QuantumPhysics.dvi

Comparison of the asymptotics of this formula with the definition of the phase shifts in (12.93), we
find, after some rearrangements, that

e^2 iδℓ(k)=

1 +iU 0 kb^2 jℓ(kb)h∗ℓ(kb)

1 −iU 0 kb^2 jℓ(kb)hℓ(kb)

(12.120)

Gottfried has a nice discussion of the physics of this problem.

12.14Resonance scattering

There is a very interesting case of scattering potentials where quasi-stable bound states exist and
affect the scattering process. This occurs when the effective potential

Veff(r) =V(r) +

ℓ(ℓ+ 1)

r^2

(12.121)

has a local minimum which is not a global minimum.

Figure 13: Potential with unstable minimum in shaded area

During a scattering process, the incident particle excitesthe potential and creates a quasi-stable
bound state. This creates a resonance effect, since it will take some time for this bound state to
decay, and the process is then referred to asresonance scattering. The scattering cross section in
this channel reaches then a maximum. Note that this will always occur, for givenℓ, whenδℓ(k)
crosses the valueπ/2. Given the partial cross section

σ(totℓ)=

4 π

k^2

(2ℓ+ 1) sin^2 δℓ(k) (12.122)

we see that this partial cross section takes its maximal value atδℓ(k) =π/2. Around this value,
we may linearize as a function of momentumk(or traditionally, more often as a function of energy
E), we have

cotgδℓ(k) = 2

E−ER

Γ

+O((E−ER)^2 ) E=

̄h^2 k^2

2 m

(12.123)