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PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 12
Topics covered: Partial waves.
1. Consider Swave scattering from a hard sphere of radius
a
. First, make the standard swave
scattering ansatz:
ψ
(
r,θ,φ
) =
e

ikr
r

(1 + 2
ikf
0
(
k
))
e
ikr
r
Then, ﬁnd the value of
f
0
(
k
) that satisﬁes the boundary condition
ψ
(
a,θ,φ
) = 0. What is the
partial amplitude
f
0
(
k
)? What is the swave phaseshift
δ
0
(
k
)?
2. For Pwave scattering from a hard sphere of radius
a
, make the ansatz
ψ
(
r,θ
) =
±²
1
kr

i
(
kr
)
2
³
e

ikr
+ (1 + 2
ikf
1
(
k
))
²
1
kr
+
i
(
kr
)
2
³
e
ikr
´
Y
0
1
(
θ.
Verify that this is an eigenstate of the full Hamiltonian for
r > a
by showing that it is a linear
superposition of two spherical Bessel functions of the thirdkind. Again solve for the partial
amplitude,
f
1
(
k
), by imposing the boundary condition
ψ
(
a,θ,φ
) = 0. What is the phaseshift
δ
1
(
k
)? Show that it scales as (
ka
)
3
in the limit
k
→
0. This is a general result that for small
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.
 Spring '11
 Moore
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