where the quantum numbers n x and n y independently assume values of 1,
2, 3, 4, and so on. The boundary conditions for this model are similar to
the one-dimensional case in that the wavefunction is zero at the edges
and outside of the rectangle. The energy levels corresponding to this
wavefunction can be written as a sum of the individual one-dimensional
energies given by Equation 4.18. Thus
E =
n
2
x h
2
8ma 2 +
n
2
y h
2
8mb 2
(4.24)
Factoring out the constants gives
E =
h
2
8m
n
2
x
a
2 +
n
2
y
b
2
(4.25)
The above equation can be used to model the energy of electrons freely
moving in a 2D quantum well of nanoscale dimension. According to
Equation 4.25, the ground state energy of an electron in the well is given
when n x = 1 and n y = 1. The next, higher-energy level occurs when one of
the n quantum numbers is 2, but exactly which one depends on the values
of a and b. If a is the larger of the two, then n x = 2 (and n y = 1) because it
will yield the smallest value of E (but still higher than the ground state
value). If the length a = b then the electron is trapped in a square of length
a. This leads to Equation 4.26, which describes the energy of an electron
in a perfect square.
E =
h
2
8m
n
2
x
a
2 +
n
2
y
a
2
=
h
2
8mL
2 n
2
x + n
2
y
À
Á
(4.26)
In the above equation, we have replaced a with L, the length of the
square. By changing the rectangle into a square, we have made our region
more symmetrical. This symmetry has a very important consequence on
the energy levels of the electron. When n x = 1 and n y = 1, the energy is
simply E = 2h
2 /8mL
2 . However, when we consider the next energy level
we find that there are two combinations of n that lead to the same energy
value, these being n x = 1, n y = 2, and n x = 2, n y = 1. These two combinations of n lead to a quantum state in which we have two levels of the
same energy (E = 5h
2 /8mL
2 ) (Figure 4.11). We refer to this state as being
twofold degenerate, or doubly degenerate, or having a degeneracy of two.
We use the term degeneracy to describe a quantum state in which we
have a number of levels with the same energy; we previously encountered
CHAPTER 4: Quantum Effects at the Nanoscale
114
2, 3, 4, and so on. The boundary conditions for this model are similar to
the one-dimensional case in that the wavefunction is zero at the edges
and outside of the rectangle. The energy levels corresponding to this
wavefunction can be written as a sum of the individual one-dimensional
energies given by Equation 4.18. Thus
E =
n
2
x h
2
8ma 2 +
n
2
y h
2
8mb 2
(4.24)
Factoring out the constants gives
E =
h
2
8m
n
2
x
a
2 +
n
2
y
b
2
(4.25)
The above equation can be used to model the energy of electrons freely
moving in a 2D quantum well of nanoscale dimension. According to
Equation 4.25, the ground state energy of an electron in the well is given
when n x = 1 and n y = 1. The next, higher-energy level occurs when one of
the n quantum numbers is 2, but exactly which one depends on the values
of a and b. If a is the larger of the two, then n x = 2 (and n y = 1) because it
will yield the smallest value of E (but still higher than the ground state
value). If the length a = b then the electron is trapped in a square of length
a. This leads to Equation 4.26, which describes the energy of an electron
in a perfect square.
E =
h
2
8m
n
2
x
a
2 +
n
2
y
a
2
=
h
2
8mL
2 n
2
x + n
2
y
À
Á
(4.26)
In the above equation, we have replaced a with L, the length of the
square. By changing the rectangle into a square, we have made our region
more symmetrical. This symmetry has a very important consequence on
the energy levels of the electron. When n x = 1 and n y = 1, the energy is
simply E = 2h
2 /8mL
2 . However, when we consider the next energy level
we find that there are two combinations of n that lead to the same energy
value, these being n x = 1, n y = 2, and n x = 2, n y = 1. These two combinations of n lead to a quantum state in which we have two levels of the
same energy (E = 5h
2 /8mL
2 ) (Figure 4.11). We refer to this state as being
twofold degenerate, or doubly degenerate, or having a degeneracy of two.
We use the term degeneracy to describe a quantum state in which we
have a number of levels with the same energy; we previously encountered
CHAPTER 4: Quantum Effects at the Nanoscale
114