise Sheet Many-Body Physics - uni-muenchen.de · Lehrstuhl f ur Theoretische Nanophysik...
Lehrstuhl fur Theoretische NanophysikProf. Dr. L. PolletDr. T. Barthel
3rd Exercise Sheet Many-Body Physics
Will be discussed on Fri May 18.
Exercise 1: Green function of a free particle in 1d
Compute the retarded Green function for a free particle in 1d with dispersionE + k = k2/(2m). Answ:
GR(xf , xi, E) =
∞∫−∞
dk
2π
eik(xf−xi)
E − k2/(2m) + i0(1)
Calculate the integral over k for E < 0. Perform analytical continuation topositive E over the upper half plane.
The retarded Green function in the (k,E) domain can be written as (show)
GR(k,E) =1
E + i0− k2/(2m)(2)
It can be thought of as the matrix elements of the operator GR(E),
〈k| 1
E + i0− H|k′〉 =
2πδ(k − k′)E + i0− k2/(2m)
(3)
but where the delta-function is absent in the matrix elements.
Exercise 2: Scattering off a δ-potential in 1d: bound states
Consider
H =−1
2m
d2
dx2+ λδ(x) = H0(x) + V (x). (4)
Introduce
GR0 =1
E + i0− H0
(5)
and the T-matrix
T = V + V G0V + V G0V G0V + . . . (6)
Evaluate
Π =
∫dk
2π
dk′
2π〈k|GR0 |k′〉 =
∫dk
2π
1
E + i0− k2/(2m), (7)
for both positive and negative energies. Show that GR = GR0 + GR0 T GR0 and
calculate 〈k|T |k′〉. When is there a bound state?
Exercise 3: Scattering off a δ-potential in 3d
Consider now the problem of scattering in 3D off a delta function potential,
H =−1
2m
d2
∇2+ gδ(r) = H0(x) + V (x). (8)
This will require evaluation of
Π =
∫Λ
d3k
(2π)3
1
E − k2/(2m) + i0, (9)
which is UV divergent. One therefore introduces a cutoff Λ for k. So, unlikethe delta function in 1d, the delta function in 3d does not represent a welldefined potential. We should hence think of a potential which is g at lowmomenta, but quickly decays to zero for momenta larger than Λ, with Λ thescale set by the inverse of the extent of the potential.
• Show that Π can be written as
Π = −mΛ
π2− 2mE
2π
√m
−2E. (10)
• Compute the scattering amplitude f = −m2πT , where T is the T-
matrix. Show that it is angle independent (fully s-wave symmetric).Obtain the scattering length a defined by a = −f(p = 0).
• When are there bound states? What happens for g = −π2/(mΛ)?