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)?