130_notes.dvi

V(r) Scattering from a Spherical Well

a r

-V 0

E

Matching the logarithmic derivative, we get

k′

[djℓ(ρ)

dρ

jℓ(ρ)

]

ρ=k′a

=k

[

Bdjdρℓ(ρ)+Cdndρℓ(ρ)

Bjℓ(ρ) +Cnℓ(ρ)

]

ρ=ka

.

Recalling that forr→∞,

jℓ→

sin(ρ−ℓπ 2 )

ρ

nℓ→

−cos(ρ−ℓπ 2 )

ρ

and that our formula with the phase shift is

R(r) ∝

sin

(

ρ−ℓπ 2 +δℓ(k)

)

ρ

=

1

ρ

[

cosδℓsin(ρ−

ℓπ

2

) + sinδℓcos(ρ−

ℓπ

2

)

]

,

we canidentify the phase shifteasily.

tanδℓ=−

C

B

We need to use the boundary condition to get this phase shift.

Forℓ= 0, we get

k′

cos(k′a)

sin(k′a)

=k

Bcos(ka) +Csin(ka)

Bsin(ka)−Ccos(ka)

k′

k

cot(k′a) (Bsin(ka)−Ccos(ka)) =Bcos(ka) +Csin(ka)

(

k′

k

cot(k′a) sin(ka)−cos(ka)

)

B=

(

sin(ka) +

k′

k

cot(k′a) cos(ka)

)

C

We can now get the phase shift.

tanδ 0 =−

C

B

=

kcos(ka) sin(k′a)−k′cos(k′a) sin(ka)

ksin(ka) sin(k′a) +k′cos(k′a) cos(ka)