Physics 742 – Graduate Quantum Mechanics 2
Solution Set L
1.
[25] A particle of mass
m
scatters from a potential
( ) ( )
Vr F r a
δ
=
−
so that the potential exists only at the surface of a narrow sphere of radius
a
(a) [4] What equation must the radial wave functions
( )
l
R
r
satisfy?
Solve this
equation in the regions
r
<
a
and
r
>
a
.
The radial wave functions must satisfy
()
() ()
(
)
22
2
2
2
12
ll
l
dl
l
l
l
m
F
rR r
U r
k
R r
r
a
k
R r
rdr
r
r
⎡⎤
⎡
⎤
++
=+
−
−
−
⎢⎥
⎢
⎥
⎣⎦
⎣
⎦
=
Away from the point
r
=
a
, the problem is simply that of a free particle, and the solution
was worked out in class.
The answer is spherical Bessel functions, and take the form
( ) ( )
l
Aj kr
Bn kr
r
a
Rr
Cj kr
Dn kr
r
a
−
<
⎧
⎪
=
⎨
−
>
⎪
⎩
The constants will generally be different in the different regions.
(b) [6] Apply appropriate boundary at
r
= 0.
Deduce appropriate boundary
conditions at
r
=
a
.
We want the radial function to be wellbehaved at
r
= 0, which implies we only
want the wellbehaved
j
l
(
r
).
Hence we demand
B
= 0.
At the boundary
r
=
a
, we must
take our radial Schrödinger equation and integrate it across the boundary.
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 Spring '04
 Unknow
 Physics, Mass, graduate quantum mechanics

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