Problem Set ---Due February 10
1) In class we discussed the separation of variables for the three dimensional time-
independent Schrödinger equation with a spherically symmetric potential. In this
problem I want you to work thorough the equivalent for a two dimension system with
an axially symmetric potential. In particular consider the equation
ψψψ
ErV
yx
m=+
∂
∂
+
∂
∂− )(
22
2
2
22
in polar coordinates. Show that the it has
solutions of the form
)()(),(
ϑϑψ
Θ= rRr
with
ϑ
ϑ
im
e=Θ )(
(integer m) and R
satisfying
)()(
2
)()(
1
22
222
rRErR
rm
m
rVrR
r
r
rrm =
++
∂
∂
∂
∂−
.
2) Consider the infinite spherical square well of radius a studied in class. We found the
energies of the l=0 states but showed that the l=1 states could only be solved numerically by
finding the roots of a transcendental equation. Find a numerical estimate for the energies of
the lowest three by solving this equation numerically.
3) The spherical harmonics are a complete orthonormal set describing smooth angular
functions. Completeness means than any smooth angular function may be written in the form
),(),(
,
,
φϑφϑ
∑
=
ml
m
lml Ycf
. Orthonormailty implies
1
,
2
,=
∑
ml
ml
c
. Find the coefficients
ml
c,
for the function
( )
)cos()2sin()sin(),(
φϑϑφϑ
+=f
.
4) In class it was claimed that the spherical harmonics were orthonormal in the sense that
( )
'.'.
*
'
'
0
2
0
),(),()sin( mmll
m
l
m
lYYdd
δδφϑφϑϑϑφ
ππ
=
∫∫
.Explicitly show that this holds for the subset
),(
2
2
φϑ
Y
,
),(
1
2
φϑ
Y
and
),(
0
2
φϑ
Y
and
),(
0
1
φϑ
Y
(where the explicit forms of these are as given
in the book or class ) by directly do the intergrals.
5) Find the following matrix elements for states labeled
ml ,
;
a)
2,22,1 z
L
b)
2,22,2 x
L
c)
1,22,2 x
L
d)
1,12,2 x
L
e)
0,22,2 2
x
L
f)
2,22,2 22
yx LL +
g)
( )
1,21,2 22
zyx LLL +
