Problems:
7.8, 7.5, 7.7
Graded
1. Starting with
ψ
(
r, θ, φ
) =
R
(
r
)
Y
(
θ, φ
) substitute into the Schr¨
odinger equation and show (using the technique
of separation of variables) that
R
satisfies

~
2
2
m
1
r
2
∂
∂r
r
2
∂
∂r
+
C
2
mr
2
+
V
(
r
)
R
=
ER
(
r
)
(1)
where
C
is a constant. The equation for
Y
(
θ, φ
) is
L
2
Y
(
θ, φ
) =
CY
(
θ, φ
)
(2)
Only for certain values of the constants
E
and
C
will the solutions be finite.
In particular it turns out that
C
=
‘
(
‘
+ 1)
~
2
with
‘
an integer.
2. The book writes the radial schrodinger equation for
R
nl
(
r
) as

~
2
2
m
1
r
2
∂
∂r
r
2
∂R
nl
∂r
+
‘
(
‘
+ 1)
~
2
R
nl
2
mr
2
=
E
nl
R
nl
(
r
)
(3)
Show that the equation for
u
nl
(
r
)
≡
√
4
π rR
nl
is given by Eq. (36). Show that the 2
p
wave functions of the
hydrogen atom satisfy the radial Schr¨
odinger equation.
3. Show that the minimum of the effective potential of the radial shr¨
odinger equation occurs at
r
=
‘
(
‘
+ 1)
a
o
4.
Graded
Consider the differences between
ψ
210
and
ψ
200
:
(a) Sketch the effective radial potential for these two wave functions
(b) Sketch radial wavefunctions
rR
nl
associated with
ψ
210
and
ψ
200
and corresponding radial probability density
P
(
r
). Despite the fact that these states have the same energy, their radial wave fuctions are qualitatively
different, explain.
(c) Determine the most likely radial position for the electron for the
ψ
200
state. Be sure to look carefully at
your graphs in part (
b
) – see also Fig 7.5 of book. (Ans:
r
= (3 +
√
5)
a
o
’
5
.
236
a
o
. Hint there are two
maxima at
r
= (3
±
√
5)
a
o
)
5. List all the levels associated of the second excited state of hydrogen. What is the energy of these states, what
are the angular momentum squared of these states, what are the zcomponent of angular momentum of these
states.
1
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2D Shr¨
odinger Equation
1. In two dimensions the Schr¨
odinger equations reads
"
P
2
x
2
m
+
P
2
y
2
m
+
V
(
x, y
)
#
Ψ(
x, y
) =
E
Ψ(
x, y
)
(4)

~
2
2
m
∂
2
∂x
2
+
∂
2
∂y
2
+
V
(
x, y
)
Ψ(
x, y
) =
E
Ψ(
x, y
)
(5)
2. For the particle in the two dimensional box the potential is
V
=
(
0
inside box

L/
2
< x, y < L/
2
∞
outside box
(6)
We solved this equation using separation of variables making an ansatz Ψ(
x, y
) =
X
(
x
)
Y
(
y
) and solving for the
functions
X
and
Y
3. We will discuss a square box
L
x
=
L
y
=
L
but you should be able to generalize this to a rectangular box and
also to three dimensions
(a) The wave functions are described by two quantum numbers
n
x
, n
y
and are
Ψ
n
x
,n
y
(
x, y
) =
X
n
x
(
x
)
Y
n
y
(
y
)
(7)
with
n
x
= 1
,
2
,
3
, . . .
and
n
y
= 1
,
2
,
3
, . . .
(8)
Where
X
n
x
(
x
) =
q
2
L
cos
(
n
x
πx
L
)
n
x
= 1
,
3
,
5
, . . .
q
2
L
sin
(
n
x
πx
L
)
n
x
= 2
,
4
,
6
, . . .
(9)
and similarly
Y
n
y
(
y
) =
q
2
L
cos
(
n
y
πy
L
)
n
y
= 1
,
3
,
5
, . . .
q
2
L
sin
(
n
y
πy
L
)
n
y
= 2
,
4
,
6
, . . .
(10)
(b) The wave functions
X
(
x
) and
Y
(
y
) satisfy the one dimensional Schr¨
odinger equation.
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 Fall '11
 Staff
 Angular Momentum, Kinetic Energy, Atomic orbital, wave functions, Rnl

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