The ground state is still nondegenerate, but when one of the n values is 2,
we are in a state in which three levels have the same energy (E = 6h
2 /
8mL
2 ). Figure 4.11 shows how the degeneracy changes for the first few
quantum states for a particle trapped in a perfect square and in a cube of
length L. In addition to knowing the energy, a probability profile can be
generated for a particle in a three-dimensional region of space. Figure
4.12b illustrates this for the state having quantum numbers n x = 2, n y = 2,
and n z = 2. We can see planes of zero probability—these are nodal planes.
Figures 4.13 and 4.14 show computer-generated three-dimensional plots
representing the probability profile for an electron in a square and cube,
respectively.
Equation 4.30 has some important applications. For example, it can be
used to calculate the optical properties of an electron trapped in a cubical
region of nanoscale dimension. Problem 4.9 goes through this exercise for
a solvated electron.
n x = 1, n y = 1
n x = 1, n y = 2
n x = 2, n y = 2
n x = 2, n y = 3
n x = 1, n y = 1
n x = 1, n y = 2
n x = 2, n y = 2
n x = 2, n y = 3
(a)
(b)
Figure 4.13 Computer generated three-dimensional plots representing (a) the wavefunction and (b) the square of
the wavefunction, for a particle in a two-dimensional plane. The plots are shown for the first four quantum states. In (b),
the amplitude of the curve indicates the probability profile of the particle.
CONFINEMENT OF ELECTRONS IN BOXES 117
we are in a state in which three levels have the same energy (E = 6h
2 /
8mL
2 ). Figure 4.11 shows how the degeneracy changes for the first few
quantum states for a particle trapped in a perfect square and in a cube of
length L. In addition to knowing the energy, a probability profile can be
generated for a particle in a three-dimensional region of space. Figure
4.12b illustrates this for the state having quantum numbers n x = 2, n y = 2,
and n z = 2. We can see planes of zero probability—these are nodal planes.
Figures 4.13 and 4.14 show computer-generated three-dimensional plots
representing the probability profile for an electron in a square and cube,
respectively.
Equation 4.30 has some important applications. For example, it can be
used to calculate the optical properties of an electron trapped in a cubical
region of nanoscale dimension. Problem 4.9 goes through this exercise for
a solvated electron.
n x = 1, n y = 1
n x = 1, n y = 2
n x = 2, n y = 2
n x = 2, n y = 3
n x = 1, n y = 1
n x = 1, n y = 2
n x = 2, n y = 2
n x = 2, n y = 3
(a)
(b)
Figure 4.13 Computer generated three-dimensional plots representing (a) the wavefunction and (b) the square of
the wavefunction, for a particle in a two-dimensional plane. The plots are shown for the first four quantum states. In (b),
the amplitude of the curve indicates the probability profile of the particle.
CONFINEMENT OF ELECTRONS IN BOXES 117