1
The
λ/r
2
term originates from the angular momentum of the electron moving through
the Coulomb potential.
U
eff
diverges as
r
→
0 because 1
/r
2
goes to zero faster than the
Coulomb term.
The boundary conditions on this wavefunction are that it must go to zero at
r
= 0
and
r
=
∞
. This is the lowest energy state for
‘
= 1 because it only goes to zero at 0
and
∞
and nowhere in between. The principle quantum number for this state is
n
= 2.
The probability density is symmetric about the zaxis and is invariant under rotations
about the zaxis. See the figure below.
2
The allowed energies will all be multiples of
E
0
≡
π
2
~
2
/
2
mL
2
. The easiest thing to do is
try different combinations of
n
2
1
+
n
2
2
+
n
2
3
/
4 and see which 5 are the lowest.
E
n
1
n
2
n
3
deg.
9
E
0
/
4
1
1
1
1
3
E
0
1
1
2
1
17
E
0
/
4
1
1
3
1
21
E
0
/
4
1
2
1
2
2
1
1
6
E
0
1
1
4
3
2
1
2
1
2
2
3
Use the expectations values,
h
L
2
i
=
~
2
‘
(
‘
+ 1)
,
h
L
z
i
=
~
m
‘
, and the constraint
m
‘
=

‘,

‘
+ 1
, . . . ,
+
‘
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 Spring '10
 SINHA
 Angular Momentum, Atomic orbital, Fundamental physics concepts, probability density, lowest energy state

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