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PH 216, Problem set 1, Due 4/6
1. In class, we estimated the groundstate energy of the 3D SHO (see Shankar 12.6.42)
using a ‘guess’ of
ψ
0
(
a
;
°r
)=
R
(
a
;
r
)
Y
0
0
(
θ,φ
)
,
by varying
a
and minimizing
°
ψ
(
a
)

H

ψ
(
a
)
±
.
Suppose we try to get the Frst energy
E
1
using the trial function
ψ
1
=
R
(
a
;
r
)
Y
0
1
,
Where the
Y
enforces orthogonality with our
ψ
0
. A trial function with zero nodes that
has the right asymptotics (
∝
r
+1
as
r
→
0 and
→
0as
r
→∞
) is
ψ
2
,
1
,
0
=
N
(
r/a
0
) exp(
−
r/
2
a
0
)cos
θ.
what then is
E
1
, and how does this compare with the Frst level of the actual 3D SHO?
2. (Shankar 16.2.8) Consider the
±
= 0 radial equation for the threedimensional Coulomb
problem. Since
V
(
r
) is singular at the turning point
r
= 0, we cannot use
n
+
3
4
as in
eq. (16.2.42) of Shankar.
(a) Will the additive constant be more or less than
3
4
? (Only heuristic reasoning is
required, but you are welcome to derive the result formally if you like.)
(b) By analyzing the exact equation near
r
= 0, it can be shown that the constant
equals 1. Using this constant, show that the WKB energy levels agree with the
exact results.
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This document was uploaded on 10/31/2011.
 Spring '09
 Energy

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