Chemistry Reference
In-Depth Information
This reduces to eq. (1.50), the equation for even confined states in an
isolated well, when the right-hand side equals zero. By re-arranging
eq. (2.38) in the form
f
(
E
)
=
sech
(κ
b
/
2
)
exp
(
−
κ
b
/
2
)
(2.39)
and then expanding
f
in a Taylor Series about
E
0
(the isolated
well ground state energy) show that
E
gs
(
E
)
, the ground state energy
in a coupled quantum well, varies for large barrier width
b
as
E
gs
(
b
)
C
e
−
κ
b
, where
C
is a constant which can in principle
be determined from eqs (2.38) and (2.39).
(
b
)
=
E
0
−
2.3
Derive an equivalent expression to eq. (2.38) for the first excited state
in a double quantumwell, and hence show that the splitting between
the ground and first excited state varies as 2
C
e
−
κ
b
for two weakly
coupled square quantum wells.
2.4
The ground state energy level in a square well of width
a
and depth
V
0
, centred at the origin, is given by
ψ(
x
)
=
A
cos
(
kx
)
,
|
x
|≤
a
/
2,
D
e
−
κ
|
x
|
ψ(
)
=
|
|≥
/
κ
and
x
for
x
a
2, where
k
and
have their usual
meanings. By evaluating
−∞
2
, calculate the magnitude of
the normalisation constants
A
and
D
in terms of
k
,
d
x
|
ψ(
x
)
|
and
a
. [This
result can be useful when applying the variational method, as in the
next question.]
κ
2.5
Using the variational wavefunction
ψ(
x
)
=
αφ
(
x
)
+
βφ
(
x
)
, where
L
R
φ
are the isolated quantum well ground state wave-
functions defined in eq. (2.12), calculate each of the integrals I, II and
IV in eq. (2.18) for the double square well potential. Hence show
that the variational method also predicts that the splitting between
the ground and first excited state energy varies as 2
C
e
−
κ
b
(
x
)
and
φ
(
x
)
L
R
for two
weakly coupled square quantum wells.
2.6
Show that the value of
C
calculated in problem 2.2 is the same as that
calculated in problem (2.5)!!
2.7 We can write the wavefunctions for an s-state and for three p
states on an isolated atom as
φ
(
r
)
=
f
s
(
r
)
,
φ
(
r
)
=
f
p
(
r
)
cos
θ
,
s
z
φ
φ)
are spherical polar coordinates centred on the atomic nucleus, and
f
s
(
r
)
=
f
p
(
r
)
sin
θ
cos
φ
and
φ
(
r
)
=
f
p
(
r
)
sin
θ
sin
φ
, where
(
r
,
θ
,
x
y
describe the radial variation of the s and p wave-
functions. Assuming that
f
s
(
r
)
and
f
p
(
r
)
(
r
)
∝
f
p
(
r
)
, show that the hybrid orbital
1
2
φ
h
(
r
)
=
[
φ
(
r
)
+
φ
(
r
)
+
φ
(
r
)
+
φ
(
r
)
]
has maximum amplitude
s
x
y
z
/
√
3
cos
−
1
along the (111) direction
(θ
=
(
1
)
,
φ
=
π/
4
)
.
As well as forming sp
3
-bonded diamond crystals, carbon can also
formsp
2
-bondedgraphite, where each carbon atomhas three nearest
2.8
