^
H = −
h
2
8π
2 m
∂
2
∂x
2 +
∂
2
∂y
2 +
∂
2
∂z
2
(4.27)
y x, y, z
ð
Þ=
ffiffiffiffiffiffiffiffi
8
abc
r
sin
n x πx
a
sin
n y πy
b
sin
n z πz
c
(4.28)
E =
h
2
8m
n
2
x
a
2 +
n
2
y
b
2 +
n
2
z
c
2
(4.29)
Equation 4.29 gives the nondegenerate quantized energy value for the
electron. When n x = n y = n z = 1, the electron is in the ground state. The
next level of higher energy occurs when one of the n quantum numbers is
2, but exactly which one depends on the values of a, b, and c. If a is the
larger of the three, then n x = 2 (and n y = n z = 1) because it will yield the
smallest value of E (but still higher than the ground state value).
Degeneracy of the energy levels emerge when we increase the symmetry
of our model so that it’s a perfect cube of length L (i.e., a = b = c). The
energy expression is now simply
E =
h
2
8mL
2 n
2
x + n
2
y + n
2
z
À
Á
(4.30)
Nodal line
Nodal plane
(a)
(b)
Figure 4.12 (a) The probability density profile for a particle in a two-dimensional region of space. The state shown
corresponds to n x = 2 and n y = 2. Regions of zero probability are defined by nodal lines. (b) The three-dimensional case
corresponding to n x = 2, n y = 2, and n z = 2. Regions of zero probability defined by nodal planes.
CHAPTER 4: Quantum Effects at the Nanoscale
116
H = −
h
2
8π
2 m
∂
2
∂x
2 +
∂
2
∂y
2 +
∂
2
∂z
2
(4.27)
y x, y, z
ð
Þ=
ffiffiffiffiffiffiffiffi
8
abc
r
sin
n x πx
a
sin
n y πy
b
sin
n z πz
c
(4.28)
E =
h
2
8m
n
2
x
a
2 +
n
2
y
b
2 +
n
2
z
c
2
(4.29)
Equation 4.29 gives the nondegenerate quantized energy value for the
electron. When n x = n y = n z = 1, the electron is in the ground state. The
next level of higher energy occurs when one of the n quantum numbers is
2, but exactly which one depends on the values of a, b, and c. If a is the
larger of the three, then n x = 2 (and n y = n z = 1) because it will yield the
smallest value of E (but still higher than the ground state value).
Degeneracy of the energy levels emerge when we increase the symmetry
of our model so that it’s a perfect cube of length L (i.e., a = b = c). The
energy expression is now simply
E =
h
2
8mL
2 n
2
x + n
2
y + n
2
z
À
Á
(4.30)
Nodal line
Nodal plane
(a)
(b)
Figure 4.12 (a) The probability density profile for a particle in a two-dimensional region of space. The state shown
corresponds to n x = 2 and n y = 2. Regions of zero probability are defined by nodal lines. (b) The three-dimensional case
corresponding to n x = 2, n y = 2, and n z = 2. Regions of zero probability defined by nodal planes.
CHAPTER 4: Quantum Effects at the Nanoscale
116