ð L
0
A n sin
nπx
L
A n sin
nπx
L
dx = 1
A
2
n
ð L
0
sin
2 nπx
L
dx = 1
Using the fact that the integral
ð
sin
2 axdx =
x
2
−
sin (2ax)
4a
we get A
2
n
x
2
−
sin 2nπx=L
ð
Þ
4nπ
L
0
= 1
substituting in the integration limits yields
A
2
n
L
2
−
sin (2nπ)
4nπ
−
0
2
−
sin (0)
4nπ
= 1
which simplifies to A
2
n
L
2
= 1
, because sin(2nπ) = 0
Thus, the normalization constant is A n =
ffiffiffi ffi
2
L
r
As mentioned in Section 4.1.4, normalization of the wavefunction ensures
that the electron has a probability of 1 of being found within the region governed by the boundary conditions. The above example showed us that the
normalized wavefunction for a particle confined along a line of length L is
y n x
ð Þ =
ffiffiffi ffi
2
L
r
sin
nπx
L
(4.14)
By applying the Hamiltonian operator (Equation 4.12) on this wavefunction, we obtain the quantized energy values of the electron in this
region (Equation 4.18):
^
Hy x
ð Þ = Ey x
ð Þ
(4.15)
^
Hy x
ð Þ = −
h
2
8π
2 m
d
2
dx
2
ffiffiffi ffi
2
L
r
sin
nπx
L
(4.16)
−
h
2
8π 2 m
ffiffiffi ffi
2
L
r
d
2
dx 2 sin
nπx
L
=
h
2
8π 2 m
n
2
π
2
L 2
ffiffiffi ffi
2
L
r
sin
nπx
L
(4.17)
∴ E n =
n
2 h
2
8mL
2
(4.18)
CONFINEMENT OF ELECTRONS IN BOXES 107